Andrián Pertout

Three Microtonal Compositions: The Utilization of Tuning Systems

in Modern Composition

Volume 1

Submitted in partial fulfilment of the requirements of the degree of Doctor of Philosophy

Produced on acid-free paper

Faculty of Music

The University of Melbourne

March, 2007

Abstract

Three Microtonal Compositions: The Utilization of Tuning Systems in

Modern Composition encompasses the work undertaken by Lou

Harrison (widely regarded as one of America’s most influential and

original composers) with regards to just intonation, and tuning and

scale systems from around the globe – also taking into account the

influential work of Alain Daniélou (Introduction to the Study of

Musical Scales), Harry Partch (Genesis of a Music), and Ben

Johnston (Scalar Order as a Compositional Resource). The

essence of the project being to reveal the compositional

applications of a selection of Persian, Indonesian, and Japanese

musical scales utilized in three very distinct systems: theory versus

performance practice and the ‘Scale of Fifths’, or cyclic division of

the octave; the equally-tempered division of the octave; and the

‘Scale of Proportions’, or harmonic division of the octave

championed by Harrison, among others – outlining their theoretical

and aesthetic rationale, as well as their historical foundations. The

project begins with the creation of three new microtonal works

tailored to address some of the compositional issues of each

system, and ending with an articulated exposition; obtained via the

investigation of written sources, disclosure of compositional

technique, mathematical analysis of relevant tuning systems,

spectrum analysis of recordings, and face-to-face discussions with

relevant key figures.

THE UNIVERSITY OF MELBOURNE Faculty of Music

TO WHOM IT MAY CONCERN

This is to certify that

(i) the thesis comprises only my original work towards the PhD except where indicated in the Preface*,

(ii) due acknowledgement has been made in the text to all other material used,

(iii) the thesis is less than 80,000 words in length, exclusive of tables, maps, bibliographies and

appendices or the thesis is [number of words] as approved by the RHD Committee.

Signature:

Name in Full: Andrián Pertout

Date: 2 March, 2007

Dedicated to my father,

the late Aleksander Herman Pertout

(b. Slovenia, 1926; d. Australia, 2000)

Acknowledgements

A special thanks to the supervisors:

Professor Brenton Broadstock (Coordinator of Composition, Faculty of Music, University of Melbourne)

and Associate Professor Neil McLachlan (School of Behavioural Science, Faculty of Medicine, Dentistry

and Health Sciences, University of Melbourne). Brenton Broadstock should be especially thanked for

being an inspirational force not only during the last four years of the PhD candidature, but throughout the

last ten years of my composition studies at the University of Melbourne. His encouragement, support,

and direction have exceeded well beyond his duties as supervisor and composition teacher, and

consequently remain forever grateful.

Professor Andrew Schultz (Dean of the Faculty of Creative Arts, University of Wollongong, NSW, Australia)

also deserving a mention with regards to L’assaut sur la raison for Symphony Orchestra (2003) and

Bénédiction d’un conquérant for Symphony Orchestra (2004), which were especially composed for

ACOF 2003 and 2004 (Australian Composers’ Orchestral Forum – Composition workshops with Brenton

Broadstock, Andrew Schultz, and the Tasmanian Symphony Orchestra).

A special thanks also to Dr. Julian Yu who was the official mentor for the 2003 and 2004 ACOF project.

A special thanks to the following people for their direct assistance to the composition folio:

Stephen Adams (Presenter, ABC Classic FM) for producing an excellent program featuring La flor en la

colina for Flute, Clarinet, Violin, Violoncello and Pianoforte (2003-2004) on his ABC Classic FM radio

program.

Susan Batten (Presenter, 3MBS FM) for producing two excellent programs featuring L’assaut sur la raison

for Symphony Orchestra (2003), Navigating the Labyrinth for String Orchestra (2002), and Aristotle’s

Rhetoric, Suite for Orchestra (2001-02, Rev. 2005) – together with an interview with the composer – on

her 3MBS FM Radio ‘Music in Melbourne’ program, in celebration of the Betty Amsden Award – 2005

3MBS FM National Composer Awards.

APRA (Australasian Performing Right Association) for recognizing L’assaut sur la raison for Symphony

Orchestra (2003) with the APRA Encouragement Award – 2004 3MBS FM National Composer Awards.

Andrew Blackburn (Artistic Director, 2007 Melbourne Town Hall Organ Project), Jean Penny and the

Melbourne City Council for commissioning Symétrie intégrante for Flute, Organ and Electronics (2005-06)

for the upcoming 2007 Melbourne Town Hall Organ Project, Melbourne, Australia.

Enmanuel Blanco (Executive Director, Festival Internacional de Música Electroacústica) for selecting

Exposiciones for Sampled Microtonal Schoenhut Toy Piano (2005) to be performed at the XI Festival

Internacional de Música Electroacústica ‘Primavera en la Habana’ 2006, 6-12 March, 2006, Habana, Cuba.

Associate Professor Jack Body (Artistic Director, 2007 Asia Pacific Festival, 26th Asian Composers League

Festival & Conference, and Associate Professor of Composition, New Zealand School of Music, Victoria

University, Wellington, New Zealand) and the festival organizers for selecting Àzàdeh for Santär and Tape

(2004, Rev. 2005) to be performed at the 2007 Asia Pacific Festival (26th Asian Composers League

Festival & Conference), 8-16 February, 2007, Wellington, New Zealand. Also, for selecting the

conference paper Theory Versus Performance Practice: Àzàdeh for Santär and Tape to be presented at

the 2007 Asia Pacific Festival ‘Tradition/Transformation’ Conference.

Warren Burt (Wollongong, NSW, Australia) for his generous support and contribution to the direction of

the PhD research, and especially with regards to Exposiciones for Sampled Microtonal Schoenhut Toy

Piano (2005).

Ao Changqun (Organizing Committee Chairman, 2005 Second Sun River Student New Composition

Competition, and President, Sichuan Conservatory of Music, Chengdu, People’s Republic of China) for

recognizing La flor en la colina for Flute, Clarinet, Violin, Violoncello and Pianoforte (2003-04) with the

Third Prize in the 2005 Second Sun River Student New Composition Competition (Chengdu, People’s

Republic of China).

Phyllis Chen for requesting a Toy Piano and Tape arrangement of Exposiciones for Sampled Microtonal

Schoenhut Toy Piano (2005), and for her incredible talent, evident in her virtuosic interpretations of the

work in Bloomington, Indiana and Chicago, Illinois, USA, as well as at the 2007 International Gaudeamus

Interpreters Competition & Chamber Music Week in Amsterdam, The Netherlands.

David Claman (Assistant Professor, Music Department, College of the Holy Cross, Worcester,

Massachusetts, USA) and Matt Malsky (Associate Professor of Music, Department of Visual and Performing

Arts, Clark University, Worcester, Massachusetts, USA) for selecting Exposiciones for Sampled Microtonal

Schoenhut Toy Piano (2005) to be part of the Extensible Toy Piano Project, 5-6 November, 2005,

Department of Visual and Performing Arts, Clark University, Worcester, Massachusetts, USA; and the

÷××× Acknowledgements

Extensible Toy Piano Festival, 4 March, 2007, Performing Arts Center, Department of Music, State University

of New York, Albany, New York, USA.

Barry Cockroft (tenor saxophone) and Adam Pinto (pianoforte) for commissioning and performing

Digressioni modali for Tenor Saxophone and Pianoforte (2003) at the Melbourne International Festival of

Single Reeds, 26-29 March, 2005, Victorian College of the Arts, Southbank, Melbourne, Australia. Also,

for recording the work for the ‘rompduo’ Crazy Logic CD release. Barry Cockroft (tenor saxophone) and

Marc Ryser (pianoforte) for performing the work at The Banff Centre, Banff, Alberta, Canada, and finally

Barry Cockroft (Reed Music) for publishing the work with Reed Music.

Professor Barry Conyngham (former Emeritus Professor of the University of Wollongong and Southern

Cross University, Lismore, NSW, Australia) for his compositional direction during his residency at the

University of Melbourne in 2005.

David Collins (Technical Officer, Faculty of Music, University of Melbourne) for technical assistance

throughout the PhD candidature, as well as invaluable advice with regards to sound diffusion concepts.

David B. Doty (Author of The Just Intonation Primer, Founder of the Just Intonation Network, and Editor of

the Network’s Journal, 1/1, San Francisco, California, USA) for making time for me during my 2004 visit to

San Francisco, California, USA, and for his compositional guidance with regards to just intonation

concepts.

Ensamble Contemporáneo (Aliocha Solovera [artistic director], Cristián Gonzáles [flute], Dante Burotto

[bass clarinet], Alexandros Jusakos [pianoforte], Davor Miric [violin], and Celso López [violoncello]) for

performing La flor en la colina for Flute, Clarinet, Violin, Violoncello and Pianoforte (2003-04) at the XV

Festival de Música Contemporánea Chilena (15th Chilean Festival of Contemporary Music), 21-27

November, 2003, Santiago, Chile.

Ivano Ercole (Presenter, Rete Italia) for producing an excellent program featuring L’assaut sur la raison for

Symphony Orchestra (2003), Navigating the Labyrinth for String Orchestra (2002), Gèrëémeler for

Amplified Èrhú, Sampled Harmonium, Cajón and Bombo (2001), Bénédiction d’un conquérant for

Symphony Orchestra (2004), An Honourable Silence for Solo Santär (2001), Renascence for Violin,

Violoncello, Piano and Percussion (2001, Rev. 2006), and Seeds of Passion for Amplified Violoncello

(1999) – together with an interview with the composer – on his Rete Italia radio program.

Acknowledgements ×Ø

The Ónix Ensamble (Alejandro Escuer [flute], Fernando Domínguez [clarinet], Abel Romero [violin],

Edgardo Espinosa [violoncello], and Krisztina Deli [pianoforte]) for selecting La flor en la colina for Flute,

Clarinet, Violin, Violoncello and Pianoforte (2003-04) to be performed at the XXIX Foro Internacional de

Música Nueva (29th International Forum of New Music), “Manuel Enríquez”, 2007, May-June, 2007,

México City, México.

Isabel Ettenauer (St. Poelten, Austria) and Goska Isphording (Eindhoven, The Netherlands) for inspiring the

arrangement of Exposiciones for Toy Piano and Spinet (2005), and for performing the work at the 2006

BMIC Cutting Edge Series, London, UK, and at Axes/Jazzpower, Eindhoven, The Netherlands.

Rodolfo Fischer (Conductor, Basel, Switzerland) for selecting Bénédiction d’un conquérant for Symphony

Orchestra (2004) to be performed by the Orquestra Petrobras Sinfônica at the Theatro Municipal do Rio

de Janeiro, in Rio de Janeiro, Brazil as part of the Orquestra Petrobras Sinfônica ‘Série Ouro Negro’ 2006

concert series, and also for his excellent direction during the rehearsals and final performance.

Robert Franz (Associate Conductor, Louisville Orchestra, Louisville, Kentucky, USA) and the Louisville

Orchestra for recognizing L’assaut sur la raison for Symphony Orchestra (2003) as the winner of the First

Prize in the 2004 ISU Contemporary Music Festival/Louisville Orchestra Composition Competition, and also

for presenting a memorable performance of the work at the Indiana State University 38th Annual

Contemporary Music Festival, ‘Plugged In: Music With an Electric Edge’, November 10-12, 2004, Terre

Haute, Indiana, USA. Indiana State University for sponsoring the award, and providing an opportunity to

conduct a lecture at the festival.

Professor Don Freund (Professor of Music Composition, Indiana University School of Music, Bloomington,

Indiana, USA) and Sandra Freund for their hospitality and enormous generosity during my weekend stay in

2004 with the Freunds in Bloomington, Indiana, USA. A further warm thanks to Don Freund for his

contribution to the development of La flor en la colina for Flute, Clarinet, Violin, Violoncello and Pianoforte

(2003-04).

Kyle Gann (Associate Professor of Music, Faculty, Bard College, Annandale-on-Hudson, New York, USA)

for his support during my 2004 visit to Bard College (Annandale-on-Hudson, New York, USA), and for his

compositional guidance with regards to just intonation concepts.

Dr Noah Getz (Instructor of Saxophone, American University, and Jazz Saxophone Instructor, Levine

School of Music, Washington, DC, USA) and the judges of the 2005 American University Saxophone

Ø Acknowledgements

Symposium Composition Contest for recognizing Digressioni modali for Tenor Saxophone and Pianoforte

(2003) as the winner of the Third Prize in the international composition competition. Noah Getz (tenor

saxophone) and John Kilkenny (marimba) for inspiring the arrangement of Digressioni modali for Tenor

Saxophone and Marimba (2003), and for performing the work in Alexandria, Virginia and Washington, DC,

USA. Noah Getz (tenor saxophone) and Laurence Gingold (pianoforte) for performing the work in

Lancaster, Pennsylvania, USA, and finally, Noah Getz (tenor saxophone) and Jeffrey Chappell (pianoforte)

for recording the work for CD release.

Brooke Green (Presenter, ABC Classic FM) for producing an excellent program featuring L’assaut sur la

raison for Symphony Orchestra (2003) on her ABC Classic FM ‘Composers Emerging’ program, together

with an interview with the composer, as part of ACOF 2003 (Australian Composers’ Orchestral Forum –

Composition workshops with Brenton Broadstock, Andrew Schultz, and the Tasmanian Symphony

Orchestra).

Dr. Stuart Greenbaum (Lecturer in Composition, Faculty of Music, University of Melbourne) for his

incredible support throughout the PhD candidature.

Alejandro Guarello (Artistic Director, XV Festival de Música Contemporánea Chilena, Instituto de Música,

Facultad de Artes, Pontificia Universidad Católica de Chile, Santiago, Chile) for selecting La flor en la colina

for Flute, Clarinet, Violin, Violoncello and Pianoforte (2003-04) to be performed at the XV Festival de

Música Contemporánea Chilena (15th Chilean Festival of Contemporary Music), 21-27 November, 2003,

Santiago, Chile.

Christian Haines (Lecturer and Unit Coordinator, Electronic Music Unit, Elder Conservatorium of Music,

University of Adelaide) for selecting Exposiciones for Sampled Microtonal Schoenhut Toy Piano (2005) to

be part of the Medi(t)ations: Computers, Music and Intermedia, Australasian Computer Music Association

Conference 2006, 11-13 July, 2006, Adelaide, Australia.

Michael Harrison (New York, NY, USA) for his demonstration of the ‘harmonic piano’ – a modified seven-

foot Schimmel grand piano – during my visit to New York, NY, USA in 2004.

The international jury of the ISCM (consisting of Stanko Horvat [Croatia], Zygmunt Krauze [Poland],

Giampaolo Coral [Italy], Frank Corcoran [Ireland/Germany], Arne Nordheim [Norway], and Berislav Šipuš

[Croatia]) for selecting La flor en la colina for Flute, Clarinet, Violin, Violoncello and Pianoforte (2003-04) to

Acknowledgements Ø×

be performed at the International Society for Contemporary Music (ISCM) World Music Days 2005 / 23rd

Music Biennale Zagreb, 15-24 April, 2005, Zagreb, Croatia.

Jerome Kitzke (New York, NY, USA) for his compositional direction during his McGeorge Fellowship

residency at the University of Melbourne in 2005.

Jennifer Logan (Co-Artistic Director, Los Angeles Sonic Odyssey, Electronic and Computer Music Concert

Series 2006, Los Angeles, California, USA) for selecting Paåc hazàr chakêà kaâ andar for Prepared Multi-

tracked Disklavier (2000), Exposiciones for Sampled Microtonal Schoenhut Toy Piano (2005), Àzàdeh for

Tape (2004, Rev. 2005), La Homa Kanto for Tape (2005), and Sonic Junk Yard for Tape (1999) to be part

of the Los Angeles Sonic Odyssey Electronic and Computer Music Concert Series 2005, 2006, and 2007

Los Angeles and Pasadena, California, USA.

Jana Haluza Lucic (Producer, HRT, Hrvatska Radio, Zagreb, Croatia) for producing an excellent program

featuring La flor en la colina for Flute, Clarinet, Violin, Violoncello and Pianoforte (2003-04) – together with

an interview with the composer – on her HRT, Hrvatska Radio (Croatian Radio) ‘World of Music’ program

in Zagreb, Croatia.

Dr. Susan McDonald (Department of Fine Arts, Philadelphia, Pennsylvania, USA) for selecting Exposiciones

for Sampled Microtonal Schoenhut Toy Piano (2005) to be performed at the ‘La Salle University:

Electroacoustic Works Inspired by Popular Music’ concert in November, 2005, Philadelphia, Pennsylvania,

USA.

Marshall McGuire (Artistic Director, Sonic Art Ensemble, Sydney, NSW, Australia) and the Sonic Art

Ensemble (Christine Draeger [flute], Margery Smith [bass clarinet], Rowan Martin [violin], Adrian Wallis

[violoncello], and Bernadette Balkus [pianoforte]) for programming the Australian premier of La flor en la

colina for Flute, Clarinet, Violin, Violoncello and Pianoforte (2003-2004) within the 2006 ‘Southern Stars’

concert in Sydney, Australia. Marshall McGuire for also inspiring and presenting the world premier of

Zambalogy for Harp (2004) in Sydney.

Pavel Mihelœiœ (Artistic Director, Ensemble MD7, and Dean of the Ljubljana Academy of Music, Ljubljana,

Slovenia) and Ensemble MD7 (Steven Loy [conductor], Anamarija Tomac [flute], Jože Kotar [clarinet], Katja

Krajnik [viola], Igor Mitrovic [violoncello], Uroš Polanc [trombone], Luca Ferrini [pianoforte], and Franci

Krevh [percussion]) for commissioning and performing Aequilibrium for Flute, Clarinet, Viola, Cello,

Ø×× Acknowledgements

Trombone, Piano and Percussion (2006) at the Ljubljana Festival 2006, 19 June – 31 August, 2006,

Ljubljana, Slovenia.

Adam Muller (Associate Professor of Saxophone, Florida International University, Miami, Florida, USA) and

Matthew Van Hoose (Accompanist in Residence, Department of Performing Arts, College of Arts and

Sciences, American University, Washington, DC, USA) for performing Digressioni modali for Tenor

Saxophone and Pianoforte at the First American University Saxophone Symposium, 26 March, 2005,

Washington, DC, USA.

Anne Norman (shakuhachi) and Peter Hagen (harpsichord) for assisting in the development of Tres

Imágenes Norteñas for Shakuhachi and Harpsichord (2006), and for performing the work at the

Melbourne Composers’ League ‘From a Silence Well’ concert as part of the 2006 Australia-Japan Year of

Exchange Celebrations.

Juan Miranda (Presenter, SBS Radio, ‘Spanish Radio’ Program) for producing an excellent program

featuring Navigating the Labyrinth for String Orchestra (2002), Seeds of Passion for Amplified Violoncello

(1999), and Bénédiction d’un conquérant for Symphony Orchestra (2004) – together with an interview

with the composer – on his SBS Radio, ‘Spanish Radio’ program.

Peter Neville (Head of Percussion, School of Music, Victorian College of the Arts) for his incredible

enthusiasm for contemporary music and Australian composition, as well as for his insight into

polyrhythmic science.

John D. Nugent (Music Editor, Oregon Literary Review: An Online Collection of Literature, Hypertext, Art,

Music, and Hypermedia, Portland, Oregon, USA) for publishing Exposiciones for Sampled Microtonal

Schoenhut Toy Piano (2005) in the Winter/Spring 2006, Vol. 1, No. 1 edition of the Oregon Literary

Review.

The Omni Ensemble (David Wechsler [flute], Paul Garment [bass clarinet], Olivier Fluchaire [violin], Deborah

Sepe [violoncello], and Jim Lahti [pianoforte]) for presenting the American premier, as well as a follow-up

performance of La flor en la colina for Flute, Clarinet, Violin, Violoncello and Pianoforte (2003-2004) in

Brooklyn and New York, NY, USA during their 2006 concert series.

José Oplustil Acevedo (Presenter, Radio Beethoven [Radioemisoras], Siglo XX, Santiago, Chile) for

producing an excellent program featuring Bénédiction d’un conquérant for Symphony Orchestra (2004),

Acknowledgements Ø×××

L’assaut sur la raison for Symphony Orchestra (2003), Görüsmeler for Amplified Èrhú, Sampled

Harmonium, Cajón and Bombo (2001), and Pañc hazar chakra kai andar for Prepared Disklavier (2000) –

together with an interview with the composer – on his Radio Beethoven (Radioemisoras) ‘Programa Siglo

XX’ program in Santiago, Chile.

Alex Pertout (Head of Improvisation, School of Music, Victorian College of the Arts) for his invaluable

advice with regards to Afro-Latin percussion, rhythm and improvisation.

Katija Farac-Pertout, my wife, for her amazing belief and understanding not only during the last four years

of the PhD degree, but throughout the last ten years of my composition studies at the University of

Melbourne.

Maritza Pertout (Library Technician, State Library of Victoria) for her assistance with Spanish grammar, as

well as countless other aspects of music publishing dilemmas.

Qmars Piraglu (formerly Siamak Noory) for his great inspiration and dedication to the realization of

Àzàdeh for Santär and Tape (2004, Rev. 2005), as well as for the performance of the work at the 2007

Asia Pacific Festival (26th Asian Composers League Festival & Conference), 8-16 February, 2007,

Wellington, New Zealand.

Glen Riddle (Coordinator, Foundation Program, Music Performance, School of Music, Victorian College of

the Arts) for the French lessons.

Hans Roels (Concert Program Manager and Producer, Logos Foundation, Ghent, Belgium) for presenting

the European premier of Exposiciones for Sampled Microtonal Schoenhut Toy Piano (2005) at the Logos

Foundation 2006 ‘Tape Tum & Heleen Van Haegenborgh’ concert in Ghent, Belgium.

Johanna Selleck for her incredible support throughout the PhD candidature.

Berislav Šipuš (Artistic Director, International Society for Contemporary Music (ISCM) World Music Days

2005 / 23rd Music Biennale Zagreb, 15-24 April, 2005, Zagreb, Croatia) for his hospitality during the

International Society for Contemporary Music (ISCM) World Music Days 2005 / 23rd Music Biennale

Zagreb, Croatia.

Ø×÷ Acknowledgements

The Sonemus Ensemble [Bosnia-Herzegovina] (Ališer Sijaric [Artistic Director], Boris Previšic [flute], Vedran

Tuce [bass clarinet], Julia Gubaidulina [pianoforte], Petar Haluza [violin], and Conradin Brodbek

[violoncello]) for the performance of La flor en la colina for Flute, Clarinet, Violin, Violoncello and

Pianoforte (2003-2004) at the International Society for Contemporary Music (ISCM) World Music Days

2005 / 23rd Music Biennale Zagreb, 15-24 April, 2005, Zagreb, Croatia.

Dr. Todd E. Sullivan (Chairperson, Department of Music, Indiana State University, Terre Haute, Indiana,

USA) for his incredible hospitality during the Indiana State University 38th Annual Contemporary Music

Festival, ‘Plugged In: Music With an Electric Edge’, November 10-12, 2004. A further warm thanks for

driving me all the way from Terre Haute to Bloomington, Indiana.

Natasha Talmacs (Presenter, SBS Radio, ‘Croatian Radio’ Program, Sydney, Australia) and Silvio Rivier

(Presenter, Narrator and Series Producer, Global Village, SBS TV, Sydney, Australia) for producing an

excellent program featuring La flor en la colina for Flute, Clarinet, Violin, Violoncello and Pianoforte (2003-

2004), L’assaut sur la raison for Symphony Orchestra (2003), and Seeds of Passion for Amplified

Violoncello (1999) – together with an interview with the composer – on her SBS Radio, ‘Croatian Radio’

program.

Antonio Tenace for his incredible support throughout the PhD candidature, and more importantly, for

fixing my scientific calculator.

Kenneth Young (Conductor, Wellington, New Zealand) and The Tasmanian Symphony Orchestra for the

performance of L’assaut sur la raison for Symphony Orchestra (2003) and Bénédiction d’un conquérant

for Symphony Orchestra (2004), which were especially composed for ACOF 2003 and 2004 (Australian

Composers’ Orchestral Forum – Composition workshops with Brenton Broadstock, Andrew Schultz, and

the Tasmanian Symphony Orchestra).

A special thanks to the following people for their general assistance, advice and support:

Betty Amsden (OAM), Celia Anderson, Dr. Jeri-Mae Astolfi (Assistant Professor, Department of Music,

Henderson State University, Arkadelphia, Arkansas, USA), Rachel Atkinson (Trio Erytheia), Peter Aviss

(Conductor and Musical Director, Oare String Orchestra, Faversham, UK), Laura Baker-Goldsmith, Pip Barry,

Natasha Bennett, Jennifer Bird (New Audience Ensemble), David Black (Rarescale, London, UK), Ellen

Bottorff (Orenunn Trio, Kansas City, Missouri, USA), Julianne Boren (Orenunn Trio, Kansas City, Missouri,

USA), Mark Boren (Orenunn Trio, Kansas City, Missouri, USA), James Bradley (Doubling Up Trio), Le Brass

Acknowledgements Ø÷

Band du Nord-Pas de Calais (Calais, France), Stuart Brownley (Doubling Up Trio), Gary Robert Buchanan

(Conductor and Musical Director, The Foundation Orchestra, Reno, Nevada, USA), David C. Bugli

(Conductor and Musical Director, Carson City Symphony, Carson City, Nevada, USA), Stuart Byrne

(Doubling Up Trio), Isin Cakmakcioglu (Trio Erytheia), José Miguel Candela (Coordinator, Comunidad

Electroacústica de Chile [CECh], Santiago, Chile), Erik Carlson (New York Miniaturist Ensemble, New York,

NY, USA), Robert Casteels (Dean of the Faculty of Performing Arts, LASALLE-SIA College of the Arts,

Singapore), Robert Chamberlain (Trio Erytheia), La Chapelle Musicale de Tournai (Tournai, Belgium),

Radiance Chen (New Audience Ensemble), Penelope Clarke (Thunder Bay, Ontario, Canada), Dr.

Christopher Coleman (Radio Television Hong Kong Radio 4, Hong Kong), Andrew Conley, Rolando Cori

(Associate Professor of Music, Facultad de Artes, Departamento de Música, Universidad de Chile, and

President, Asociacion Nacional de Compositores de Chile, Santiago, Chile), Nicholas Cowall (Conductor,

Victorian Youth Symphony Orchestra), Patricia Da Dalt (Quinteto CEAMC, Buenos Aires, Argentina), Lerida

Delbridge (The Tin Alley String Quartet), Madonna Douglas (Thunder Bay, Ontario, Canada), Eve Duncan

(President, The Melbourne Composers’ League), Shannon Ebeling, Mark Engebretson (Conference Chair,

2005 Society of Composers [SCI] National Conference, School of Music, University of North Carolina at

Greensboro, Greensboro, North Carolina, USA), Ed Ferris (New Audience Ensemble), Barbara Finch

(Thunder Bay, Ontario, Canada), The Foundation Orchestra (Reno, Nevada, USA), Johannes Fritzsch

(Nürnberg, Germany), Steve Gibson (Open Space Art Society, Victoria, British Columbia, Canada), Yves

Gigon (Canadian Electroacoustic Community [CEC], Montréal, Québec, Canada), Ian Godfrey (Lecturer in

Music and Education, Faculty of Music, University of Melbourne), Ben Goudy, Alejandro Guarello (Artistic

Director, XIII Festival de Música Contemporánea Chilena, Instituto de Música, Facultad de Artes, Pontificia

Universidad Católica de Chile, Santiago, Chile), Elías Gurevich (Quinteto CEAMC, Buenos Aires, Argentina),

Steven Heyman (The Syracuse Ensemble, Syracuse, New York, USA), Nancy Hosking, Luke Howard,

Ashley Hribar (Speak Percussion), Frédéric Inigo (Artistic Director, 3èmes Rencontres Musiques Nouvelles,

Lunel, France), Jason Kenner, Danae Killian, Victoria Jacono (3 Lines String Trio, Sydney), Jérôme Joy

(Coordinator, Locus Sonus – Audio in Art, École Nationale Supérieure d’Art de Nice-Villa Arson, Nice,

France), Stijn Kuppens (Artistic Director, Violoncello 2005, Brussels, Belgium), Laura Lentz (Crossroads Trio,

New York, NY, USA), Jennifer Logan (Co-Artistic Director, Los Angeles Sonic Odyssey, Electronic and

Computer Music Concert Series 2005, Los Angeles, California, USA), Phillipe Lorthios (Conductor, Le Brass

Band du Nord-Pas de Calais, Calais, France), Eric Lyon (Assistant Professor, Dartmouth College, Hanover,

New Hampshire, USA), George Macero (The Syracuse Ensemble, Syracuse New York, USA), Briony

Mackenzie (New Audience Ensemble), Marco Antonio Mazzini (Duo Dicto and Diversity, Ghent, Belgium),

John McMurtery (Doctoral Fellow, The Juilliard School of Music, New York, NY, USA), Nyssa McPhail, The

Melbourne University Orchestra, Natsuko Mineghishi, Patrick Murphy (3 Lines String Trio, Sydney), Simona

Musiani (Crossroads Trio, Rome, Italy), Tom Nelson (Southhampton, UK), Cliff Ojala (Thunder Bay, Ontario,

Ø÷× Acknowledgements

Canada), Jorge Pérez (Quinteto CEAMC, Buenos Aires, Argentina), Sonni Petrovski (Musical Director, The

Alea Contemporary Music Ensemble, Skopje, Republic of Macedonia), Marina Phillips (3 Lines String Trio,

Sydney), Timothy Phillips (Speak Percussion), Judy Pile, Vladimir Pritsker (The Syracuse Ensemble, Syracuse

New York, USA), Aleksander Pusz, Ryszard Pusz, Sabina Rakcheyeva (Diversity, Ghent, Belgium), Carla Rees

(Rarescale, London, UK), Darlene Chepil Reid (President, New Music North, Thunder Bay, Ontario,

Canada), Dr. James Romig (Co-Musical Director, The Society for Chromatic Art, New York, NY, USA),

Joelene Rzepisko, Guillermo Sánchez (Quinteto CEAMC, Buenos Aires, Argentina), Naomi Sato (The

Netherlands), Ginevra Schiassi (Ensemble Octandre, Bologna, Italy), José Schiller (Rádio MEC ‘Concerto

das Américas’, Rio de Janeiro, Brazil), Sam Schmetterer (New Audience Ensemble), Phillip Schroeder

(Associate Professor, Department of Music, Henderson State University, Arkadelphia, Arkansas, USA),

Haydée Schvartz (Quinteto CEAMC, Buenos Aires, Argentina), Johanna Selleck, Gemma Sherry, Tarko

Sibbel, Robert Sipos-Ori, Frank Sita (Plenty Valley FM), Emma Skillington (The Tin Alley String Quartet), Laura

Sullivan, Gabriella Swallow (Rarescale, London, UK), Matt Tait, Gaspare Tirincanti (Ensemble Octandre,

Bologna, Italy), Jo To, Eugene Ughetti (Speak Percussion), Josephine Vains, Amy Valent, Carlos Vera

(Santiago, Chile), Lauren Van Der Werff, Orchestra Victoria, The Victorian Youth Symphony Orchestra,

Professor Cirilo Vila Castro (Facultad de Artes, Departamento de Música, Universidad de Chile), Ward de

Vleeschhouwer (Duo Dicto, Ghent, Belgium), Carina Voly (Crossroads Trio, Buenos Aires, Argentina), Cory

Wagstaff, Koen Walraevens (Diversity, Ghent, Belgium), Russell Ward, Anneliese Weibel (Artistic Director,

2004 Society of Composers [SCI] Region II Conference, University of New York, School of Performing

Arts, Geneseo, New York, USA), Larissa Weller (New Audience Ensemble), Justin Williams (The Tin Alley

String Quartet), Elissa Wilson, Michelle Wood (The Tin Alley String Quartet), Larry Zimmerman (Minneapolis,

Minnesota, USA).

Acknowledgements Ø÷××

Table of Contents

Volume 1

Introduction ............................................................................................................................................................ 1

Microtonality ..................................................................................................................................................... 1

Pitch Audibility and Discrimination ................................................................................................................ 3

Three Microtonal Compositions .................................................................................................................... 4

Folio of Compositions ..................................................................................................................................... 5

Methodology ................................................................................................................................................... 6

Interval Nomenclature and Notation System ............................................................................................... 9

1. Theory Versus Performance Practice: Àzàdeh for Santñr and Tape ..................................... 11

A Brief History of Persian Classical Music .................................................................................................... 11

The Seventeen-Note Gamut ........................................................................................................................ 12

Persian Musical Scholarship in the Twentieth Century .............................................................................. 15

The Twenty-Four Equally-Tempered Quarter-Tone Scale ........................................................................ 16

The Pythagorean Division of the Octave ..................................................................................................... 18

Alain Daniélou’s Scale of Fifths ..................................................................................................................... 22

The Twenty-Two Note Division of the Octave .......................................................................................... 26

The Theory of Flexible Intervals .................................................................................................................... 27

Àzàdeh for Santñr and Tape ....................................................................................................................... 30

The Artist ......................................................................................................................................................... 30

The Instrument ................................................................................................................................................ 31

The Persian Modal System ............................................................................................................................ 32

Tuning Analysis Protocols ............................................................................................................................. 36

Tuning of the Santñr ...................................................................................................................................... 38

Spectrum Analysis Results............................................................................................................................. 42

Analysis of Variance ...................................................................................................................................... 49

Tuning System Comparison .......................................................................................................................... 52

Performance Practice and Tuning ................................................................................................................ 53

The Piano Tuner’s Octave and Inharmonicity ............................................................................................. 55

The Tuning of Unisons ................................................................................................................................... 56

Climate and Tuning ........................................................................................................................................ 58

Gušes of Dastgàh-e Segàh ........................................................................................................................... 60

Sampling of the Santñr and Vocals ............................................................................................................. 63

2. The Equally-Tempered Archetype: Exposiciones for Sampled Microtonal Schoenhut

Toy Piano ....................................................................................................................................................... 67

Equal Temperaments .................................................................................................................................... 67

Studies of Microtonal Equal Temperaments .............................................................................................. 68

Nicolas Mercator’s Fifty-Three-Tone Equally-Tempered Division of the Octave .................................. 71

Pietro Aron’s Quarter-Comma Meantone Tempered Division of the Octave ....................................... 74

Joseph Sauveur’s Forty-Three-Tone Equally-Tempered Division of the Octave ................................... 80

Origins of Equal Temperament .................................................................................................................... 83

The Twelve-Tone Equally-Tempered Division of the Octave .................................................................. 87

Exposiciones for Sampled Microtonal Schoenhut Toy Piano ................................................................. 92

A Brief History of the Toy Piano ................................................................................................................... 92

The Schoenhut Toy Piano Sample .............................................................................................................. 94

Sound Diffusion ............................................................................................................................................. 95

Polyrhythmic Theory ...................................................................................................................................... 96

Alain Daniélou’s Scale of Proportions ......................................................................................................... 99

Notation for the Twenty-Four Equal Temperaments .............................................................................. 102

Sléndro and Pélog Scales .......................................................................................................................... 104

One-Tone Equal Temperament ................................................................................................................. 107

Two-Tone Equal Temperament ................................................................................................................. 109

Three-Tone Equal Temperament ............................................................................................................... 111

Four-Tone Equal Temperament ................................................................................................................. 113

Five-Tone Equal Temperament .................................................................................................................. 115

Six-Tone Equal Temperament .................................................................................................................... 117

Seven-Tone Equal Temperament .............................................................................................................. 119

Eight-Tone Equal Temperament ................................................................................................................ 121

Nine-Tone Equal Temperament................................................................................................................. 123

Ten-Tone Equal Temperament .................................................................................................................. 127

Eleven-Tone Equal Temperament ............................................................................................................. 129

Twelve-Tone Equal Temperament ............................................................................................................ 132

Thirteen-Tone Equal Temperament ........................................................................................................... 135

ØØ Table of Contents

Fourteen-Tone Equal Temperament......................................................................................................... 139

Fifteen-Tone Equal Temperament ............................................................................................................. 144

Sixteen-Tone Equal Temperament ........................................................................................................... 150

Seventeen-Tone Equal Temperament ..................................................................................................... 153

Eighteen-Tone Equal Temperament ......................................................................................................... 157

Nineteen-Tone Equal Temperament ........................................................................................................ 162

Twenty-Tone Equal Temperament ........................................................................................................... 168

Twenty-One-Tone Equal Temperament .................................................................................................. 171

Twenty-Two-Tone Equal Temperament .................................................................................................. 174

Twenty-Three-Tone Equal Temperament ................................................................................................ 179

Twenty-Four-Tone Equal Temperament .................................................................................................. 184

Blackwood’s Dictum .................................................................................................................................. 188

3. The Harmonic Consideration: La Homa Kanto for Harmonically Tuned Synthesizer

Quartet ........................................................................................................................................................ 189

Just Intonation ............................................................................................................................................. 189

The Harmonic and Subharmonic Series .................................................................................................. 190

The Monochord ......................................................................................................................................... 198

Combinational Tones ................................................................................................................................. 200

Periodicity Pitch ........................................................................................................................................... 201

Prime Numbers, Primary Intervals, and Prime Limits ................................................................................ 202

The Just Diatonic Scale .............................................................................................................................. 202

The Just Chromatic Scale ........................................................................................................................... 205

Ben Johnston’s Fifty-Three-Tone Just Intonation Scale .......................................................................... 209

Harry Partch’s Forty-Three-Tone Just Intonation Scale ........................................................................... 212

Adriaan Daniël Fokker’s Thirty-One-Tone Equally-Tempered Division of the Octave ........................ 216

La Homa Kanto for Harmonically Tuned Synthesizer Quartet ............................................................... 218

The Harpsichord Sample ........................................................................................................................... 221

Ben Johnston’s System of Notation ......................................................................................................... 222

Compositional Strategy .............................................................................................................................. 223

Composing With Melodicles ..................................................................................................................... 224

Three-Limit Just Intonation ......................................................................................................................... 231

Five-Limit Just Intonation ............................................................................................................................ 236

Seven-Limit Just Intonation ........................................................................................................................ 243

Table of Contents ØØ×

Eleven-Limit Just Intonation ........................................................................................................................ 252

Thirteen-Limit Just Intonation ...................................................................................................................... 259

Seventeen-Limit Just Intonation ................................................................................................................. 266

Nineteen-Limit Just Intonation .................................................................................................................... 272

Twenty-Three-Limit Just Intonation ............................................................................................................ 278

Twenty-Nine-Limit Just Intonation.............................................................................................................. 284

Thirty-One-Limit Just Intonation ................................................................................................................. 289

Johnston’s Dictum ....................................................................................................................................... 296

Conclusion ........................................................................................................................................................... 297

‘Manual’ of Microtonal Composition ........................................................................................................ 297

A Vast Universe of Subtle Intervallic Relationships .................................................................................. 297

Bibliography ....................................................................................................................................................... 301

Appendices ........................................................................................................................................................ 311

Appendix A: Comparative Table of Musical Intervals ............................................................................ 311

Appendix B: Microtonal Notation Font .................................................................................................... 345

Volume 2

Recordings – Folio of Compositions 2003-2007: Volume 2 .......................................................................... vii

1. Àzàdeh for Santär and Tape, no. 389 (2004, Rev. 2005) .................................................................. 1

2. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, no. 392 (2005) ........................... 47

3. La Homa Kanto for Harmonically Tuned Synthesizer Quartet, no. 393 (2005) .......................... 91

4. Symétrie intégrante for Flute, Organ and Electronics, no. 394 (2005-2006) ......................... 153

5. Tres Imágenes Norteñas for Shakuhachi and Harpsichord, no. 396 (2006) .......................... 203

ØØ×× Table of Contents

Volume 3

Recordings – Folio of Compositions 2003-2007: Volume 3 .......................................................................... vii

1. L’assaut sur la raison for Symphony Orchestra, no. 386 (2003) .................................................... 1

2. Digressioni modali for Tenor Saxophone and Pianoforte, no. 387 (2003) ................................ 71

3. La flor en la colina for Flute, Clarinet, Violin, Violoncello and Pianoforte, no. 388 .................. 97

(2003, Rev. 2004)

4. Bénédiction d’un conquérant for Symphony Orchestra, no. 390 (2004) ........................... 175

5. Zambalogy for Harp, no. 391 (2004) ............................................................................................... 245

6. Aequilibrium for Flute, Clarinet, Viola, Cello, Trombone, Piano and Percussion, no. 395 ..... 257

(2006)

Table of Contents ØØ×××

Introduction

Microtonality

In a Perspectives of New Music article, Douglas Keislar states that the term microtonality “conjures up

images of impossibly minute intervals, daunting instruments with hundreds of notes per octave, and wildly

impractical performance instructions,” but that “such difficulties in fact characterize only a small

percentage of the music that uses tunings other than standard twelve-note equal temperament.” Keislar

then suggests that American composer Ivor Darreg’s proposal of the Greek term ‘xenharmonic’ or

‘unfamiliar modes’ is perhaps better suited to music utilizing “radically different tunings.”1 Alternative

language for the term ‘microtonal’ is presented by Lydia Ayers in Exploring Microtonal Tunings: A

Kaleidoscope of Extended Just Tunings and their Compositional Applications, with the following list of

expressions: “tuning; microintervals; macrointervals or macrotones, such as 5-tone, 7-tone, and 10-tone

equal temperaments; omnitonal; omnisonics; neoharmonic; xenharmonic; ‘exploring the sonic spectrum’;

and non-twelve.” Although in spite of Ayers’s general attraction to the broadness of ‘omnitonal’,

‘microtonal’ is nevertheless espoused for its universality.2

The actual term ‘microtonal’ is generally reserved for music utilizing “scalar and harmonic resources”

outside of Western traditional twelve-tone equal temperament, with “music which can be performed in

twelve-tone equal temperament without significant loss of its identity” not considered “truly microtonal”

by some theorists. Most non-western musical traditions (intonationally disengaged from contemporary

Western musical practice) almost certainly accommodate this description. In the online Encyclopedia of

Microtonal Music Theory, Joe Monzo provides the following discussion about the etymology of

‘microtonal’:

“Strictly speaking, as can be inferred by its etymology, ‘microtonal’ refers to small intervals. Some theorists

hold this to designate only intervals smaller than a semitone (using other terms, such as ‘macrotonal’, to

describe other kinds of non-12-edo intervals), while many others use it to refer to any intervals that deviate

from the familiar 12-edo scale, even those which are larger than the semitone – the extreme case being

exemplified by Johnny Reinhard, who states that all tunings are to be considered microtonal.”3

In the West, the concept of microtonality was notably given prominence to during the Renaissance by

Italian composer and theorist Nicola Vicentino (1511-1576), in response to “theoretical concepts and

1 Douglas Keislar, “Introduction,” Perspectives of New Music 29.1 (Winter, 1991): 173.

2 Lydia Ayers, “Exploring Microtonal Tunings: A Kaleidoscope of Extended Just Tunings and their Compositional

Applications,” (DMA diss., U. of Illinois, Urbana-Champaign, 1994, PA 9512292) 1-2.

3 Joe Monzo, “Encyclopedia of Microtonal Music Theory,” Microtonal, Just Intonation Electronic Music Software,

2005, Tonalsoft, 17 Nov. 2006, <http://www.tonalsoft.com/>.

materials of ancient Greek music,”4 and later, by music theorists R. H. M. Bosanquet (1841-1912), as well

as Hermann L. F. Helmholtz (1821-1894), and his “translator and annotator” Alexander John Ellis (1814-

1890).5 With regards to the adoption of microtonality by composers in more recent times, according to

The New Harvard Dictionary of Music:

“The modern resurgence of interest in microtonal scales coincided with the search for expanded tonal

resources in much 19th-century music. Jacques Fromental Halévy was the first modern composer to

subdivide the semitone, in his cantata Prométhée enchâiné (1847). The first microtonal piece to use

Western instrumental forms is a string quartet by John Foulds (1897); and the earliest known published

quarter-tone composition, Richard Stein’s Zwei Konzertstücke, op. 26 (1906), is for cello and piano.”6

Gardner Read offers the following historical perspective:

“The history of microtonal speculation during the first half of the twentieth century displays six names above

all others: Julián Carrillo, Adriaan Fokker, Alois Hába, Harry Partch, Ivan Wyschnegradsky, and Joseph Yasser.

All six contributed extensive studies on microtones – historical, technical, and philosophical – and all but

Yasser composed a significant body of music based on their individual explorations into microtonal

fragmentation of the traditional twelve-tone chromatic scale. Later theorist-composers – notably Easley

Blackwood, Ben Johnston, Rudolf Rasch, and Ezra Sims – have extended those explorations into various

tuning systems and temperaments, and each has devised a personal notation for various unorthodox

divisions of the octave.”

Read identifies five essential strategies for the procurement of microtonal intervals, which include: quarter-

and three-quarter-tones, or the division of the octave into twenty-four equal intervals; eighth- and

sixteenth-tones, or forty-eight and ninety-six equal intervals; third-, sixth-, and twelfth-tones, or eighteen,

thirty-six, and seventy-two equal intervals; and fifth-tones, or thirty-one equal intervals; as well as

“extended and compressed microtonal scales” with forty-three, fifty-three, sixty, seventy-two, or more

equal or unequal intervals in the octave.7 J. Murray Barbour on the other hand pronounces Pythagorean

(“excellent for melody, unsatisfactory for harmony”), just intonation (“better for harmony than for

melody”), meantone (“a practical substitute for just intonation, with usable triads all equally distorted”),

and equal temperament (“good for melody, excellent for chromatic harmony”) as the “four leading tuning

4 Accounts of the arcicembalo (a two-manual harpsichord capable of producing thirty-six distinct pitches per

octave) and arciorgano (organ adaptation) were presented by Nicola Vicentino in his treatises L’antica musica ridotta a la

moderna prattica of 1555 and Descrizione dell’ arciorgano (1561). For a further discussion, see Don Michael Randel, ed.,

The New Harvard Dictionary of Music (Cambridge, Mass.: Belknap Press of Harvard U Press, 1986) 47.

5 John H. Chalmers, Divisions of the Tetrachord: A Prolegomenon to the Construction of Musical Scales (Hanover,

NH: Frog Peak Music, 1993) 1-2.

6 Randel, ed., The New Harvard Dictionary of Music 492.

7 Gardner Read, 20th-Century Microtonal Notation (Westport, CT: Greenwood Press, 1990) 2-127.

2 Introduction

systems,” or the “Big Four.” Barbour also makes mention of the “more than twenty varieties of just

intonation,” and “six to eight varieties of the meantone temperament,” as well as the “geometric,

mechanical, and linear divisions of the line” for the mathematical approximation of equal temperament.8

According to Barbour, tuning systems may be classified into two distinct classes: the first being ‘regular’,

where all fifths but one are equal in size; and the second, ‘irregular’, where more than one fifth is unequal

in size. The former includes Pythagorean, meantone, and equal temperament, while the latter (as

classified by Barbour) excludes just intonation.9 Pitch Audibility and Discrimination Although it may be stated that the human ear has a general capacity to hear frequencies between the

ranges of 16Hz and 16,000Hz (equal to 16 to 16,000 cycles per seconds, and approximately C0 and

B9), it must be noted that numerous factors influence the actual outcomes. The 16Hz lower limit is

dependent on two principal factors, being wave intensity and shape; with the inclusion and exclusion of

pure tones displacing the figures for the lower limit to anywhere between 12Hz and 100Hz

(approximately Gþ0 and G2). The 16,000Hz upper limit is generally reserved for a healthy population

under the age of forty, with adolescent capacity as high as 25,000Hz (approximately G10); a supposed

‘normal hearing’ population in some cases not surpassing a 5,000Hz (approximately DÚ8) upper limit; and

another probable large percentage incapable of hearing beyond 10,000Hz (approximately DÚ9).10 The

frequency range of the 88-key pianoforte is between 27.5Hz and 4,186Hz, or A0 to C8, and therefore

encompasses pitch material with a range of over seven octaves. The seven-octave range additionally

represents the range embodied within the collection of instruments that constitute the traditional

symphony orchestra.11

The pitch discrimination threshold for an average adult is around 3Hz at 435Hz, which is

approximately one seventeenth of an equal tone, or 11.899 cents, although a “very sensitive ear can hear

as small a difference as 0.5Hz or less” (approximately a hundredth of a tone, or 1.989 cents). Tests

conducted in 1908 by Norbert Stücker (Zeitschrift für Sinnesphysiologie 42: 392-408) of sixteen

professional musicians in the Viennese Royal Opera conclude a pitch discrimination threshold between

one five-hundred-and-fortieth (0.1Hz) and one forty-ninth of a tone (1.1Hz), or 0.370 and 4.082 cents,

8 J. Murray Barbour, “Irregular Systems of Temperament,” Journal of the American Musicological Society 1.3

(Autumn, 1948): 20. 9 J. Murray Barbour, Tuning and Temperament: A Historical Survey (New York: Dover Publications, 2004) x-xi

10 Carl E. Seashore, Psychology of Music (New York: Dover Publications, 1967) 54-55. 11 Harry F. Olson, Music, Physics and Engineering, 2nd ed. (New York: Dover Publications, 1967) 123.

Introduction 3

with an average of 0.556Hz (approximately a hundredth of a tone), or 2.060 cents.12 In Tuning, Timbre,

Spectrum, Scale William A. Sethares adds the following to the discussion:

“The Just Noticeable Difference (JND) for frequency is the smallest change in frequency that a listener can

detect. Careful testing such as that of E. Zwicker and H. Fastl (Psychoacoustics, Springer-Verlag, Berlin

[1990]) has shown that the JND can be as small as two or three cents, although actual abilities vary with

frequency, duration and intensity of the tones, training of the listener, and the way in which JND is

measured.”13

Three Microtonal Compositions Three Microtonal Compositions: The Utilization of Tuning Systems in Modern Composition encompasses

the work undertaken by Lou Harrison (widely regarded as one of America’s most influential and original

composers) with regards to just intonation, and tuning and scale systems from around the globe – also

taking into account the influential work of Alain Daniélou (Introduction to the Study of Musical Scales),

Harry Partch (Genesis of a Music), and Ben Johnston (Scalar Order as a Compositional Resource). The

essence of the project being to reveal the compositional applications of a selection of Persian,

Indonesian, and Japanese musical scales utilized in three very distinct systems: theory versus performance

practice and the ‘Scale of Fifths’, or cyclic division of the octave; the equally-tempered division of the

octave; and the ‘Scale of Proportions’, or harmonic division of the octave championed by Harrison,

among others – outlining their theoretical and aesthetic rationale, as well as their historical foundations.

The project begins with the creation of three new microtonal works tailored to address some of the

compositional issues of each system, and ending with an articulated exposition; obtained via the

investigation of written sources, disclosure of compositional technique, mathematical analysis of relevant

tuning systems, spectrum analysis of recordings, and face-to-face discussions with relevant key figures.

The three microtonal works discussed in the thesis include Àzàdeh for santñr and tape, no 389

(2004, Rev. 2005) – composed for Iranian santñrist Qmars Piraglu (formerly Siamak Noory) – which

features the Persian santär (72-string box zither), and serves as a practical study of Persian tuning systems,

with its presentation of both ‘theoretical’ and ‘performance practice’ tunings; an ‘acousmatic’ work

entitled Exposiciones for sampled microtonal Schoenhut toy piano, no. 392 (2005), which attempts to

explore the equally-tempered sound world within the context of a sampled microtonal Schoenhut

model 6625, 25-key toy piano, a complex polyrhythmic scheme, and sequential tuning modulations

12 “Pitch discrimination is measured by sounding two pure tones in quick succession and gradually reducing the

difference in frequency until the observer is unable to tell which of the two tones is higher. The steps usually employed in

such a series are 30, 23, 17, 12, 8, 5, 3, 2, 1, and 0.5Hz, at the level of international (standard) pitch.” For a further

discussion, see Seashore, Psychology of Music 56-57.

13 William A. Sethares, Tuning, Timbre, Spectrum, Scale, 2nd ed. (London: Springer-Verlag, 2005) 44.

4 Introduction

featuring the first twenty-four equally-tempered divisions of the octave; and La Homa Kanto (or ‘The

Human Song’ in Esperanto) for harmonically tuned synthesizer quartet, which derives its pitch material

from Lou Harrison’s five-tone scales (presented in Lou Harrison’s Music Primer: Various Items About Music

to 1970) and features ten distinct tuning modulations: 3-limit through to 31-limit just intonation systems

based on the third, fifth, seventh, eleventh, thirteenth, seventeenth, nineteenth, twenty-third, twenty-ninth,

and thirty-first partials of the harmonic series.

The aim of the dissertation is to present an articulated exposition of three ‘original’ and unique

microtonal composition models individually exploring the expanded tonal resources of Pythagorean

intonation, equal temperament, and just intonation. It is also proposed that the thesis outlines their

theoretical and aesthetic rationale, as well as their historical foundations, with mathematical analysis of

relevant tuning systems, and spectrum analysis of recordings providing further substance to the project.

Theory versus performance is also taken into account, and the collaboration with an actual performer is

intended to deliver the corporeal perspective. It is anticipated that the thesis will not represent current

acoustic and psychoacoustic research at any great depth, and therefore should not be seen to serve as a

comprehensive study of physics and music. It will nevertheless provide a foundation for the exploration

of tuning systems, and additionally, present a composer’s perspective – as opposed to a musicological

or ethnomusicological study – of microtonal music composition. Folio of Compositions Other works incorporated into volume two and three of ‘Folio of Compositions 2003-07’ include: Symétrie

intégrante for Flute, Organ and Electronics, no. 394 (2005-06); Aequilibrium for flute, clarinet, viola, cello,

trombone, piano and percussion, no. 395 (2006); Tres imágenes norteñas for shakuhachi and harpsichord,

no. 396 (2006); L’assaut sur la raison for symphony orchestra, no. 386 (2003); Digressioni modali for tenor

saxophone and pianoforte, no. 387 (2003); La flor en la colina for flute, clarinet, violin, violoncello and

pianoforte, no. 388 (2003-04); Bénédiction d’un conquérant for symphony orchestra, no. 390 (2004);

and Zambalogy for harp, no. 391 (2004). These works do not represent the microtonal models of the first

three compositions, yet certainly adhere to an exploration of alternative scalar and harmonic materials,

and their application in contemporary compositional practice. Pitch material for these works has been

generated via a selection of methods such as multi-octave grouping (pitch material based on multi-

octave scales constructed of dissimilar tetrachords), modality (modes generated by the major, in,

hirajoshi and kumoijoshi scales), aleatoric formation (pitch material generated via indeterminate means),

pitch class set theory (pitch material derived from the 208 basic pitch-class sets of set theory), synthetic

symmetry (hexatonic and octatonic major and minor scales), cluster generation (pitch material derived

from five-note chords and inversions), physical and psychological concepts of consonance and

Introduction 5

dissonance (the harmonic language of the twelve primary intervals), polymodal and polytonal

juxtaposition (multiple scales and tonalities), as well as cross-cultural abstraction (non-Western music

theoretical concepts).

Methodology Chapter one (theory versus performance practice) begins with a brief history of Persian music, and is

followed by the presentation of Éafå al-Dån Urmawå’s seventeen-note gamut and division of the whole-

tone, and an explanation of the significance of the tetrachord in the construction of melodic and

harmonic structures. A discussion of Persian musical scholarship in the twentieth century then introduces

the three separate theories on intervals and scales of Persian music proposed in the twentieth century:

the twenty-four equally-tempered quarter-tone scale proposed by Ali Naqi Vaziri in the 1920s, the

alternative twenty-two-note scale proposed by Mehdi Barkešli in the 1940s based on Pythagorean

principles, as well as the theory of the five primary intervals of performance practice presented by

Hormoz Farhat in the 1990 publication of his doctoral thesis The Dastgàh Concept in Persian Music.14 The

division of the octave into twenty-four equally-tempered quarter-tones is given a historical perspective,

as well as a mathematical exposition, while the concept of Pythagorean intonation is firstly illustrated via

the construction of a twenty-seven-note Pythagorean scale with the necessary intervals to facilitate the

general modulations of Western tonal music; and secondly, via Daniélou’s ascending ‘scale of fifths’, or

cyclic division of the octave, which presents a series of fifty-nine consecutive fifths, or sixty lü. The BCE

Chinese origins of Pythagoreanism and its philosophical significance according to theorist King Fâng are

also subsequently discussed.15 The development of the seventeen-note gamut by Mehdi Barkešli into a

twenty-two-note Pythagorean scale is then presented, which is followed by Farhat’s theory of flexible

intervals, or of the five primary intervals of performance practice – advocated by Farhat in opposition to

both twenty-four-tone equally-tempered, and twenty-two-note Pythagorean scales of Vaziri and

Barkešli.16

The work, Àzàdeh for santñr and tape, is then introduced, together with a brief biography of the

artist, Qmars Piraglu; a description of instrument, the Persian santär (a 72-string [or 18 quadruple-stringed]

box zither); and a discussion of the essence of the Persian modal system. Following the establishment of

the tuning analysis protocols, a detailed exposition of the tuning process of the santñr for dastgàh-e

segàh (on F) is presented. Spectrum analysis results collected on three separate occasions (with a

14 Hormoz Farhat, The Dastgàh Concept in Persian Music (New York: Cambridge U. Press, 1990) 7.

15 Alain Daniélou, Music and the Power of Sound: The Influence of Tuning and Interval on Consciousness

(Rochester, VT: Inner Traditions, 1995) 35-37.

16 Farhat, The Dastgàh Concept in Persian Music 15-16.

6 Introduction

periodicity of 3-6 months) for each of the twenty-seven sets of strings are then analyzed with regards to

the intervallic size of octaves, perfect fifths, perfect fourths, tempered perfect fourths, and neutral thirds.

An analysis of variance is then conducted with the data collected, which in turn produces mean

measurements with the capacity to characterize tuning characteristics. A tuning system comparison then

concludes a relationship between Farhat’s and Piraglu’s division of the whole-tone, with Farhat’s theory of

flexible intervals accorded as the most plausible hypothesis.

In view of the fact that stretched, as well as compressed octaves are a common occurrence in

Piraglu’s tuning of the santñr, the theory of the ‘piano tuner’s octave’ is discussed, along with the natural

phenomenon of inharmonicity – a factor especially affecting plucked and struck strings (along with other

musical sounds with a short decay).17 A comparison is also made between the tuning of a triple-string

unison of a piano and a quadruple-string unison of a santñr. Climate and its effects on tuning are then

considered, and especially in order to substantiate Piraglu’s claims of the climatic conditions of

Melbourne, Australia being “unsatisfactory” for the tuning of the santär in comparison to Tehran, Iran. The

twenty-four gušes for dastgàh-e segàh according to a prominent radif associated with Mñsà Marñfi are

then presented, followed by the pitch organization of the adopted six most prominent elements of the

radif of dastgàh-e segàh. Finally, the structural scheme of the work and its basis on ‘golden mean’

proportions are explained, as well as the sampling process of the santär and vocals, and digital

processing that culminates in the tape element of Àzàdeh for santñr and tape.

Chapter two (the equally-tempered archetype) begins with a discussion about Partch’s notion of

two distinct classes of equal temperaments: those that produce equal third-tones, quarter-tones, fifth-

tones, sixth-tones, eighth-tones, twelfth-tones, and sixteenth-tones; as opposed to those that divide the

octave into nineteen, thirty-one, forty-three, and fifty-three equally-tempered intervals.18 This is followed

by a brief history of some important studies of the equally-tempered paradigm, namely by Julián Carrillo

Trujillo, Ferruccio Busoni, Ramon Fuller, and Easley Blackwood, with the latter two serving as benchmarks

for the establishment of the criteria to properly assess the musical virtues of a particular equal

temperament. The deviation of basic equally-tempered intervals from just intonation, Fuller’s eight best

equal temperaments, and Blackwood’s concept of ‘recognizable diatonic tunings’ are then discussed.

Nicolas Mercator’s fifty-three-tone equally-tempered division of the octave, which is Fuller’s

recommendation for a temperament with the capacity to approximate just intervals, is consequently

presented, along with an opposing view by Dirk de Klerk.

In order to illustrate the principal evolutionary markers leading up to the adoption of equal

temperament in the West – from Pythagorean intonation, meantone and well temperament, to equal

17 Lloyd, and Boyle, Intervals, Scales and Temperaments: An Introduction to the Study of Musical Intonation 166-67.

18 Harry Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments, 2nd ed. (New York:

Da Capo, 1974) 425.

Introduction 7

temperament – Pietro Aron’s quarter-comma meantone temperament is introduced, as well as Joseph

Sauveur’s forty-three-tone equal temperament, which approximates fifth-comma meantone temperament.

The origins of equal temperament are then traced back to 1584 China, and Prince Chu Tsai-yü’s

monochord. What follows is a discussion of the geometrical and numerical approximations of Marin

Mersenne and Simon Stevin, which culminate in Johann Faulhaber’s monochord, and the first printed

numerical solution to equal temperament based on the theory of logarithmic computation.19 The

mathematical formula for twelve-tone equal temperament, the equally-tempered monochord, and

beating characteristics of the twelve-tone equally-tempered major and minor triads are then sequentially

presented, which are followed by the equal thirds, sixths, fifths, and fourths in piano tuning.

The work, Exposiciones for Sampled Microtonal Schoenhut Toy Piano, is then introduced, together

with a brief history of the toy piano, the Schoenhut toy piano sample, as well as concepts of sound

diffusion and polyrhythmic theory utilized in the composition. In order to illustrate the design of the

proposed notation for the twenty-four equal temperaments, Daniélou’s ‘scale of proportions’, or

harmonic division of the octave, which presents a series of sixty-six unique intervals is introduced. Paul

Rapoport’s Pythagorean notation then provides an alternative to the system of notation based on

Daniélou’s subdivision of the whole-tone. Sléndro and pélog scales are then discussed from a historical

perspective, with the gamelan gedhé sléndro and pélog tunings from Sri Wedhari theatre auditorium in

Solo, Central Java serving as the ‘performance practice’ model. The harmonic characteristics of the

sléndro and pélog scales are then presented in accordance to five-limit intonation principles. What

follows is a systematic exposition of the compositional application of each equal temperament between

one and twenty-four.

Chapter three (the harmonic consideration) begins with a basic outline of just intonation and ‘extended

just intonation’, or the incorporation of partials beyond the sixth harmonic.20 A historical and scientific

perspective of the harmonic series is then presented, together with examples of the beating characteristics of

the first eight partials of the harmonic series, as well as of the mistuned and properly tuned unison, and

mistuned and properly tuned octave. Dissonance, with special reference to the theory of beats, is defined

according to James Tenney, Helmholtz, Bosanquet, and Johnston. The complement or mirror image of the

harmonics series, or the ‘subharmonic series’, is also discussed, together with Partch’s theory of ‘otonalities’

(pitches derived from the ascending series) and ‘utonalities’ (pitches derived from the descending series).21 A

comparative table of intonation then provides interval, ratio, and cents data for the twelve basic intervals of just

intonation, Pythagorean intonation, meantone temperament, and equal temperament.

19 Barbour, Tuning and Temperament: A Historical Survey 78.

20 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters,” 106-07.

21 David D. Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation, 3rd ed.

(San Francisco: Other Music, 2002) 28-30.

8 Introduction

In order to illustrate the basic principles of proportions and string lengths, the traditional structure and

function of the monochord is explained, with the generation of simple octaves and fifths utilized to

demonstrate the theoretical basis for the Pythagorean monochord. A table depicting all the intervals of

the harmonic series from the first partial through to the one-hundred-and-twenty-eighth partial is then

presented. Combinational tones, or differential and summation tones, are also subsequently explained,

together with their implications on the intervals of the octave, just perfect fifth, just perfect fourth, just

major third, just minor sixth, just minor third, and just major sixth. This is followed by a discussion of

periodicity pitch, and its theoretical significance in relation to JND, or Just Noticeable Difference. The

relationship of prime numbers, primary intervals, and prime limits to just intonation principles is

subsequently explained.

The concept of just intonation is then illustrated via the construction of a seven-note just diatonic

scale, and the presentation of the beating characteristics of the just major triad. This is followed by the

construction of a twenty-five-note just enharmonic scale, and its development into Johnston’s fifty-three-

tone just intonation scale. Harry Partch’s forty-three-tone just intonation scale, and his rationale for the

consequential harmonic expansion to eleven-limit is then explained. The twenty unique triads, fifteen

unique tetrads, and six unique pentads made possible via the inclusion of the eleven-limit intervals are

additionally presented. The final octave division discussed in the chapter is Adriaan Daniël Fokker’s thirty-

one-tone equally-tempered division of the octave, and in view of its capability to approximate the tonal

resources of seven-limit just intonation.

The work, La Homa Kanto for Harmonically Tuned Synthesizer Quartet, is then introduced, together

with a presentation of Harrison’s five pentatonic scales, which serve as the pitch material, the ‘1967

William Dowd French Double Harpsichord’ sample, and Johnston’s system of notation, which serves as

the system of notation utilized in the score. Compositional strategy is then discussed, together with

Harrison’s concept of composing with melodicles, or neumes, which is adopted and developed into a

system incorporating three categories of motivic manipulation: melodic transformation of motive, rhythmic

transformation of motive, and harmonic transformation of motive. What follows is a systematic exposition

of the compositional application of each just intonation limit between three and thirty-one. Interval Nomenclature and Notation System Intervals based on Pythagorean intonation have been simply named according to their cyclical position,

and therefore follow an either ascending 3/2 incremental progression from natural, sharp, double sharp,

to triple sharp; or a descending 4/3 incremental progression from natural, flat, double flat, to triple flat.

The procedure is exemplified via the twenty-seven-note Pythagorean scale, which incorporates fifteen

intervals generated by an ascending series of fifths, or the pitches C, G, D, A, E, B, F!, C!, G!, D!, A!, E!,

Introduction 9

B!, F#, C#, and G#; and another eleven intervals, by a descending series, or C, F, B", E", A", D", G", C", F",

B$, E$, and A$. The method adopted in equal temperament on the other hand is a nomenclature

based on the comma approximations to Daniélou’s ‘scale of proportions’, or sixty-six-note just intonation

scale, with every interval not characterized by the equal semitones and quarter-tones of 12-et and 24-et

further indentified via its origin (for example: 5-et supermajor second, 7-et grave or small tone, and 9-et

great limma, or large half-tone). Exceptions to this rule include 31-et, 43-et, and 53-et, which because

are not discussed in the thesis with relation to other intervals, do not require a differential prefix with the

same conditions. Adriaan Daniël Fokker’s thirty-one-tone equally-tempered division of the octave

introduces a further element to intervallic nomenclature. The system, which was developed by David C.

Keenan, involves the prefixes: double diminished, subdiminished, diminished, sub, perfect, super,

augmented, superaugmented, and double augmented for unisons, fourths, fifths, and octaves; while

subdiminished, diminished, subminor, minor, neutral, major, supermajor, augmented, and

superaugmented for seconds, thirds, sixths, sevenths, and ninths. Perfect and major, or “the ones implied

when there is no prefix,” represent the central position of a range based on comma or diesis increments

from ß4 to +4 (for example: diminished third, subminor third, minor third, neutral third, major third,

supermajor third, and augmented third).22 For intervals beyond five-limit intonation, James B. Peterson’s

recommendations for the naming of bases has been adopted, which results in the following additional

prefixes for seven-, eleven-, thirteen-, seventeen-, nineteen-, twenty-three-, twenty-nine-, and thirty-one-

limit: septimal, undecimal, tridecimal, septendecimal, nonadecimal, trivigesimal, nonavigesimal, and

untrigesimal (for example: septimal superfifth, undecimal subfifth, tridecimal subfifth, septendecimal

superfifth, nonadecimal superfifth, trivigesimal superfifth, nonavigesimal subfifth, and untrigesimal

superfifth).23 The classification of 724 unique intervals incorporated into the comparative table of musical

intervals (see Appendix A) includes all the intervals cited in the current study.

The notation symbols utilized in the thesis include the five standard accidental signs of Western

music; four common quarter-tone and three-quarter-tone symbols; twenty-three unique symbols based

on Daniélou’s division of the whole-tone; Ali Naqi Vaziri’s notation system, or four accidentals of Persian

music; Johnston’s system of notation, which contains twenty-three unique symbols for the notation of just

intonation up to the thirty-first harmonic; as well as Fokker’s nine symbols for the notation of thirty-one

equal temperament. All these symbols have been incorporated into a 238-character microtonal notation

PostScript Type 1 font (see Appendix B), which was created via the modification of a selection of

symbols in the Coda Music Finale’s Maestro font utilizing CorelDraw 13.0 and FontMonger 1.0.8.

22 David C. Keenan, “A Note on the Naming of Musical Intervals,” David Keenan’s Home Page, 3 Nov. 2001, 22 Nov.

2006, <http://users.bigpond.net.au/d.keenan/Music/IntervalNaming.htm>. 23 James B. Peterson, “Names of Bases,” The Math Forum: Ask Dr. Math, 15 Apr. 2002, Drexel U., Philadelphia, PA,

22 Nov. 2006, <http://mathforum.org/library/drmath/view/60405.html>.

10 Introduction

1. Theory Versus Performance Practice: Àzàdeh for Santñr and Tape

A Brief History of Persian Classical Music Modern Persian scholarship on the theory of intervals and scales may be mainly attributed to the

theoretical writings of medieval music scholars Éafå al-Dån ‘Abd al-Mu’min al-Urmawå’ (d. 1294) and Quðb

al-Dån ‘Maämñd ibn Mas’ñd al-Shåràzå’ (1236-1311). “The latter half of the thirteenth century constitutes

one of the most important periods in the history of Arab and Persian musical theory,” notes Owen Wright.

“It witnessed the emergence of a corpus of theoretical writings that not only demonstrate a considerable

degree of originality, but also provided the framework within which all the major theorists of the following

two centuries were to operate.” Éafå al-Dån in particular is acknowledged for founding the ‘Systematist

school’ with his two influential treatises: Kitàb al-adwàr of 1252 (‘Book of Cycles’) and Risàla al-sharafiyya fi

al-nisàb al-ta’lifiyya of 1267 (‘Sharafian Treatise on Intervallic Relations’), while Quðb al-Dån for his further

contribution to the theory within a section about music contained in his encyclopedia Durrat al-tàj (‘Pearl

of the Crown’), published circa 1300.24 In the spirit of their predecessors – Al-Kindå (d. 873), and

celebrated author of Kitàb al-mñsåqå al-kabår (‘Great Book on Music’), Abu Nasr Fàràbå (872-950); as well

as Ibn Sånà (980-1037) – their findings were essentially based on the musical theories of the classical

Greeks; from Pythagoras of Samos (fl. 530 B.C.) to Aristoxenus of Tarentum (fl. 400 B.C.).25 Cultural links

between Persia and Ancient Greece existed between 500 B.C. and 300 A.D., and were further infused

by Alexander the Great’s conquest of the Achaemenid Empire in 330 B.C. that generated the hundred

years of Greek rule in Persia. As a consequence, “the works of Euclid, Aristoxenus, Ptolemy, and others

translated into Arabic at Baghdad during the ninth century,26 served as models for the great Islamic

theorists,” notes Ella Zones.27

24 Owen Wright, The Modal System of Arab and Persian Music A.D. 1250-1300, London Oriental Series, vol. 28

(Oxford: Oxford U. Press, 1978) 1-20. 25 Hormoz Farhat, “Iran: Classical Traditions,” The New Grove Dictionary of Music and Musicians, ed. Stanley Sadie

and John Tyrrell, 2nd ed., vol. 12 (London: Macmillan Reference, 2001) 531.

26 “During the Abbasid period (750-1258) many branches of Islamic scholarship developed rapidly, among them

medicine, astronomy, alchemy, geography, mathematics, and also music theory. This development was stimulated by

contact with ancient Greek writings which became available to Islamic scholars through translations done in the Bait al-

Äikma (House of Wisdom), a library, astronomical observatory, and translation institute established in Baghdad by caliph al

Ma’mñn.” For a further discussion, see by Josef M. Pacholczyk, “Secular Classical Music in the Arabic Near East,” Musics of

Many Cultures (Berkeley, CA: U. of California Press, 1980) 255. 27 Ella Zonis, “Contemporary Art Music in Persia,” The Music Quarterly 51.4 (Oct., 1965): 636-37.

The Seventeen-Note Gamut In Kitbag al-adware, Éafå al-Dån proposes that a Pythagorean whole-tone (equal to the frequency ratio of

9/8, or 203.910 cents) should only be subdivided into either one Pythagorean limma (256/243, or

90.225 cents), or two Pythagorean limmas (equal to a Pythagorean diminished third, 65536/59049, or

180.450 cents). This in effect generates a theoretical basis for a whole-tone constructed from the sum of

two limmas and a Pythagorean comma (531441/524288, or 23.460 cents), and a tetrachord made up of

two whole-tones and a limma that is implemented in an octave as two conjunct tetrachords, plus an

additional whole-tone. The result is a seventeen-note scale,28 and modality based on two conjunct

tetrachords, which may be theoretically referred to as a bitetrachordal system.29

Table 1. Éafå al-Dån Urmawå’s division of the whole-tone

PATTERN L L + L L + L + C

INTERVAL Pythagorean

limma

Pythagorean

diminished third

just

major tone

RATIO 256/243 65536/59049 9/8

CENTS 90.225 180.450 203.910

The tetrachord (a four-note series enclosed within the range of a perfect fourth) is a concept borrowed

from ancient Greek music, where as part of the ‘Greater Perfect System’ – a two-octave system made up

of four conjunct and disjunct tetrachords (Hypaton, Meson, Diezeugmeson and Hyperbolaeon), as well

as an additional whole-tone (Proslambanomenos) to complete the lower part of the range – was “the

basic building block of Greek music,” and therefore at the core of Greek theory on intervals and scales.30

The essence of Ancient Greek music and its proponents is summarized by R. P. Winnington-Ingram thus:

Ancient Greek music was purely or predominantly melodic; and in such music subtleties of intonation

count for much. If our sources of information about the intervals used in Greek music are not always easy

to interpret, they are at any rate fairly voluminous. On the one hand we have Aristoxenus, by whom

musical intervals were regarded spatially and combined and subdivided by the processes of addition and

subtraction; for him the octave consisted of six tones, and the tone was exactly divisible into fractions such

28 Owen Wright, “Arab Music: Art Music,” The New Grove Dictionary of Music and Musicians, ed. Stanley Sadie and

John Tyrrell, 2nd ed., vol. 12 (London: Macmillan Reference, 2001) 806.

29 Dariush Talai, “A New Approach to the Theory of Persian Art Music: The Radåf and the Modal System,” ed. Virginia

Danielson, Scott Marcus and Dwight Reynolds, The Garland Encyclopedia of World Music: The Middle East, vol. 6 (New

York: Routledge, 2002) 871.

30 André C. Barbera, “Greece,” ed. Don Michael Randel, The New Harvard Dictionary of Music (Cambridge, Mass.:

Belknap Press of Harvard U Press, 1986) 347-49.

12 Theory Versus Performance Practice

as the half and quarter, so that the fourth was equal to two tones and a half, the fifth to three tones and a

half, and so on. On the other hand we have preserved for us in Ptolemy’s Harmonics the computations of

a number of mathematicians, who realized correctly that intervals could only be expressed as ratios (e.g. of

string-lengths), that the octave was less than the sum of six whole tones and that this tone could not be

divided into equal parts. These authorities are Archytas, the Pythagorean of the early fourth century,

Eratosthenes (third century), Didymus (first century), and Ptolemy himself (second century A.D.). To these

we must add the scale of Plato’s Timaeus (35B) and, closely related to it, the computations of the pseudo-

Philolaus (ap. Boethium, Mus. III, 8) and of Boethius himself (IV, 6).31

With regards to the function of tetrachords in the construction of melodic and harmonic structures, John

H. Chalmers presents the following discussion:

“Tetrachords are modules from which more complex scalar and harmonic structures may be built. These

structures range from the simple heptatonic scales known to the classical civilizations of the eastern

Mediterranean to experimental gamuts with many tones. Furthermore, the traditional scales of much of the

world’s music, including that of Europe, the Near East, the Catholic and Orthodox churches, Iran, and India,

are still based on tetrachords. Tetrachords are thus basic to an understanding of much of the world’s

music.”

Chalmers then further expands on the issue with the subsequent definition:

“The tetrachord is the interval of a perfect fourth, the diatessaron of the Greeks, divided into three

subintervals by the interposition of two additional notes. The four notes, or strings, of the tetrachord were

named hypate, parhypate, lichanos, and mese in ascending order from 1/1 to 4/3 in the first tetrachord of

the central octave of the ‘Greater Perfect System’, the region of the scale of most concern to theorists.

Ascending through the second tetrachord, they were called paramese, trite, paranete, and nete.”32

Stringed instruments are recognized as a major factor in the design of tuning and scale systems. “The

fretting and tuning of stringed instruments was directly connected to the development of modes. We

can deduce this from the fact that stringed instruments have been used to study intervals and tetrachords

from antiquity, and from the fact that in the past, musicians were poet-singers first of all, accompanying

their poetry and song with stringed instruments,” explains Dariush Talai. It is interesting to note that “the

tetrachord also corresponds to a physical area on the neck of instruments such as the ‘ñd, tàr, and setàr,

where the fingers can reach the notes without changing position.”33

31 R. P. Winnington-Ingram, “Aristoxenus and the Intervals of Greek Music,” The Classical Quarterly 26.3/4 (Jul.-Oct.,

1932): 195.

32 Chalmers, Divisions of the Tetrachord: A Prolegomenon to the Construction of Musical Scales 4.

33 Talai, “A New Approach to the Theory of Persian Art Music: The Radåf and the Modal System,” The Garland

Encyclopedia of World Music: The Middle East 868-69.

Theory Versus Performance Practice 13

Table 2. The seventeen-note gamut.34

TRADITIONAL

PERSIAN SYSTEM

PHONETIC

TRANSCRIPTION

PYTHAGOREAN

NOTATION

CONTEMPORARY

NOTATION

RATIO

(FRACTION)

CENTS

� � � � � � � � ��

� �� �� �� �� ��

��

A

b

j

d

h

v

z

ä

ë

y

yà

yeb

yej

yed

yeh

yu

yez

yeä

C

D"

E$

D

E"

F"

E

F

G"

A$

G

A"

B$

A

B"

C"

B

C

C

D"

Dî

D

E"

Eî

E

F

G"

Gî

G

A"

Aî

A

B"

Bî

B

C

1/1

256/243

65536/59049

9/8

32/27

8192/6561

81/64

4/3

1024/729

262144/177147

3/2

128/81

32768/19683

27/16

16/9

4096/2187

243/128

2/1

0.000

90.225

180.450

203.910

294.135

384.360

407.820

498.045

588.270

678.495

701.955

792.180

882.405

905.865

996.090

1086.315

1109.775

1200.000

Illustrated in following music example is the scale of Éafå al-Dån, which is referred to as the seventeen-note

gamut and based on Al-Fàràbå’s first tetrachord division on the Khorasan long-necked lute, or ðunbñr

khuràsànå. The seventeen-note gamut (utilizing G as the tonic) is further characterized by the following

intervallic pattern: L, L, C – L, L, C – L – L, L, C – L, L, C – L + L, L, C. The transposition to C of the intervallic

pattern reorganized thus: L, L, C – L, L, C – L + L, L, C – L, L, C – L, L, C – L.35

Ex. 1. Subdivision of the tetrachord and the seventeen-note gamut (Éafå al-Dån Urmawå, 13th century)

34 Taqi Bineš, The Short History of Persian Music (Tehran: Àrvin Publication, 1995) 106.

35 Wright, “Arab Music: Art Music,” The New Grove Dictionary of Music and Musicians 806.

14 Theory Versus Performance Practice

Although the theoretical notion of a seventeen-note division of the octave became widely adopted by

the Islamic world for centuries to come, there is much speculation amongst musicologists today about

the actual implementation of this synthetic scale by musicians of the time in the context of performance

practice, as the employment of an intuitive system of flexible intervals seems more probable. It should

be noted that no mode has ever contained all seventeen notes (heptatonic being the dominant

character), and that the Pythagorean comma was not utilized as an actual successive pitch in a scale

formation but rather as an addition to another intervallic value to form a composite interval.36 From a

historical perspective, with regards to the adoption of a theoretical scale by the Arabs and Persians in the

fourteenth century and the “so-called messel or octave of seventeen third tones,”37 A. J. Hipkins writes:

“The arithmetical reasonings of philosophers who sought to explain the musical scale could never have

been, excepting in the larger intervals, the practical art of the musicians.”38

Persian Musical Scholarship in the Twentieth Century In the sixteenth century, with Shi’a rule imposed by the Sadavid dynasty (1501-1722) generating a certain

hostility towards music, Persian musical scholarship enters its period of decline, and it is not until the

beginning of the twentieth century during the Pahlavi dynasty (1925-1979) that we begin to see the

resurgence of theoretical research into the Persian theory of intervals and scales.39

“In the beginning of the twentieth century, Iran entered a period during which the Western world was

idealized. Western music had already been taught in Iran for over three decades; thus, Iranian musicians

were thoroughly educated in it,” explains Talai. “In fact, some people began to question the value of

Iranian music as serious and dignified, doubting that its instruction should be institutionalized. Others,

motivated in part by a spirit of nationalism, tried to revive Persian music. They wanted to create a theory

relating it to the perspective of what they regarded as the more progressive, Western classical music.”40

36 Farhat, The Dastgàh Concept in Persian Music 12-13.

37 “La Borde, Villoteau, and Kiesewetter, studying Arab music from obscure treatises of medieval Arab philosophers

such as Khalil, Al-Kindå, Ibn Khaldñn, and Al-Fàràbå, maintained that Arab music is based on the so-called messel or octave

of seventeen third tones, and such was purely Oriental and fundamentally different from the Greek diatonic system, which

is based on tones and semitones. They proceed, therefore, to see the origin of Arab music in Persia, where we find a

scale of semitones, demi-semitones, and even semidemi-semitones, their theory being that Persia, after being conquered

by the Arabs, had imposed its music upon the conquerors.” For a further discussion, see Joseph Reider, “Jewish and

Arabic Music,” The Jewish Quarterly Review 7.4 (Apr., 1917): 640.

38 Francesco Salvador-Daniel, and Henry George Farmer, The Music and Musical Instruments of the Arab: With

Introduction on How to Appreciate Arab Music (Portland, ME: Longwood Press, 1976) 182.

39 Farhat, The Dastgàh Concept in Persian Music 5.

40 Dariush Talai, Traditional Persian Art Music: The Radif of Mirza Abdollah, trans. Manoochehr Sadeghi (Coasta Mesa,

CA: Mazda, 2000) 7-8.

Theory Versus Performance Practice 15

This ideological shift in effect being the impetus for three separate theories on intervals and scales of

Persian music proposed in the twentieth century: the twenty-four equally-tempered quarter-tone scale

proposed by Ali Naqi Vaziri in the 1920s, the alternative twenty-two-note scale proposed by Mehdi

Barkešli in the 1940s based on Pythagorean principles, as well as the theory of the five primary intervals of

performance practice presented by Hormoz Farhat in the 1990 publication of his doctoral thesis The

Dastgàh Concept in Persian Music.41 In spite of the almost universal rejection of the proposal for a twenty-

four-tone equally-tempered scale, one cannot underestimate Vaziri’s influence to Persian musical

scholarship in the twentieth century, as some of the reforms implemented to the art form by the theorist

did gain wide acceptance.

Vaziri was born in Tehran, in 1887, and following his traditional studies in classical Persian music,

went on to study piano, harmony, and voice at the École Supérieure de Musique in Paris, and

counterpoint and composition at the Hochschule für Musik in Berlin. Upon his return to Iran in 1923, he

founded his conservatory, the Madresse-ye Ali-ye Musiqi, and amongst his many achievements is today

credited for the first published transcriptions of Persian music in Western staff notation.42 As theorist, his

most notable writings on the twenty-four equally-tempered quarter-tone scale and the Persian modal

system are contained within his 1934 publication Musiqi-ye Nazari.43 The Twenty-Four Equally-Tempered Quarter-Tone Scale The concept of the division of the octave into twenty-four equally-tempered quarter-tones is not a new

phenomena to the region, with Lebanese theorist Måkhà’ål Måshàqàh (1800-1889)44 proposing an Arabian

quarter-tone system in the previous century.45 Earlier examples include Zalzal (d. after 842) – a prominent

ñd teacher of the ninth century who reformed Pythagorean intonation of the time with his introduction of

eleven-limit ratios to the fretting of the lute (an intonation system limited to the eleventh harmonic with

intervals closely resembling those of equal temperament). The Zalzal intervals included the undecimal

grave or small neutral second (12/11, or 150.637 cents), undecimal subfifth (16/11, or 648.682 cents),

and undecimal grave or small neutral sixth (18/11, or 852.592 cents); and represents a system subscribed

41 Farhat, The Dastgàh Concept in Persian Music 10.

42 Ella Zonis, “Classical Iranian Music,” Musics of Many Cultures (Berkeley, CA: U. of California Press, 1980) 186-87.

43 Farhat, The Dastgàh Concept in Persian Music 9.

44 Habib Hassan Touma, The Music of the Arabs, trans. Laurie Schwartz (Portland, OR: Amadeus Press, 1957) 19.

45 Måshàqàh’s treatise proposing the twenty-four equally-tempered division of the octave for Turko-Arabian music

translated into English by American protestant missionary and scholar Eli Smith (1801-57), and then consequently

published in the Journal of the American Oriental Society (1849). For a further discussion, see Alexander J. Ellis, and

Alfred J. Hipkins, “Tonometrical Observations on Some Existing Non-Harmonic Musical Scales,” Proceedings of the Royal

Society of London 37 (1884): 371.

16 Theory Versus Performance Practice

to for seven centuries prior to the reforms of fourteenth century theorists Quðb al-Dån and Abd al Qàdir

al-Maràghå (d 1435).46

The twenty-four-tone equally-tempered division of the octave is based on the Western system of

twelve-tone equal temperament – the division of the octave into twelve equal intervals, technically

referred to as tempered half-tones, while the division of the octave into twenty-four equal intervals,

technically referred to as tempered quarter-tones; the frequency ratio of each quarter-tone therefore

mathematically representing the twenty-fourth root of two, or in different terms, the distance between

any two tones representing twenty-four times the logarithm on the base of two of the frequency ratio.47

The ratio of the equally-tempered quarter-tone may be expressed in mathematical terms as

1: 24 2 =1:1.029302237, or the fraction approximations: 527/512 (17×31/2ù) and 35/34 (5×7/2×17).48

The unit of a cent being the method to further subdivide the semitone, and equal to the twelve-

hundredth root of two; with 12 semitones per octave, 1200 cents per octave, and hence, 100 cents per

semitone. The ratio of the cent may be expressed in mathematical terms as 1: 1200 2 =1:1.000577789, or

approximately 1731/1730. The calculation of cents (a system of measurement devised by Alexander J.

Ellis [1814-90]) is obtained via the formula: 1200ïlog2Ïlogf=cents.49

The methodology involved in the construction of a twenty-four equally-tempered quarter-tone

scale requires the establishment of twenty-four degrees via the equation f= 24 2 ; the equation producing

the figure 1.029302237 (the frequency ratio of one equally-tempered quarter-tone), which when

multiplied by the powers of 2, 3, 4, etc., generates the ratios for the remainder of the scale degrees.50 It

should be noted that the multiplication by the power of 18 ( 1824 ]2[ ) generates the frequency ratio of

46 “In view of the fact that the division into ‘quarter-tones’ gives two of the ten ratios of 11 with a falsity of only 0.6

cent (12/11, or 150.637 cents; and 11/6, or 1049.363 cents), two with a falsity of only 1.3 cents (11/8, or 551.318

cents; and 16/11, or 648.682 cents), and two with a falsity of 2.6 cents (11/9, or 347.408 cents; and 18/11, or 852.592

cents), the probabilities are that Måshàqàh’s claims for Arabic folk melodies are valid. But the fate of the other ratios of 11,

and of the 5 and 7 identities (the 5th and 7th partials of the harmonic series) in ‘quarter-tones’ is another story.” For a

further discussion see, Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 426.

47 Olson, Music, Physics and Engineering 46-47.

48 Alain Daniélou, Tableau Comparatif des Intervalles Musicaux (Pondichéry, India: Institut Français d’Indologie,

1958) 14.

49 Hermann L. F. Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music, 2nd ed.

(New York: Dover Publications, 1954) 431.

50 “‘Quarter-tones’ are obtained by a simple duplication of the twelve-tone tempered scale a ‘quarter-tone’ higher.

The original, since it is the usual scale, has good 3 identities (the 3rd partial of the harmonic series) and two fairly good 9

identities (the 9th partial of the harmonic series), but poor 5 identities (13.7 and 15.6 false). The doubling, a ‘quarter-

tone’ higher, gives six of the ten 11 identities (the 11th partial of the harmonic series) almost perfectly, but fails to

represent the other four at all; it comes closer to the 7 identities than the original, but not close enough, since they show

a maximum falsity of 18.8 cents; it improves the 5s not at all.” For a further discussion see, Partch, Genesis of a Music: An

Account of a Creative Work, its Roots and its Fulfilments 428.

Theory Versus Performance Practice 17

A=440Hz (1.68179283), or equal major sixth ( 34 ]2[ , approximately 37/22, or 900.000 cents),51 while

440ï1.68179283 presents the relative frequency of middle C (261.6255654Hz), or 1/1. Once data is

available for all frequency ratios, it simply becomes a matter of multiplying every ratio by 261.6255654Hz

(1/1) in order to obtain the frequency values in hertz (Hz) of every scale degree. The mathematical

formula for cents is then utilized to generate the cent values of every scale degree, although unnecessary

in equal temperament because of the obvious outcomes.

The Pythagorean Division of the Octave

‘Pythagorean’ intonation, and the generation of a scale based on a series of ‘pure’ fifths (3/2s), is in

reference to Greek philosopher Pythagoras (570-504 B.C.), who according to Llewelyn Southworth

Lloyd, and Hugh Boyle is “usually credited with the discovery that a vibrating string, stopped at two thirds

or one-half of its length, sounds the fifth (3/2) or octave (2/1) of the note it produces when vibrating

freely (1/1).”52 The system of intonation “prevailed in China and Arabia almost to the exclusion of anything

else,” notes Partch, and “was for many centuries the only system in ecclesiastical Roman and Byzantine

music.”53 According to Fritz A. Kuttner, the Pythagorean conceptual basis for the construction of a scale

of twelve semitones via the generation of twelve consecutive fifths was recognized in China “several

centuries before Pythagoras, probably in the ninth century B.C., or even earlier.” Kuttner explains:

“The twelve semitones (lüs) received pitch names which are mentioned for the first time in the Kuo Yü, a

work usually dated towards the end of the Chou Dynasty (c. 1030-722 B.C). However, the Kuo Yü is

believed to contain a good deal of much ‘earlier material from ancient written sources,’ so the origin of the

pitch names must be assigned to about 900 B.C. or earlier.”54

In order to construct a Pythagorean scale with the necessary intervals to facilitate the general modulations

of Western tonal music one must produce twenty-seven distinct pitches.55 Fifteen intervals are generated

by an ascending series of fifths, or the pitches C, G, D, A, E, B, F!, C!, G!, D!, A!, E!, B!, F#, C#, and G#;

while the other eleven intervals, by a descending series, or C, F, B", E", A", D", G", C", F", B$, E$, and A$.

The mathematical procedure thus begins with the generation of the first ascending fifth, G; which has a

ratio of 3/2 (the just and Pythagorean perfect fifth, measuring 701.955 cents) in relation to the

51 Daniélou, Tableau Comparatif des Intervalles Musicaux 79.

52 Llewelyn Southworth Lloyd, and Hugh Boyle, Intervals, Scales and Temperaments: An Introduction to the Study

of Musical Intonation (London: McDonald and Jane’s, 1978) 2.

53 Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 399.

54 Fritz A. Kuttner, “A Musicological Interpretation of the Twelve Lüs in China’s Traditional Tone System,”

Ethnomusicology 9.1 (Jan., 1965): 22.

55 Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 433.

18 Theory Versus Performance Practice

Tabl

e 3.

The

twen

ty-fo

ur-to

ne e

qua

lly-te

mp

ered

div

ision

of t

he o

ctav

e Re

lativ

e Pi

tch:

A4=

440H

z / C

4 (m

idd

le C

)=26

1.62

5565

4Hz

DEG

REE

NUM

BER

NO

TE

INTE

RVA

L FA

CTO

R RA

TIO

(D

ECIM

AL)

FR

EQUE

NCY

(H

ERTZ

) CE

NTS

TU

NIN

G

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÑÓ

ÑÔ

ÑÕ

ÑÖ

Ñ×

ÑØ

ÑÙ

ÒÐ

ÒÑ

ÒÒ

ÒÓ

ÒÔ ÐÑ

CCË

/ Dì

CÚ

/ DÝ

CÍ

/ Dí

D

DË /

Eì

DÚ /

EÝ

DÍ /

Eí

E EË

/ Fí

F

FË / G

ì FÚ

/ GÝ

FÍ

/ Gí

G

GË /

Aì

GÚ /

AÝ

GÍ /

Aí

A AË

/ Bì

AÚ

/ BÝ

AÍ

/ Bí

B

BË /

Cí

C

uniso

n eq

ual q

uarte

r-ton

e eq

ual s

emito

ne, o

r min

or s

econ

d

equa

l thr

ee-q

uarte

r-ton

e eq

ual t

one

five

equa

l qua

rter-

tone

s eq

ual a

ugm

ente

d s

econ

d, o

r min

or th

ird

seve

n eq

ual q

uarte

r-ton

es

equa

l maj

or th

ird

nine

eq

ual q

uarte

r-ton

es

equa

l per

fect

four

th

elev

en e

qua

l qua

rter-t

ones

eq

ual t

riton

e, a

ugm

ente

d fo

urth

, or d

imin

ished

fifth

thirt

een

equa

l qua

rter-t

ones

eq

ual p

erfe

ct fi

fth

fifte

en e

qua

l qua

rter-t

ones

eq

ual a

ugm

ente

d fi

fth, o

r min

or s

ixth

se

vent

een

equa

l qua

rter-t

ones

eq

ual m

ajor

six

th

nine

teen

eq

ual q

uarte

r-to

nes

equa

l aug

men

ted

six

th, o

r min

or s

even

th

twen

ty-o

ne e

qua

l qua

rter-t

ones

eq

ual m

ajor

sev

enth

tw

enty

-thre

e eq

ual q

uarte

r-ton

es

oc

tave

1/1

242

12

2

82

6

2 5

24)

2(

4

2 7

24)

2(

3

2 3

8)

2(

5

12)

2(

11

24)

2(

2

2 13

24)

2(

7

12)

2(

5

8)

2(

2

3)

2(

17

24)

2(

3

4)

2(

19

24)

2(

5

6)

2(

7

8)

2(

11

12)

2(

23

24)

2(

2/1

1.00

0000

1.

0293

02

1.05

9463

1.

0905

08

1.12

2462

1.

1553

53

1.18

9207

1.

2240

54

1.25

9921

1.

2968

40

1.33

4840

1.

3739

54

1.41

4214

1.

4556

53

1.49

8307

1.

5422

11

1.58

7401

1.

6339

15

1.68

1793

1.

7310

73

1.78

1797

1.

8340

08

1.88

7749

1.

9430

64

2.

0000

00

261.

626

269.

292

277.

183

285.

305

293.

665

302.

270

311.

127

320.

244

329.

628

339.

286

349.

228

359.

461

369.

994

380.

836

391.

995

403.

482

415.

305

427.

474

440.

000

452.

893

466.

164

479.

823

493.

883

508.

355

52

3.25

1

0.00

0 50

.000

10

0.00

0 15

0.00

0 20

0.00

0 25

0.00

0 30

0.00

0 35

0.00

0 40

0.00

0 45

0.00

0 50

0.00

0 55

0.00

0 60

0.00

0 65

0.00

0 70

0.00

0 75

0.00

0 80

0.00

0 85

0.00

0 90

0.00

0 95

0.00

0 10

00.0

00

1050

.000

11

00.0

00

1150

.000

1200

.000

+00

+

50

+00

+

50

+00

+

50

+00

+

50

+00

+

50

+00

+

50

+00

+

50

+00

+

50

+00

+

50

+00

+

50

+00

+

50

+00

+

50

+

00

Theory Versus Performance Practice 19

fundamental C (1/1). 2/1 (measuring 1200.000 cents) is consequently subtracted from intervals

exceeding the 2/1 ratio in order to bring the interval within the octave. These procedures are repeated

consecutively, producing the following series of fifteen ascending fifths:

i. G (just and Pythagorean perfect fifth) generated by ratio 3/2

ii. D (just major tone) by the equation (3/2×3/2)/2=9/8

iii. A (Pythagorean major sixth) by 9/8×3/2=27/16

iv. E (Pythagorean major third, or ditone) by (27/16×3/2)/2=81/64

v. B (Pythagorean major seventh) by 81/64×3/2=243/128

vi. F! (Pythagorean tritone, or augmented fourth) by (243/128×3/2)/2=729/512

vii. C! (Pythagorean apotome) by (729/512×3/2)/2=2187/2048

viii. G! (augmented fifth) by 2187/2048×3/2=6561/4096

ix. D! (augmented second) by (6561/4096×3/2)/2=19683/16384

x. A! (augmented sixth) by 19683/16384×3/2=59049/32768

xi. E! (augmented third) by (59049/32768×3/2)/2=177147/131072

xii. B! (Pythagorean comma) by (177147/131072×3/2)/2=531441/524288

xiii. F# (double augmented fourth) by 531441/524288×3/2=1594323/1048576

xiv. C# (double augmented octave) by (1594323/1048576×3/2)/2=4782969/4194304

xv. G# (double augmented fifth) by 4782969/4194304×3/2=14348907/8388608

The process is then reversed, with the generation of the first descending fifth (F) by the mathematical

equation of 4/3 (the just and Pythagorean perfect fourth, measuring 498.045 cents). These procedures

are repeated consecutively, producing the following concluding series of eleven descending fifths:

i. F (just and Pythagorean perfect fourth) generated by the ratio 4/3, or 2/3×2=4/3

ii. B" (just minor seventh) by the equation (4/3×2/3)×2=16/9

iii. E" (Pythagorean minor third, or trihemitone) by 16/9×2/3=32/27

iv. A" (Pythagorean minor sixth) by (32/27×2/3)×2=128/81

v. D" (Pythagorean limma) by 128/81×2/3=256/243

vi. G" (diminished fifth) by (256/243×2/3)×2=1024/729

vii. C" (diminished octave) by (1024/729×2/3)×2=4096/2187

viii. F" (diminished fourth) by 4096/2187×2/3=8192/6561

ix. B$ (diminished sixth) by (8192/6561×2/3)×2=32768/19683

x. E$ (diminished third) by 32768/19683×2/3=65536/59049

xi. A$ (diminished sixth) by (65536/59049×2/3)×2=262144/177147

20 Theory Versus Performance Practice

Tabl

e 4.

The

Pyt

hago

rean

div

ision

of t

he o

ctav

e Re

lativ

e Pi

tch:

A4=

440H

z / C

4 (m

iddl

e C)

=26

1.62

5565

4Hz

DE

GRE

E N

UMBE

R N

OTE

IN

TERV

AL

RATI

O

(FRA

CTIO

N)

RATI

O

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

TUN

ING

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÑÓ

ÑÔ

ÑÕ

ÑÖ

Ñ×

ÑØ

ÑÙ

ÒÐ

ÒÑ

ÒÒ

ÒÓ

ÒÔ

ÒÕ

ÒÖ

Ò× ÐÑ

C BÚ

DÝ

CÚ

EÞ

D CÛ

EÝ

DÚ

FÝ E F EÚ

GÝ

FÚ

AÞ

G FÛ

AÝ

GÚ

BÞ

A GÛ

BÝ

AÚ

CÝ B C

uniso

n Py

thag

orea

n co

mm

a (A

) XI

I

Pyth

agor

ean

limm

a, o

r dia

toni

c se

mito

ne (D

) V

Pyth

agor

ean

apot

ome,

or c

hrom

atic

sem

itone

(A)

VII

Pyth

agor

ean

dim

inish

ed th

ird (D

) X

just

maj

or to

ne (A

) II (

9th

harm

onic

) Py

thag

orea

n d

oubl

e au

gmen

ted

oct

ave

(A)

XIV

Pyth

agor

ean

min

or th

ird, o

r trih

emito

ne (D

) III

Pyth

agor

ean

augm

ente

d s

econ

d (A

) IX

Pyth

agor

ean

dim

inish

ed fo

urth

(D)

VIII

Pyth

agor

ean

maj

or th

ird, o

r dito

ne (A

) IV

(81s

t har

mon

ic)

just

and

Pyt

hago

rean

per

fect

four

th (D

) I

Pyth

agor

ean

augm

ente

d th

ird (A

) XI

Pyth

agor

ean

dim

inish

ed fi

fth (D

) V

I Py

thag

orea

n tri

tone

, or a

ugm

ente

d fo

urth

(A)

VI

Pyth

agor

ean

dim

inish

ed s

ixth

(D)

XI

just

and

Pyt

hago

rean

per

fect

fifth

(A)

I (3r

d h

arm

onic

) Py

thag

orea

n d

oubl

e au

gmen

ted

four

th (A

) XI

II

Pyth

agor

ean

min

or s

ixth

(D)

IV

Pyth

agor

ean

augm

ente

d fi

fth (A

) V

III

Pyth

agor

ean

dim

inish

ed s

even

th (D

) IX

Pyth

agor

ean

maj

or s

ixth

(A)

III (2

7th

harm

onic

) Py

thag

orea

n d

oubl

e au

gmen

ted

fifth

(A)

XV

Pyth

agor

ean

min

or s

even

th (D

) II

Pyth

agor

ean

augm

ente

d s

ixth

(A)

X

Pyth

agor

ean

dim

inish

ed o

ctav

e (D

) V

II

Pyth

agor

ean

maj

or s

even

th (A

) V

oc

tave

1/1

5314

41/5

2428

8 25

6/24

3 21

87/2

048

6553

6/59

049

9/8

4782

969/

4194

304

32/2

7 19

683/

1638

4 81

92/6

561

81/6

4 4/

3 17

7147

/131

072

1024

/729

72

9/51

2 26

2144

/177

147

3/2

1594

323/

1048

576

128/

81

6561

/409

6 32

768/

1968

3 27

/16

1434

8907

/838

8608

16

/9

5904

9/32

768

4096

/218

7 24

3/12

8

2/1

1.00

0000

1.

0136

43

1.05

3498

1.

0678

71

1.10

9858

1.

1250

00

1.14

0349

1.

1851

85

1.20

1355

1.

2485

90

1.26

5625

1.

3333

33

1.35

1524

1.

4046

64

1.42

3828

1.

4798

11

1.50

0000

1.

5204

65

1.58

0247

1.

6018

07

1.66

4787

1.

6875

00

1.71

0523

1.

7777

78

1.80

2032

1.

8728

85

1.89

8438

2.00

0000

261.

626

265.

195

275.

622

279.

382

290.

367

294.

329

298.

344

310.

075

314.

305

326.

663

331.

120

348.

834

353.

593

367.

496

372.

510

387.

156

392.

438

397.

792

413.

433

419.

074

435.

551

441.

493

447.

517

465.

112

471.

458

489.

995

496.

680

52

3.25

1

0.00

0 23

.460

90

.225

11

3.68

5 18

0.45

0 20

3.91

0 22

7.37

0 29

4.13

5 31

7.59

5 38

4.36

0 40

7.82

0 49

8.04

5 52

1.50

5 58

8.27

0 61

1.73

0 67

8.49

5 70

1.95

5 72

5.41

5 79

2.18

0 81

5.64

0 88

2.40

5 90

5.86

5 92

9.32

5 99

6.09

0 10

19.5

50

1086

.315

11

09.7

75

12

00.0

00

+00

+

23

ß10

+

14

ß20

+

04

+27

ß

06

+18

ß

16

+08

ß

02

+22

ß

12

+12

ß

22

+02

+

25

ß08

+

16

ß18

+

06

+29

ß

04

+20

ß

14

+10

+00

Theory Versus Performance Practice 21

Alain Daniélou’s Scale of Fifths An alternative method for deriving a Pythagorean scale is Alain Daniélou’s ‘scale of fifths’ (the cyclic

division of the octave), as presented by Daniélou in Music and the Power of Sound: The Influence of

Tuning and Interval on Consciousness (1995) – the current edition of his 1943 monumental work

Introduction to the Study of Musical Scales. The theoretical basis for the ‘cyclic division of the octave’ is

the ascending scale of fifths.56

The theory states that when a length of bamboo (called ‘81 parts’ for symbolic reasons) is

reduced by a third (2/3), it will produce the perfect fifth (3/2, or 54 parts). If the process is continued

by increasing the length of bamboo by a third (4/3), and then alternated between the two ratios, it will

result in the following five-note series of pipe lengths and frequency ratios: 81 (1/1), 54 (3/2), 72 (9/8),

48 (27/16), and 64 (81/64). The juxtaposition of two sets of six lü, or pitch pipes a Pythagorean major

half-tone apart generates a Pythagorean twelve-tone scale, or twelve lü (alternatively achieved via the

simple expansion of the series of consecutive fifths initially forming a heptatonic scale). Han Dynasty

historian Ssu-ma Chi’en (145-86 B.C.) attributes the “mathematical formula for the pentatonic scale to Ling

Lun, minister or court musician under Emperor Huang-Ti, of the twenty-seventh century B.C.”57 Chinese

theorist King Fâng (c. 45 B.C.) followed the series up to the sixtieth sound (equal to fifty-nine consecutive

fifths), although mainly for symbolic reasons. The fifty-third fifth is the cyclic octave (3õó/2øô, or 3.615

cents), but the series being of an infinitive nature, it is musically impractical to follow it beyond this given

point, so it is rounded off to the ratio of 2/1. Fang based his system of the sixty lü on the “analogy of the

eight gua or mystical trigrams of the I ching, which united two by two, form sixty-four distinct

combinations.” Sixty-four hexagrams allow for the “representation of all aspects of existence.” The

twelve original lü, multiplied by five (the number of elements), also equal to sixty lü. Daniélou makes the

following statement with regards to the infinite nature of the series:

“However, after the 52nd fifth, the octave is filled up and the 53rd fifth (note 54) comes out of the octave

and inserts itself between the octave C (2/1) and the twelfth fifth C¢ (531441/524288), thus forming,

above C (1/1), a small interval of 0.84 savarts (3.349 cents [1 savart = 3.986313725 cents, or 1 cent =

56 “Alain Daniélou (1907-94), the founder of the International Institute for Comparative Musicology in Berlin,

elucidated for tens of thousands of readers the meanings of the arts and religious traditions of both East and West. He

was an accomplished player on the vånà and taught in the music department at the University of Benares. His numerous

books, the product of a career spanning six decades, include The Myths and Gods of India; Gods of Love and Ecstasy;

While the Gods Play; Virtue, Success, Pleasure, and Liberation; The Phallus; Mastering the Secrets of Matter and the

Universe; and The Complete Kàma Sñtra.” For a further discussion, see Daniélou, Music and the Power of Sound: The

Influence of Tuning and Interval on Consciousness 35.

57 Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 362.

22 Theory Versus Performance Practice

0.250858329 savarts]).58 Thus begins a new cycle, which in its turn, with a period of 53 fifths, will divide

the octave into small intervals of 0.84 savarts. The next cycle will appear during the seventh series of 53,

when the 359th fifth (note 360) comes out of the octave and forms, with C (1/1), an interval of 0.47

savarts (1.874 cents). The next cycle is of 666 notes, with a basic interval of 0.035 savarts (0.140 cents).

Then comes a cycle of 25,524 notes with a basic interval of 0.0021174 savarts (0.008 cents). This cycle is

very near to that of the precession of the equinoxes, or the Pythagorean great year, which is of 25,920

solar years. The small difference between the twelfth fifth and octave, similar to that of the lunar and the

solar year, leaves the door open for further cycles.”59

Table 5. Some correspondences of the first twelve lü NOTE NAME OF

LÜ

SEVEN

BEGINNINGS

FUNDAMENTAL

FOR SACRIFICES

ANNUAL

CYCLE

GENDER MOON

C

D¸

D

EÝ

E¢

F¢

F¥

G

AÝ

A¢

BÝ

B¢

huang zhong (I)

da lü (VIII)

tai cu (male form

of jia zhong (III)

jia zhong or

yuan zhong (X)

gu xian (V)

zhong lü (XIII)

rui bin (VII)

lin zhong or

han zhong (II)

yi tze (IX)

nan lü (IV)

wu yi (XI)

ying zhong (VI)

heaven

man

spring

summer

earth

autumn

winter

to the yellow emperor,

to heaven

to the white emperor,

to ancestors

to the green emperor,

to the first agriculturists

to the red emperor,

to earth

to the black emperor, to

the empress, to the moon

winter

solstice

spring

equinox

summer

solstice

autumn

equinox

masculine

feminine

masculine

feminine

masculine

feminine

masculine

feminine

masculine

feminine

masculine

feminine

11th (December)

12th (January)

1st (February)

2nd (March)

3rd (April)

4th (May)

5th (June)

6th (July)

7th (August)

8th (September)

9th (October)

10th (November)

58 “An alternative method (to cents) of interval measurement, but one used far less frequently, is the savart. Named

after the French scientist Félix Savart (1791-1841), this method assigns a total of 25 savarts to each semitone (or

25.08583297), one savart being approximately equal to four cents.” For a further discussion, see Read, 20th-Century

Microtonal Notation 7.

59 Daniélou, Music and the Power of Sound: The Influence of Tuning and Interval on Consciousness 20-50.

Theory Versus Performance Practice 23

Tabl

e 6.

Ala

in D

anié

lou’

s sc

ale

of fi

fths

(the

cyc

lic d

ivisi

on o

f the

oct

ave)

Re

lativ

e Pi

tch:

A4=

440H

z / C

4 (m

iddl

e C)

=26

1.62

5565

4Hz

DE

GRE

E N

UMBE

R N

OTE

IN

TERV

AL

RATI

O

(FRA

CTIO

N)

RATI

O

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

TUN

ING

ÐÑ ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ ÑÑ

ÑÒ

ÑÓ

ÑÔ

ÑÕ

ÑÖ

Ñ×

ÑØ

ÑÙ ÒÐ

ÒÑ

ÒÒ

ÒÓ

ÒÔ

ÒÕ

ÒÖ

C C¢

C£

CÚ

D¹

D¸

DÝ

D³

D²

D D¢

D£

DÚ

E¸

EÝ

E³

E² E E¢ E£

F³

F² F F¢

F£

FÚ

uniso

n cy

clic

oct

ave

(A)

LIII

Pyth

agor

ean

com

ma

(A)

XII

grea

t die

sis (A

) XX

IV

cycl

ic g

rave

or s

mal

l chr

omat

ic s

emito

ne, o

r min

or h

alf-t

one

(A)

XXXV

I

cycl

ic P

ytha

gore

an li

mm

a (A

) XL

VIII

Pyth

agor

ean

apot

ome,

dia

toni

c se

mito

ne, o

r maj

or h

alf-t

one

(A)

VII

cycl

ic g

reat

lim

ma,

acu

te o

r lar

ge h

alf-t

one

(A)

XIX

cycl

ic g

rave

or s

mal

l ton

e (A

) XX

XI

cycl

ic m

inor

tone

(A)

XLIII

just

maj

or to

ne (A

) II (

9th

harm

onic

) 55

th c

yclic

fifth

(A)

LV

acut

e or

larg

e to

ne (A

) XI

V

cycl

ic s

uper

maj

or s

econ

d (A

) XX

VI

cycl

ic a

ugm

ente

d se

cond

(A)

XXXV

III

cycl

ic P

ytha

gore

an m

inor

third

, or t

rihem

itone

(A)

L

cycl

ic m

inor

third

(A)

IX

cycl

ic n

eutra

l thi

rd (A

) XX

I

cycl

ic g

rave

or s

mal

l maj

or th

ird (A

) XX

XIII

cycl

ic m

ajor

third

(A)

XLV

Pyth

agor

ean

maj

or th

ird, o

r dito

ne (A

) IV

(81

st h

arm

onic

) 57

th c

yclic

fifth

(A)

LVII

cycl

ic a

cute

or l

arge

maj

or th

ird (A

) XV

I

cycl

ic s

ubfo

urth

(A)

XXV

III

cycl

ic g

rave

or s

mal

l fou

rth (A

) XL

cycl

ic p

erfe

ct fo

urth

(A)

LII

cycl

ic a

cute

or l

arge

four

th (A

) XI

cycl

ic s

uper

four

th (A

) XX

III

cycl

ic g

rave

or s

mal

l aug

men

ted

four

th (A

) XX

XV

1 ñ/1

ñ3õ

ó/2øô

3ñò/2

ñù 3ò

ô/2óø

3óö/2

õ÷ 3ô

ø/2÷ö

3÷/2

ññ

3 ñù/2

óð 3 ó

ñ/2ôù

3ôó/2

öø 3ò

/2ó

3õõ/2

ø÷ 3ñ

ô/2òò

3òö/2

ôñ 3ó

ø/2öð

3õð/2

÷ù 3ù

/2ñô

3ò

ñ/2óó

3óó/2

õò 3ô

õ/2÷ñ

3ô/2

ö 3õ

÷/2ùð

3ñö/2

òõ 3ò

ø/2ôô

3ôð/2

öó 3õ

ò/2øò

3ññ/2

ñ÷ 3ò

ó/2óö

3óõ/2

õõ

1.00

0000

1.

0020

90

1.01

3634

1.

0274

73

1.04

1491

1.

0557

00

1.06

7871

1.

0824

40

1.09

7208

1.

1121

78

1.12

5000

1.

1273

52

1.14

0349

1.

1559

07

1.17

1677

1.

1876

63

1.20

1355

1.

2177

45

1.23

4359

1.

2512

00

1.26

5625

1.

2682

71

1.28

2892

1.

3003

95

1.31

8137

1.

3361

20

1.35

1524

1.

3699

64

1.38

8654

261.

626

262.

172

265.

195

268.

813

272.

481

276.

198

279.

382

283.

194

287.

058

290.

974

294.

329

294.

944

298.

344

302.

415

306.

541

310.

723

314.

305

318.

593

322.

940

327.

346

331.

120

331.

812

335.

637

340.

217

344.

858

349.

563

353.

593

358.

417

363.

307

0.00

0 3.

615

23.4

60

46.9

20

70.3

80

93.8

40

113.

685

137.

145

160.

605

184.

065

203.

910

207.

525

227.

370

250.

830

274.

290

297.

750

317.

595

341.

055

364.

515

387.

975

407.

820

411.

435

431.

280

454.

740

478.

200

501.

660

521.

505

544.

965

568.

425

+00

+

04

+23

+

47

ß30

ß

06

+14

+

37

ß39

ß

16

+04

+

08

+27

+

51

ß26

ß

02

+18

ß

59

ß35

ß

12

+08

+

11

+31

ß

45

ß22

+

02

+22

+

45

ß32

24 Theory Versus Performance Practice

DEG

REE

NUM

BER

NO

TE

INTE

RVA

L RA

TIO

(F

RACT

ION

) RA

TIO

(D

ECIM

AL)

FR

EQUE

NCY

(H

ERTZ

) CE

NTS

TU

NIN

G

Ò×

ÒØ ÒÙ

ÓÐ

ÓÑ

ÓÒ ÓÓ

ÓÔ

ÓÕ

ÓÖ

Ó×

ÓØ

ÓÙ

ÔÐ

ÔÑ ÔÒ

ÔÓ

ÔÔ

ÔÕ

ÔÖ

Ô×

ÔØ

ÔÙ

ÕÐ ÕÑ

ÕÒ

ÕÓ ÐÑ

F ¤ F¥ GÝ

G³

G²

G G¢

G£

GÚ

A¸

AÝ

A³

A²

A A¢ A£

AÚ

B¹

B¸

BÝ

B³

B² B B¢ B£

C³

C² C

cycl

ic tr

itone

, or a

ugm

ente

d fo

urth

(A)

XLV

II

Pyth

agor

ean

trito

ne, o

r aug

men

ted

four

th (A

) V

I

59th

cyc

lic fi

fth (A

) LIX

cycl

ic a

cute

or l

arge

dim

inish

ed fi

fth (A

) XV

III

cycl

ic s

ubfif

th (A

) XX

X

cycl

ic g

rave

or s

mal

l fift

h (A

) XL

II

just

and

Pyt

hago

rean

per

fect

fifth

(A)

I (3r

d h

arm

onic

) 54

th c

yclic

fifth

(A)

LIV

cycl

ic a

cute

or l

arge

fifth

(A)

XIII

cycl

ic s

uper

fifth

(A)

XXV

cycl

ic a

ugm

ente

d fi

fth (A

) XX

XVII

cycl

ic m

inor

six

th (A

) XL

IX

Pyth

agor

ean

min

or s

ixth

(A)

VIII

cycl

ic n

eutra

l six

th (A

) XX

cycl

ic g

rave

or s

mal

l maj

or s

ixth

(A)

XXXI

I

cycl

ic m

ajor

six

th (A

) XL

IV

Pyth

agor

ean

maj

or s

ixth

(A)

III (2

7th

harm

onic

) 56

th c

yclic

fifth

(A)

LVI

cycl

ic a

cute

or l

arge

maj

or s

ixth

(A)

XV

cycl

ic a

ugm

ente

d six

th (A

) XX

VII

cycl

ic g

rave

or s

mal

l min

or s

even

th (A

) XX

XIX

cycl

ic m

inor

sev

enth

(A)

LI

cycl

ic a

cute

or l

arge

min

or s

even

th (A

) X

cycl

ic n

eutra

l sev

enth

(A)

XXII

cycl

ic g

rave

or s

mal

l maj

or s

even

th (A

) XX

XIV

cycl

ic d

iato

nic

maj

or s

even

th (A

) XL

VI

Pyth

agor

ean

maj

or s

even

th (A

) V

58th

cyc

lic fi

fth (A

) LV

III

cycl

ic a

cute

or l

arge

maj

or s

even

th (A

) XV

II

cycl

ic s

uboc

tave

(A)

XXIX

cycl

ic g

rave

or s

mal

l oct

ave

(A)

XLI

oc

tave

3 ô÷/2

÷ô 3 ö

/2ù

3õù/2

ùó 3ñ

ø/2òø

3óð/2

ô÷ 3ô

ò/2öö

3ñ/2

ñ 3õ

ô/2øõ

3ñó/2

òð 3ò

õ/2óù

3ó÷/2

õø 3ô

ù/2÷÷

3ø/2

ñò

3òð/2

óñ 3ó

ò/2õð

3ôô/2

öù 3ó

/2ô

3õö/2

øø 3ñ

õ/2òó

3ò÷/2

ôò 3ó

ù/2öñ

3õñ/2

øð 3ñ

ð/2ñõ

3òò/2

óô 3ó

ô/2õó

3ôö/2

÷ò 3õ

/2÷

3õø/2

ùñ 3ñ

÷/2òö

3òù/2

ôõ 3ô

ñ/2öô

2ñ/1

ñ

1.40

7600

1.

4238

28

1.42

6804

1.

4432

54

1.46

2944

1.

4829

04

1.50

0000

1.

5031

35

1.52

0465

1.

5412

09

1.56

2236

1.

5835

50

1.60

1807

1.

6236

61

1.64

5813

1.

6682

67

1.68

7500

1.

6910

27

1.71

0523

1.

7338

60

1.75

7516

1.

7814

94

1.80

2032

1.

8266

18

1.85

1539

1.

8768

00

1.89

8480

1.

9024

06

1.92

4338

1.

9505

93

1.97

7205

2.00

0000

368.

264

372.

510

373.

289

377.

592

382.

744

387.

966

392.

438

393.

259

397.

792

403.

220

408.

721

414.

297

419.

074

424.

791

430.

587

436.

461

441.

493

442.

416

447.

517

453.

622

459.

811

466.

084

471.

458

477.

890

484.

410

491.

019

496.

680

497.

718

503.

456

510.

325

517.

287

52

3.25

1

591.

885

611.

730

615.

345

635.

190

658.

650

682.

110

701.

955

705.

570

725.

415

748.

875

772.

335

795.

795

815.

640

839.

100

862.

560

886.

020

905.

865

909.

480

929.

325

952.

785

976.

245

999.

705

1019

.550

10

43.0

10

1066

.470

10

89.9

30

1109

.775

11

13.3

90

1133

.235

11

56.6

95

1180

.155

1200

.000

ß08

+

12

+15

+

35

ß41

ß

18

+02

+

06

+25

+

49

ß28

ß

04

+16

ß

61

ß37

ß

14

+06

+

09

+29

ß

47

ß24

+

00

+20

ß

57

ß34

ß

10

+10

+

13

+33

ß

43

ß20

+00

Theory Versus Performance Practice 25

The Twenty-Two Note Division of the Octave The twenty-two note division of the octave was conceptualized by Persian scholar and physicist Barkešli

in the 1940s, following his personal analysis of contemporary tradition, and is based on the Pythagorean

cycle of fifths, as well as the writings of medieval theorists Éafå al-Dån Urmawå and Abu Nasr Fàràbå. After

scientific analysis of intervals in recordings of vocal music by five respected traditional musicians, Barkešli

concluded that the intervallic structure of the whole-tone and semitone in Persian music were reasonably

fixed, and closely resembled the first tetrachord of the Pythagorean diatonic model with the figures:

206+204+89=499 cents. In exact Pythagorean terms this is expressed as 203.910 (two limmas and a

comma) + 203.910 + 90.225 (limma) = 498.045, or 9/8×9/8×256/243=4/3. In other tetrachords he

also identified a “120-cent interval as the characteristic interval of Persian music”, which is close to a

Pythagorean apotome (2187/2048, or 113.685 cents), as well another less popular interval measuring

181 cents similar to a Pythagorean diminished third (65536/59049, or 180.450 cents).

Éafå al-Dån’s seventeen-note gamut, with its whole-tone pattern of L, L + L, and L + L + C is

therefore extended by Barkešli into one exhibiting the following intervallic pattern: L, L + C, L + L, and L

+ L + C; or in more elaborate terms as: 256/243, 256/243×531441/524288=2187/2048 (Pythagorean

apotome), 256/243×256/243=65536/59049 (Pythagorean diminished third), and L + L + C = 9/8, or

just major tone.

Table 7. Mehdi Barkešli’s division of the whole-tone

PATTERN L L + C L + L L + L + C

INTERVAL Pythagorean

limma

Pythagorean

apotome

Pythagorean

diminished third

just

major tone

RATIO 256/243 2187/2048 65536/59049 9/8

CENTS 90.225 113.685 180.450 203.910

It may therefore be summarized that the first eight intervals of the twenty-two-note division of the octave

are derived from the construction of a Pythagorean C mixolydian scale (the pattern of a ratio of 3/2, or

ascending fifths generating the pitches G, D, A, E, and B, while the ratio 4/3, or descending fifths

generating the pitches F and B"). The basic design essentially allows for the formation of both major (C,

D, E, and F) and minor (G, A, B", and C) tetrachords. Another fourteen intervals are then derived from the

construction of a whole-tone based on the four-microtone model of L, L + L, and L + L + C. The

twenty-two-note division of the octave is characterized by the following intervallic pattern: L, C, DDT, C –

L, C, DDT, C – L + L, C, DDT, C – L, C, DDT, C – L, C, DDT, C – L. ‘DDT’, or Pythagorean double diminished

26 Theory Versus Performance Practice

third (134217728/129140163, or 66.765 cents) represents the interval between the Pythagorean

diminished third (65536/59049) and apotome (2187/2048).

Ex. 2. Subdivision of the tetrachord and the 22-note division of the octave (Mehdi Barkešli, 20th century)

The Theory of Flexible Intervals The third proposition for a theory on intervals and scales of Persian music – the theory of flexible intervals,

or of the five primary intervals of performance practice – advocated by Farhat in opposition to both

twenty-four-tone equally-tempered, and twenty-two-note Pythagorean scales of Vaziri and Barkešli.

Farhat offers the following critical assessment:

“Both theories suffer equally from a tendency to accommodate certain Western concepts. Each theory, by

suggesting very exact intervals, remains oblivious to the fluidity and flexibility of Persian intervals. Vaziri did

not take account of this instability, as his apparent objective was to make Persian music adhere to a process

of equal temperament so that it can be harmonized. Barkešli, on the other hand, was not interested in the

Westernisation of Persian music but was committed to prove that today’s music is still rooted in the

medieval system. He has taken the exactness of the medieval theory very seriously – as have many others –

and has proposed a system vested with even greater precision.”

Utilizing a stroboconn and a melograph as measurement devices, and fretted instruments (tàrs and setàrs)

as the sources, Farhat identifies five primary intervals, with semitones and whole-tones representative of

relatively fixed intervals, while the rest, of a flexible nature. His classification of Persian intervals include: (1)

the semitone or minor second, measuring approximately 90 cents; (2) the small neutral tone, measuring

between 125 and 145 cents, or the average of 135 cents; (3) the large neutral tone, measuring between

150 and 170 cents, or the average of 160 cents; (4) the whole-tone or major second, measuring

approximately 204 cents; and (5) the plus-tone, measuring approximately 270 cents.60

60 Farhat, The Dastgàh Concept in Persian Music 10-16.

Theory Versus Performance Practice 27

Tabl

e 8.

Meh

di B

arke

šli’s

twen

ty-tw

o-no

te d

ivisi

on o

f the

oct

ave

Rela

tive

Pitc

h: A

4=44

0Hz

/ C4 (

mid

dle

C)=

261.

6255

654H

z

DEG

REE

NUM

BER

NO

TE

INTE

RVA

L RA

TIO

(FRA

CTIO

N)

RATI

O

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

TUN

ING

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÑÓ

ÑÔ

ÑÕ

ÑÖ

Ñ×

ÑØ

ÑÙ

ÒÐ

ÒÑ

ÒÒ ÐÑ

C CÎ

CÚ

Dî

D DÎ

DÚ

Eî E F FÎ

FÚ

Gî

G GÎ

GÚ

Aî

A BÝ

AÚ

Bî B C

uniso

n

sup

eroc

tave

[6] (

D)

V

Pyth

agor

ean

apot

ome,

or c

hrom

atic

sem

itone

[7] (

A)

VII

neut

ral s

econ

d [8

] (D

) X

just

maj

or to

ne (A

) II (9

th h

arm

onic

)

sup

erm

ajor

sec

ond

[9] (

D) II

I

Pyth

agor

ean

augm

ente

d s

econ

d [1

0] (

A)

IX

neut

ral t

hird

[11]

(D)

VIII

Pyth

agor

ean

maj

or th

ird, o

r dito

ne (A

) IV

(81s

t har

mon

ic)

just

and

Pyt

hago

rean

per

fect

four

th (D

) I

sup

erfo

urth

[12]

(D

) V

I

Pyth

agor

ean

trito

ne, o

r aug

men

ted

four

th [1

3] (

A)

VI

subf

ifth

[14]

(D)

XI

just

and

Pyt

hago

rean

per

fect

fifth

(A)

I (3r

d h

arm

onic

)

sup

erfif

th [1

] (D

) IV

Pyth

agor

ean

augm

ente

d fi

fth [2

] (A

) V

III

neut

ral s

ixth

[3] (

D) IX

Pyth

agor

ean

maj

or s

ixth

(A)

III (2

7th

harm

onic

)

Pyth

agor

ean

min

or s

even

th (

D)

II

Pyth

agor

ean

augm

ente

d s

ixth

[4] (

A)

X

neut

ral s

even

th [5

] (D

) V

II

Pyth

agor

ean

maj

or s

even

th (A

) V

octa

ve

1/1

256/

243

2187

/204

8

6553

6/59

049

9/8

32/2

7

1968

3/16

384

8192

/656

1

81/6

4

4/3

1024

/729

729/

512

2621

44/1

7714

7

3/2

128/

81

6561

/409

6

3276

8/19

683

27/1

6

16/9

5904

9/32

768

4096

/218

7

243/

128

2/1

1.00

0000

1.05

3498

1.06

7871

1.10

9858

1.12

5000

1.18

5185

1.20

1355

1.24

8590

1.26

5625

1.33

3333

1.40

4664

1.42

3828

1.47

9811

1.50

0000

1.58

0247

1.60

1807

1.66

4787

1.68

7500

1.77

7778

1.80

2032

1.87

2885

1.89

8438

2.00

0000

261.

626

275.

622

279.

382

290.

367

294.

329

310.

075

314.

305

326.

663

331.

120

348.

834

367.

496

372.

510

387.

156

392.

438

413.

433

419.

074

435.

551

441.

493

465.

112

471.

458

489.

995

496.

680

523.

251

0.00

0

90.2

25

113.

685

180.

450

203.

910

294.

135

317.

595

384.

360

407.

820

498.

045

588.

270

611.

730

678.

495

701.

955

792.

180

815.

640

882.

405

905.

865

996.

090

1019

.550

1086

.315

1109

.775

1200

.000

+00

ß10

+14

ß20

+04

ß06

+18

ß16

+08

ß02

ß12

+12

ß22

+02

ß08

+16

ß18

+06

ß04

+20

ß14

+10

+00

28 Theory Versus Performance Practice

Table 9. Five-limit approximation of Hormoz Farhat’s division of the whole-tone

PATTERN L L + TSC L + GJCS L + L + C

INTERVAL Pythagorean

limma

great limma, acute or

large half-tone

grave or

small tone

just

major tone

RATIO 256/243 27/25 800/729 9/8

CENTS 90.225 133.238 160.897 203.910

The five-limit approximation of Farhat’s division of the whole-tone exhibits the following intervallic pattern:

L, L + TSC, L + GJCS, and L + L + C. ‘TSC’, or two syntonic commas (also referred to as the Mathieu

superdiesis, and produced by the factors 0.58081 and 6561/6400, or 43.013 cents), and ‘GJCS’, or grave

just chromatic semitone (25/24, or 70.672 cents) represent the intervals between the Pythagorean great

limma (27/25, or 133.238 cents) and limma (256/243), and grave or small tone (800/729) and limma

(256/243) respectively. The interval between the great limma (27/25) and limma (256/243) is

recognized by Manuel Op de Coul as the minimal diesis (20000/19683, or 27.660 cents).61

Farhat’s research concludes that “in Persian music, intervals are often unstable: they tend to

fluctuate, within a certain latitude, depending on the mode and according to the performer’s tastes and

inclination.” Farhat also acknowledges that the organization of melodic material beyond a tetrachord or

pentachord, and the actual notion of a ‘Persian scale’ has “no practical applications” in Persian music, and

is therefore misleading.62 The statement that, “these theories were written by eminent scholars who, in

most cases, were not practicing musicians,” additionally suggestive of the divide between theorists and

practitioners; a point further marked by Farhat’s insistence that “today’s musical traditions do not support

the exactitude of those theories.”63 With regards to the relationship between frequency ratios, string

lengths, or cents and the performer, it is interesting to note that as Scott Marcus suggests:

“The term ‘comma’ is not meant to signify an interval of any specific size and is thus never defined in terms

of frequency ratios, string lengths, or cents. Rather, like the term æruti as it occurs in North Indian music

today, kñmà is used to refer to slight changes in the pitch of a given note. Thus when a specific note is said

to be ‘minus a comma’, it means only that the note is slightly lowered from what is perceived to be the

note’s standard position.”64

61 Manuel Op de Coul, “Huygens-Fokker: List of intervals,” Huygens-Fokker Foundation: Centre for Microtonal Music,

2006, Huygens-Fokker Foundation, 22 Nov. 2006, <http://www.xs4all.nl/~huygensf/doc/intervals.html>.

62 Farhat, “Iran: Classical Traditions,” The New Grove Dictionary of Music and Musicians 532.

63 Farhat, The Dastgàh Concept in Persian Music 15.

64 Scott Marcus, “The Interface Between Theory and Practice: Intonation in Arab Music,” Asian Music 24.2 (Spring-

Summer, 1993): 41.

Theory Versus Performance Practice 29

Àzàdeh for Santñr and Tape Àzàdeh for santñr and tape – composed for Iranian santñrist Qmars Piraglu – features the Persian santär (a

72-string [or 18 quadruple-stringed] box zither), and will hereby serve as a practical study of Persian

tuning systems, with its presentation of both ‘theoretical’ and ‘performance practice’ tunings. Within the

context of an analysis of the tuning methodology of performer Qmars Piraglu, an attempt will be made to

make comparisons not just with the three propositions outlined previously, but also with other existing

alternative tuning systems. This discussion presented in order to generate a greater understanding of the

musical processes that govern the theory of Persian intervals, at the same time disclosing some of the

strategies employed in the general contemporary ‘cross-cultural’ compositional process. The enquiry into

the basic structure of the Persian modal system reveals some of this music’s essential characteristics, while

also providing the necessary organisational framework for the composition.

The Artist Iranian artist Qmars Piraglu (formerly Siamak Noory) was born in Tehran in 1961, and initiated his musical

education at the age of five via Orff instruments (children’s instruments designed by German composer

Carl Orff [1895-1982] such as the xilophon, metalophon, and vibraphon).65 Following a seven-year

period, he then adopted the Persian santär (a 72-string [or 18 quadruple-stringed] box zither), initially

learning the instrument under the guidance of Ms Farzaneh Noshad (a student at the Persian Traditional

Music Conservatory in Tehran). Three years later, he is accepted as a student of “one of Iran’s greatest

contemporary composers and master santärists,” Ostàd Faràmarz Pàyvar (1933-). For the next eight years

Qmars develops his knowledge and understanding of the complete Persian classical music repertoire. As

well as this, he synchronically studies Western music – the pianoforte with Taher Djalili (a pianist from the

Music Conservatory), and for a year, the bassoon with Khosrow Soltani.

In 1979, the Islamic revolution in Iran bestowed the study of music with an illegal status,

consequently bringing about the premature closure of the Music Conservatory. “Because of the Islamic

revolution in 1979, which caused an undemocratic situation, more than three million Iranians became

obliged to escape from the country. Of these three million nearly sixty thousand came to Sweden during

the 1980s. More than a million people fled to USA, and the rest mostly went to Germany, France, and

other European countries,” explains Qmars. After the fact, for his own ‘anti-establishment’ beliefs in

65 German composer Carl Off (1895-1982) developed Orff-Schilwerk, “a system of music education intended for

groups of children singing and playing together,” which emphasized “the development of creativity and the ability to

improvise.” For a further discussion, see Randel, ed., The New Harvard Dictionary of Music 577-78.

30 Theory Versus Performance Practice

artistic freedom, he too becomes a victim of the ongoing institutionalised persecution, and in 1981 is

consequently arrested and imprisoned for a period of two years.

Qmars ultimately escaped from Iran in 1986 (during the Iran-Iraq war of the eighties), arriving in

Turkey as a refugee, to then temporarily settle in Belgium, where he studied the piano at the Music

Academy in Antwerp with Hedvig Vanvarenberg, before establishing a permanent base in Sweden in

1989. Here he studied the piano for two years at the Birkagårdens Folkhögskolan in Stockholm with Stella

Tchaikowsky, and went on to graduate from the University of Göteborg with a Bachelor of Science in

Musicology. Since 2001, Qmars has been living in Melbourne, Australia. In 2003, he completed a Master

of Music (MMus) degree at the University of Melbourne, and is currently undertaking a Doctor of

Philosophy (PhD) degree at Monash University.66

Fig. 1. Andrián Pertout, Qmars Piraglu.67

The Instrument The Persian or Iranian santär is an integral part of the traditional orchestra, sharing the lute repertoire of the

tàr and setàr. It is also utilized in the motrebi ‘entertainment music’ genre, but folk styles are excluded.

An article by Jean During, Scheherazade Q. Hassan, and Alastair Dick describes the santär’s construction

thus:

66 Andrián Pertout, “Siamak Noory: The Santurist – Part 1,” interview with Siamak Noory (Qmars Piraglu) Mixdown

97 (May 2002): 25.

67 Qmars Piraglu, personal photograph of Andrián Pertout, 22 Oct. 2006.

Theory Versus Performance Practice 31

“The santär consists of a trapeziform case made of walnut wood, approximately 90cm wide at the broad

end, 35cm wide at the narrow end and 6cm deep. The sides form an angle of 45 degrees to the wider

end. The strings are fixed to hitch-pins along the left-hand side and wound round metal wrest-pins on the

right by means of which they are tuned with a tuning-key. Each quadruple set of strings rests on a movable

bridge of hardwood (kharak). These bridges are aligned almost parallel with the sides of the case. The

right-hand rank corresponds to the bass strings and that on the left to the treble strings. In the centre of the

santär the low-pitched strings on the right cross the high-pitched strings on the left.”

Adding to this, the instrument features three courses of strings (the bass strings, made of brass, while the

treble ones, steel), with a total of 72 strings, or 18 groups of strings, capable of producing 27 different

pitches, and is played “by striking the strings with two hammers (mezràb) held in three fingers of each

hand.”68

Ex. 3. Range of the santñr

Persian 72-string (or 18 quadruple-stringed) box zither

The Persian Modal System An important aspect of understanding the basic concept behind the Persian modal system is the radif

(row, series), being the term used to describe the complete collection of melodies that constitute the

repertoire of Persian traditional music, as well as the separate issue of melodic patterns associated with

each individual mode represented within the subsystem of the twelve dastgàhs.69 According to Talai:

“To understand the radif, we must first understand that it is something different from the modal system. This

repertory is not like Western art music, which is composed and intended to be played exactly as written.

The radif consists of traditional melodies, many of which are derived from popular and folk sources, and

whose origins have been obscured with the passage of time.”70

On the other hand, the notion of dastgàh (organisation, system) – the subsystem of the radif – further

delineates the twelve groupings of modes, with their own collection of associated melodies, or guêes

(corner, section, piece). The introductory section of each mode referred to as the daràmad (opening,

introduction), while the conclusive, as the forñd (descent, cadence). Another factor of note includes 68 Jean During, Scheherazade Q Hassan, and Alastair Dick, “Santñr,” The New Grove Dictionary of Musical

Instruments, ed. Stanley Sadie, vol. 3 (London: Macmillan Reference, 1984) 291-92.

69 Farhat, The Dastgàh Concept in Persian Music 21.

70 Talai, Traditional Persian Art Music: The Radif of Mirza Abdollah 4.

32 Theory Versus Performance Practice

the specific role of certain tones within a mode, which include the ist, or ‘stop’ note used to conclude

phrases; the šàhed (witness), or prominent note; as well as the àqàz (beginning), or introductory note in

improvisation. The moteqayyer (changeable), or variable note further represents a tone reintroduced to

a mode in an altered form. It is quite common for example for a mode to utilize A koron (a microtonal

inflection on the tone of A) in the bottom registers, while A flat, in the top; or for there to be two

microtonal possibilities for a note within the same register, with the use of one or both pitches governed

by attributes of a particular guše.71 Habib Hassan Touma offers the following discussion with regards to

the compositional and improvisational nature of the dastgàh system in the music of the Middle East:

“The realization of a truly convincing and original maqam requires a creative faculty like that of a composer

of genius. Nevertheless, this phenomenon can be considered only partly as a composed form, because

no maqam, makam, mugam, or dastgàh can be identical to any other. Each time it is re-created as a new

composition. The compositional aspect is demonstrated in the pre-determined tonal-spatial organization

of a fixed number of tone-levels without repetitions, while the improvisation aspect unfolds itself freely in

the rhythmic-temporal scheme. Thus, the interplay of composition and improvisation is one of the most

distinctive features of the maqam phenomenon.”72

The presentation of the twelve dastgàhs that follows (the basic pitch material of the modes) adheres to

the range of the santñr, and utilizes Vaziri’s system of accidentals, which was originally intended for the

microtonal sharpening and flattening of tones in equal quarter-tone ( 24 2 ) increments, but consequently

adopted as a general standard for Persian music notation.73

Fig. 2. The Ali Naqi Vaziri notation system (the accidentals of Persian music)

sori – raised by one Pythagorean limma (90.225 cents)

sharp – raised by one limma and one Pythagorean comma (113.685 cents)

koron – lowered by one Pythagorean limma (90.225 cents)

flat – lowered by one limma and one Pythagorean comma (113.685 cents)

71 Farhat, The Dastgàh Concept in Persian Music 19-26.

72 Habib Hassan Touma, “The Maqam Phenomenon: An Improvisation Technique in the Music of the Middle East,”

Ethnomusicology 15.1 (Jan., 1971): 47.

73 Zonis, Classical Persian Music: An Introduction 56-57.

Theory Versus Performance Practice 33

Ex. 4. The twelve dastgàhs i. Šhñr

ii. Dašhtå

iii. Abñ atà

iv. Màhñr

v. Ràst-Panjgàh

34 Theory Versus Performance Practice

vi. Homàyñn

vii. Bayàt-e Eéfahàn

viii. Bayàt-e Tork

ix. Segàh

x. Áahàrgàh

Theory Versus Performance Practice 35

xi. Afšhàrå

xii. Navà

Tuning Analysis Protocols The obvious consequential procedure in the compositional process now being the selection of the

appropriate dastgàh, which is an exercise delivering an outcome based on definite instinctive principles,

although assisted somewhat by the audition of modes in workshops organized with Piraglu. Upon the

decision to select dastgàh-e segàh, it simply becomes a matter of tuning the santñr to concert

performance standard so as to acquire twenty-seven naturally decaying (approximately six seconds

each) samples. The sound of every string therefore captured in the digital domain (in this case utilizing an

Audio-Technica AT4050/CM5 condenser microphone [with switchable cardioid, omnidirectional, or

figure-8 operation, and a frequency response from 20Hz-20kHz], Mackie 32×8×2 8-Bus Mixing Console,

and a Fostex D-160 Digital Multitrack Recorder), to be then digitally transferred (via optical cable) to a

digital audio editor PC software environment (Sony Sound Forge 8.0). The strategy adopted in order to

obtain the frequencies (in hertz) of each string with the aid of spectrum analyzing software, which will

provide an invaluable source of data for later extensive analysis. This data (collected on three separate

occasions), once subjected to an analysis of variance (utilizing average and standard deviation criteria)

will then form a ‘mean’ scale, which in essence will represent the intervallic nature of dastgàh-e segàh

according to Persian performer Qmars Piraglu. The first collection of samples will additionally serve as the

source material for the creation of a multi-sampled santñr for the Akai S3000XL Midi Stereo Digital

Sampler.

In view of the fact that Sony Sound Forge 8.0 is unable to provide accurate enough frequency

readings, a series of tests is then implemented in order to locate the appropriate software package.

These tests include the multiplatform readings of 1Hz, 10Hz, 100Hz, 1KHz, 10KHz, and 440Hz sine wave

36 Theory Versus Performance Practice

sync tones, with the results nominating Adobe Audition 2.0 as the most error free vehicle. Utilizing the

frequency analysis window function of the program it is now possible to generate FFT (Fast Fourier

Transform) computations that translate frequency and amplitude analysis into a simple display of the mean

frequency of the most prominent frequency peak, or the fundamental.74 The Adobe Audition manual

describes the FFT process thus: “Fourier Theory states that any waveform consists of an infinite sum of sin

and cos functions, allowing frequency and amplitude to be quickly analyzed.” The program uses an

algorithm based on the Fourier Theory for filtering, spectral view, and frequency analysis. The FFT size

menu includes 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, and 65536 sample

options; while the FFT type menu provides eight type of FFT windows – Triangular, Hanning, Hamming,

Blackmann, Welch (Gaussian), Blackmann-Harris, Kaiser (120dB), and Kaiser (180dB). According to the

manual:

“The Triangular window gives a more precise frequency estimate, but it’s also the noisiest, meaning that

other frequencies will be shown as present, even though they may be much lower in volume. At the other

extreme, the Blackmann-Harris window has a broader frequency band, which isn’t as precise, but the

sidelobes are very low, making it easier to pick out the major frequency components.”

An analysis of the test tone readings (all one hundred percent accurate apart from the 0.94257Hz result

for 1Hz) reveal that a FFT size of 65536 and a Hamming75 type is required. At 44.1KHz, or 44,100

samples per second, a 65536 FFT sample window length allows for the equivalent of 673 milliseconds of

sample time. One further test procedure is nevertheless implemented in order to confirm the above

results, and this involves obtaining data (utilizing a smaller 4096 FFT sample window length, or 93

milliseconds) from all Cs (C3, C4, C5, and C6) at multiple start points (in 50 millisecond intervals beginning

from 0 and ending on 500 milliseconds). The vibrating strings are shown to be extremely stable over

time, with approximately ß99.923564% and +100.122296% deviations from the 65536 FFT sample

window length reading of the first sample (C3), which represents fluctuations within 0.26Hz, or 3.440

cents.

74 “Frequency, which is a measurement expressed in hertz (Hz), is derived from German physicist Heinrich Rudolf

Hertz (1857-94), and represents the rate of vibrations per second of a sound wave, with each cycle a sonic structure

incorporating positive and negative amplitudes.” For a further discussion, see Arthur H. Benade, Fundamentals of Musical

Acoustics, 2nd ed. (New York: Dover Publications, 1990) 159.

75 The Hamming window is “a very popular compromise between simplicity and effectiveness. It’s named after

Richard W. Hamming, a pioneer in the application of computers to practical computation. Mathematically, it consists of a

single cycle of a cosine, raised and weighted so that it drops to 0.08 at the end-points and has a peak value of one:

ht=0.54ß0.46cos(2�t/nß1), nt <0 � .” For a further discussion, see Ken Steiglitz, A DSP Primer: with Applications to

Digital Audio and Computer Music (Menlo Park, CA: Addison-Wesley Pub., 1995) 206.

Theory Versus Performance Practice 37

Tuning of the Santñr

The tuning of the santñr for dastgàh-e segàh (on F) presents the following series of pitches: C3, F3, G3,

Aù3, B"3, C4, Dù4, E"4, F4 (first position); Eù4, F4, G4, Aù4, B"4, C5, Dù5, E"5, F5 (second position); and Eù5,

F5, G5, A"5, B"5, C6, Dù6, E6, F6 (third position). The ‘position’ is in reference to each of the instrument’s

three courses of strings. The instrument’s design incorporates eighteen quadruple set of strings, with the

division via movable bridges of the second set of nine strings generating the second and third courses

(two individual pitches generally tuned octaves apart), to produce twenty-seven distinct pitches in total.

The santñrist generally commences with the tuning of the first front string, which then becomes the basis

for the generation of unison intervallic relationships (removing beating as much as possible) with the other

three strings of each set. An interesting point of note is that although the primary structure of dastgàh-e

segàh can be essentially stated to be heptatonic, in performance practice the scale is expanded into a

three-octave scale with dissimilar tetrachords. Some points of interest include the low brass string tuned

to C'; the A’s in the first two octaves tuned to Aù, while in the third, tuned to A"; E" and Eù alternatives

provided in two octaves; and the duplication of F pitches in the second and third octaves. The design

of the instrument – presenting a total of nine strings per octave, with two strings, E and F, tunable to either

E', E" or Eù; and F' or F! respectively – allows for the performance of heptatonic scales (inclusive of their

auxiliary pitches) from the complete Persian modal system.76

Dastgàh-e segàh is mode number nine of the twelve dastgàhs (according to Piraglu’s own

codification derived from the santñr tradition of Pàyvar), and its name (“literally ‘third place’”) is in

reference to “the positioning of the central pitch relative to other modes.” According to Zonis, “se is the

Persian word for ‘three’; gàh, the word for ‘time’ or ‘place’,” and alludes to the fact that “this kind of

designation was formerly used both in Persia and in the Arab countries to indicate the degrees of the

scale.”77 The guêe nomenclature and design of the dastgàh is extensively interconnected with dastgàh-e

áahàrgàh (“literally ‘fourth place’”).78 “Segàh and áahàrgàh allied to a degree unparalleled in any other

two dastgàhs,” notes Farhat. “Virtually every piece performed in segàh can be performed in áahàrgàh,

although áahàrgàh includes a few guêes peculiar to that dastgàh and not performed in segàh.”79 The

following three examples present dastgàh-e segàh’s basic structure, as well as the tuning of the santñr,

and harmonic characteristics (in Pythagorean terms).

76 Ella Zonis, Classical Persian Music: An Introduction (Cambridge, MA: Harvard U. Press, 1973) 57-58.

77 Zonis, Classical Persian Music: An Introduction 88.

78 Laudan Nooshin, “The Song of the Nightingale: Processes of Improvisation in Dastgàh Segàh (Iranian Classical

Music),” British Journal of Ethnomusicology 7 (1998): 76.

79 Farhat, The Dastgàh Concept in Persian Music 56.

38 Theory Versus Performance Practice

Ex. 5. Dastgàh-e segàh i. Basic mode

ii. Tuning of the santñr for dastgàh-e segàh (on F)

iii. The harmonic characteristics of dastgàh-e segàh

The tuning of the santñr for dastgàh-e segàh (on F) involves a specific tuning scheme, which begins with

the acquisition of a fundamental C' in second position (C5) calibrated to A=440Hz. The notion of

standard pitch only idealistic as in actuality on one occasion demonstrating a 10.223 cent deviation from

A=440Hz (or A=442.606Hz), while on other occasions, 10.387 cents (or A=442.648Hz), and 1.584

cents (or A=440.403Hz). In spite of Piraglu’s insistence of possessing perfect pitch and therefore the

ability to tune to a sung high C at standard pitch,80 something that becomes clearly evident via this

experience is that in this particular context the solo performer is rarely that concerned about the

procurement of A=440Hz in literal terms, hence it is a musical pitch certainly not in the domain of

obligatory criteria. Although it must be noted that in a performance setting involving the santñr in

80 “Absolute pitch, while extremely rare in the general population (its incidence has been estimated as 1 in

10,000), is in reality common among professional musicians, and to a limited extent may serve as a marker for musicality.

But as N. Slonimsky, in his autobiography, Perfect Pitch, writes: ‘The lack of it does not exclude musical talent, or even

genius. Neither Wagner nor Tchaikovsky had absolute pitch, while a legion of mediocre composers possessed it to the

highest degree.’ There is a greatly heightened occurrence of absolute pitch in some other populations: among the autistic

the incidence may be about one in 20; and among those with savant syndrome, more than a third have musical gifts –

and all musical savants, apparently, have absolute pitch.” For a further discussion, see Oliver Sacks, G. Schlaug, L. Jäncke,

Y. Huang, and H. Steinmetz, “Musical Ability,” Science 268.5211 (May 5, 1995): 621.

Theory Versus Performance Practice 39

conjunction with Western traditional instruments, some form of pitch calibration will be required in order

to accommodate the limitations of some of the instruments. A performance setting involving a Persian

ensemble on the other hand will require the whole group to calibrate their pitch to the santñr, for the

simple reason of practicality – the instrument requiring the tuning of seventy-two strings. It is interesting to

note that following a discussion with the performer about the rationale of the last measurement of

A=440.403Hz, and its obvious discrepancy in comparison with the mean of the first two (442.627Hz,

and therefore a difference of almost nine cents), it is revealed that a duet performance with a flautist had

required the santñr to be calibrated down to A=440Hz.

Stage one of the tuning process encapsulates the following procedures:

I. i. C5 is tuned to A=440Hz

ii. C4 to C5

iii. C3 to C4

iv. C6 to C5

II. i. F4 (2nd position) is tuned to C5, generating a perfect fifth

ii. F3 to F4 (2nd position)

iii. F5 (3rd position) to F4 (2nd position)

iv. F5 (2nd position) to F5 (3rd position)

v. F6 to F5 (2nd position)

vi. F4 (1st position) to F4 (2nd position)

III. i. G4 is tuned to C5, generating a perfect fourth

ii. G3 to G4

iii. G5 to G4

IV. i. B"4 is tuned to F4, generating a tempered perfect fourth

ii. B"3 to B"4

iii. B"5 to B"4

V. i. Aù4 is tuned to first tetrachord melodic patterns (F4, G4, Aù4 and B"4)

ii. Aù3 to Aù4

VI. i. Dù5 is tuned to Aù4, generating a tempered perfect fourth

ii. Dù4 to Dù5

iii. Dù6 to Dù5

VII. i. E"5 is tuned to B"4, generating a tempered perfect fourth

ii. E"4 to E"5

iii. E"6 to E"5

VIII. i. Eù4 is tuned to Aù4, generating a tempered perfect fourth

40 Theory Versus Performance Practice

ii. Eù5 to Eù4

IX. i. A"5 is tuned to second tetrachord melodic patterns (Eù5, F5, G5 and A"5)

It may be further noted that although the technique applied to the tuning of Aù4 has a basis of first

tetrachord melodic patterns (F4, G4, Aù4, and B"4), the interval is additionally calculated according to the

evaluation of the perfect fourth simultaneous sonorities of F4 (2nd position) and B"4, as well as F4 (2nd

position) and Aù4. As a matter of interest, it must be stated that the perfect fourth is “the smallest

invariable interval” in Persian music, and recognized as the most important.81 The duplication of Fs (F4 in

first and second positions, and F5 in second and third positions) serve merely as alternatives for the

equivalent pitches, although in performance practice, F4 (first position) and F5 (second position) are

generally avoided in dastgàh-e segàh (on F) due to their inferior timbral quality (the strings, shorter in this

region of the instrument, hence dynamically weaker). In the context of the pitch material requirements of

other dastgàhs, this additional pitch serves to provide a mandatory chromatic alternation. The

synchronous utilization of Eù and E" in dastgàh-e segàh (on F) illustrates the application of that principle.

The fact that the integrity of second and third position pitches are determined by the placement of

individual bridges dividing the string into specific ratios, these bridges may also have to be manually

adjusted in order to produce the desired sonorities. Third position A"5 is produced via this method,

with the adjustment of the bridge dividing the relevant string essentially simultaneously producing second

position Aù4 and third position A"5. This also holds true for other pitches acquired during the dipartite

tuning process not conforming to the aesthetics of dastgàh-e segàh (on F), which may have to be

alternatively fine-tuned via the adjustments of the bridge. The final pitch of interest in need of some

explanation is C3 in first position, which according to Piraglu, is a relatively modern phenomenon –

adopted post 1980 by a new generation of santñr players such as Parviz Meshkàtiàn (1955-) and

Pashang Kàmkàr (1951-). The practice favours the tuning of the low string to C or D (depending on the

dastgàh), as opposed to the earlier schools of Abol-Äasan Éabà (1902-1957) and Faràmarz Pàyvar

(1933-) subscribing to the acquisition of E or Eù. The C3 in first position of course provides the facility for

the instrument to produce both perfect fourth (between C3 and F3) and perfect fifth (between C3 and

G3) dyads in the low tessitura of the instrument.82

Stage two of the tuning process – the evaluation of simultaneous sonorities (perfect fifths, perfect

fourths, tempered perfect fourths, and neutral thirds) – then encapsulates the following procedures:

I. i. F4 (2nd position) is evaluated with C5, generating a perfect fifth

81 Talai, Traditional Persian Art Music: The Radif of Mirza Abdollah 10.

82 Qmars Piraglu, “The Persian Music Tradition (In Iran) under Qajar and Pahlavi Dynasties,” diss. Göteborg U., Swed.,

1998, 33.

Theory Versus Performance Practice 41

ii. C4 with F4 (2nd position), generating a perfect fourth

iii. C4 with F4 (1st position), generating a perfect fourth

iv. C5 with F5 (3rd position), generating a perfect fourth

v. C5 with F5 (2nd position), generating a perfect fourth

vi. F5 (3rd position) with C6, generating a perfect fifth

vii. F3 with C4, generating a perfect fifth

viii. C3 with F3, generating a perfect fourth

ix. C6 with F6, generating a perfect fourth

II. i. C4 is evaluated with G4, generating a perfect fifth

ii. G4 with C5, generating a perfect fourth

iii. C5 with G5, generating a perfect fifth

iv. G5 with C6, generating a perfect fourth

v. G3 with C4, generating a perfect fourth

vi. C3 with G3, generating a perfect fifth

III. i. F4 (2nd position) is evaluated with B"4, generating a tempered perfect fourth

ii. F4 (2nd position) with Aù4, generating a neutral third

iii. Aù4 with Dù5, generating a tempered perfect fourth

iv. F3 with B"3, generating a tempered perfect fourth

v. F3 with Aù3, generating a neutral third

vi. Aù3 with Dù4, generating a tempered perfect fourth

vii. F5 (3rd position) with B"5, generating a tempered perfect fourth

IV. i. B"4 is evaluated with E"5, generating a tempered perfect fourth

ii. B"3 with E"4, generating a tempered perfect fourth

iii. B"5 with E"6, generating a tempered perfect fourth

V. i. Eù4 is evaluated with Aù4, generating a tempered perfect fourth Spectrum Analysis Results The following three tables present the spectrum analysis results collected on three separate occasions

(with a periodicity of 3-6 months) for each of the twenty-seven sets of strings, and therefore denotes all

data for string set, pitch, order, frequency (Hz), ratio (decimal), and cents. The calculation of frequency

represents prime (not normalized) data, and therefore ratio and cents are the only two comparative

frames of reference. Ratios have been calculated from the relationship of frequencies to the base pitch

of the second position C5 (the reference pitch, or 526.350Hz, 526.400Hz, and 523.730Hz respectively

for each of the three tunings), while cents are a derivative of ratio data.

42 Theory Versus Performance Practice

Table 10. Tuning of the santñr for dastgàh-e segàh (on F) – tuning no. 1 1st position

STRING SET NOTE ORDER FREQUENCY (HERTZ) RATIO (DECIMAL) CENTS

ÐÑ ÐÒ ÐÓ ÐÔ ÐÕ ÐÖ Ð× ÐØ ÐÙ

C3

F3

G3

Aî3

BÝ3

C4

Dî4

EÝ4

F4

3 (I-iii)

6 (II-ii)

12 (III-ii)

18 (V-ii)

15 (IV-ii)

2 (I-ii)

20 (VI-ii)

23 (VII-ii)

10 (II-vi)

130.830

174.990

196.480

213.800

232.060

261.390

286.780

310.080

351.200

1.988487

1.329838

1.493151

1.624774

1.763541

1.986435

1.089693

1.178227

1.334473

1190.005

493.500

694.032

840.287

982.171

1188.218

148.706

283.942

499.524

2nd position

STRING SET NOTE ORDER FREQUENCY (HERTZ) RATIO (DECIMAL) CENTS

ÑÐ ÑÑ ÑÒ ÑÓ ÑÔ ÑÕ ÑÖ Ñ× ÑØ

Eî4

F4

G4

Aî4

BÝ4

C5

Dî5

EÝ5

F5

25 (VIII-i)

5 (II-i)

11 (III-i)

17 (V-i)

14 (IV-i)

1 (I-i)

19 (VI-i)

22 (VII-i)

8 (II-iv)

322.540

350.090

394.970

429.610

465.950

526.350

578.200

620.720

703.100

1.225572

1.330256

1.500788

1.632412

1.770495

1.000000

1.098509

1.179291

1.335803

352.147

494.044

702.865

848.406

988.983

0.000

162.655

285.504

501.249

3rd position

STRING SET NOTE ORDER FREQUENCY (HERTZ) RATIO (DECIMAL) CENTS

ÑÙ ÒÐ ÒÑ ÒÒ ÒÓ ÒÔ ÒÕ ÒÖ Ò×

Eî5

F5

G5

AÝ5

BÝ5

C6

Dî6

EÝ6

F6

26 (VIII-ii)

7 (II-iii)

13 (III-iii)

27 (IX-i)

16 (IV-iii)

4 (I-iv)

21 (VI-iii)

24 (VII-iii)

9 (II-v)

645.220

697.240

788.260

827.400

930.310

1049.300

1144.400

1246.000

1406.200

1.225838

1.324670

1.497597

1.571958

1.767474

1.993540

1.087109

1.183623

1.335803

352.522

486.759

699.179

783.075

986.027

1194.399

144.596

291.852

501.249

Theory Versus Performance Practice 43

Table 11. Tuning of the santñr for dastgàh-e segàh (on F) – tuning no. 2 1st position

STRING SET NOTE ORDER FREQUENCY (HERTZ) RATIO (DECIMAL) CENTS

ÐÑ ÐÒ ÐÓ ÐÔ ÐÕ ÐÖ Ð× ÐØ ÐÙ

C3

F3

G3

Aî3

BÝ3

C4

Dî4

EÝ4

F4

3 (I-iii)

6 (II-ii)

12 (III-ii)

18 (V-ii)

15 (IV-ii)

2 (I-ii)

20 (VI-ii)

23 (VII-ii)

10 (II-vi)

130.790

174.260

197.400

213.740

231.990

260.930

286.690

310.050

351.170

1.987690

1.324164

1.500000

1.624164

1.762842

1.982751

1.089248

1.178002

1.334233

1189.311

486.098

701.955

839.637

981.484

1185.004

147.999

283.610

499.212

2nd position

STRING SET NOTE ORDER FREQUENCY (HERTZ) RATIO (DECIMAL) CENTS

ÑÐ ÑÑ ÑÒ ÑÓ ÑÔ ÑÕ ÑÖ Ñ× ÑØ

Eî4

F4

G4

Aî4

BÝ4

C5

Dî5

EÝ5

F5

25 (VIII-i)

5 (II-i)

11 (III-i)

17 (V-i)

14 (IV-i)

1 (I-i)

19 (VI-i)

22 (VII-i)

8 (II-iv)

322.490

348.840

394.670

428.470

465.940

526.400

578.150

620.620

703.090

1.225266

1.325380

1.499506

1.627926

1.770289

1.000000

1.098309

1.178989

1.335657

351.714

487.687

701.385

843.642

988.782

0.000

162.341

285.061

501.060

3rd position

STRING SET NOTE ORDER FREQUENCY (HERTZ) RATIO (DECIMAL) CENTS

ÑÙ ÒÐ ÒÑ ÒÒ ÒÓ ÒÔ ÒÕ ÒÖ Ò×

Eî5

F5

G5

AÝ5

BÝ5

C6

Dî6

EÝ6

F6

26 (VIII-ii)

7 (II-iii)

13 (III-iii)

27 (IX-i)

16 (IV-iii)

4 (I-iv)

21 (VI-iii)

24 (VII-iii)

9 (II-v)

645.160

697.280

788.220

829.380

930.230

1048.700

1144.200

1254.800

1406.000

1.225608

1.324620

1.497378

1.575570

1.767154

1.992211

1.086816

1.191869

1.335486

352.197

486.694

698.927

787.049

985.714

1193.245

144.129

303.871

500.838

44 Theory Versus Performance Practice

Table 12. Tuning of the santñr for dastgàh-e segàh (on F) – tuning no. 3 1st position

STRING SET NOTE ORDER FREQUENCY (HERTZ) RATIO (DECIMAL) CENTS

ÐÑ ÐÒ ÐÓ ÐÔ ÐÕ ÐÖ Ð× ÐØ ÐÙ

C3

F3

G3

Aî3

BÝ3

C4

Dî4

EÝ4

F4

3 (I-iii)

6 (II-ii)

12 (III-ii)

18 (V-ii)

15 (IV-ii)

2 (I-ii)

20 (VI-ii)

23 (VII-ii)

10 (II-vi)

131.160

174.570

197.130

212.300

231.460

262.250

285.400

311.140

346.900

1.001738

1.333282

1.505585

1.621446

1.767781

1.001470

1.089875

1.188169

1.336185

3.005

497.979

708.389

836.737

986.328

2.543

148.995

298.489

501.743

2nd position

STRING SET NOTE ORDER FREQUENCY (HERTZ) RATIO (DECIMAL) CENTS

ÑÐ ÑÑ ÑÒ ÑÓ ÑÔ ÑÕ ÑÖ Ñ× ÑØ

Eî4

F4

G4

Aî4

BÝ4

C5

Dî5

EÝ5

F5

25 (VIII-i)

5 (II-i)

11 (III-i)

17 (V-i)

14 (IV-i)

1 (I-i)

19 (VI-i)

22 (VII-i)

8 (II-iv)

320.500

348.970

394.930

427.390

462.800

523.730

569.000

617.650

698.550

1.223913

1.332633

1.508144

1.632101

1.767323

1.000000

1.086438

1.179329

1.333798

349.801

497.136

711.328

848.076

985.879

0.000

143.526

285.560

498.648

3rd position

STRING SET NOTE ORDER FREQUENCY (HERTZ) RATIO (DECIMAL) CENTS

ÑÙ ÒÐ ÒÑ ÒÒ ÒÓ ÒÔ ÒÕ ÒÖ Ò×

Eî5

F5

G5

AÝ5

BÝ5

C6

Dî6

EÝ6

F6

26 (VIII-ii)

7 (II-iii)

13 (III-iii)

27 (IX-i)

16 (IV-iii)

4 (I-iv)

21 (VI-iii)

24 (VII-iii)

9 (II-v)

638.520

695.250

791.640

824.560

925.770

1051.000

1141.700

1242.500

1396.800

1.219178

1.327497

1.511542

1.574399

1.767647

1.003380

1.089970

1.186203

1.333512

343.090

490.450

715.226

785.761

986.197

5.84102

149.146

295.621

498.276

Theory Versus Performance Practice 45

An analysis of simultaneous sonorities based on stage one of the tuning process for the three tunings

reveal octaves with a mean value of 1201.019 cents, a range between 1181.941 and 1218.810 cents,

and a standard deviation of 7.771183. These results effectively pronounce the octave as being in the

vicinity of the just perfect octave (2/1), although in the range from the double tritone (2025/1024, or

1180.447 cents) and acute or large octave (81/40, or 1221.506 cents) with a falsity of +1.494 and

ß2.696 cents on each count. In comparison with the just perfect octave (2/1), the octave is within the

range of ß18.059 and +18.810 cents.83

Table 13. Stage one of tuning process – octaves

NUMBER NOTES (POSITION) RANGE (CENTS) AVERAGE (CENTS) STANDARD DEVIATION

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÑÓ

ÑÔ

ÑÕ

ÑÖ

C3 and C4

F3 and F4(2)

G3 and G4

Aî3 and Aî4

BÝ3 and BÝ4

C4 and C5

Dî4 and Dî5

EÝ4 and EÝ5

Eî4 and Eî5

F4(2) and F5(3)

G4 and G5

BÝ4 and BÝ5

C5 and C6

Dî5 and Dî6

EÝ5 and EÝ6

F5(2) and F6

1195.693 � 1199.538

1199.157 � 1201.589

1199.430 � 1208.833

1204.005 � 1211.339

1199.551 � 1207.298

1197.457 � 1214.996

1194.532 � 1214.343

1187.071 � 1201.563

1193.289 � 1200.483

1192.715 � 1199.007

1196.314 � 1203.897

1197.044 � 1200.318

1193.245 � 1205.841

1181.941 � 1205.620

1206.347 � 1218.810

1199.628 � 1200.000

1197.814

1200.430

1203.734

1207.821

1204.554

1208.078

1207.608

1196.695

1198.049

1195.012

1199.251

1198.098

1197.828

1189.783

1211.740

1199.802

1.953313

1.220009

4.751527

3.675986

4.339307

9.337949

11.325798

8.334857

4.122885

3.472527

4.070141

1.923469

6.963106

13.715187

6.398768

0.187035

Simultaneous sonorities based on stage two of the tuning process for the three tunings on the other hand

reveal perfect fifths with a mean value of 706.894 cents, a range between 694.718 and 716.381 cents,

and a standard deviation of 6.537279. The primary perfect fifths, or those unique and sequentially

superior are represented by F4 (2nd position) and C5, C4 and G4, and the mean values of 707.044 and

713.271 cents respectively. The range of these fifths being between 698.927 and 715.226 cents

indicates an inclination to represent the meantone perfect fifth (3/2× 48180 , approximately 154/103, or

696.578 cents), just perfect fifth (3/2), and equal perfect fifth ( 712 ]2[ , approximately 767/512, or

700.000 cents) with a falsity of +2.274, ß3.028, and ß1.073 cents at the lower end of the scale; while

83 Lloyd, and Boyle, Intervals, Scales and Temperaments: An Introduction to the Study of Musical Intonation ix.

46 Theory Versus Performance Practice

the sléndro acute or large fifth ( 35 ]2[ , approximately 97/64, or 720.000 cents), just perfect fifth (3/2),

and equal perfect fifth ( 712 ]2[ ) with a falsity of ß4.774, +13.271, and +15.226 cents at the higher

end.84 The sléndro acute or large fifth is the identical interval found in five-tone equal temperament, and

is produced by the ratio 1: 35 )2( =1:1.515717.85

Table 14. Stage two of tuning process – perfect fifths

NUMBER NOTES (POSITION) RANGE (CENTS) AVERAGE (CENTS) STANDARD DEVIATION

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

C3 and G3

F3 and C4

C4 and G4

F4(2) and C5

C5 and G5

F5(3) and C6

704.027 � 712.644

694.718 � 704.565

708.785 � 716.381

702.864 � 712.313

698.927 � 715.226

706.550 � 715.391

707.351

699.396

713.271

707.044

704.444

709.860

4.633233

4.941669

3.980471

4.817387

9.338180

4.820301

An analysis of simultaneous sonorities based on stage two of the tuning process for the three tunings

reveal perfect fourths with a mean value of 497.436 cents, a range between 483.049 and 514.208 cents,

and a standard deviation of 7.640715. The primary perfect fourths are represented by C4 and F5 (2nd

position), and G4 and C5, and the mean values of 501.034 and 494.807 cents respectively. The range of

these fourths being between 488.672 and 505.826 indicates an inclination to represent the sléndro

grave or small fourth ( 25 ]2[ , approximately 128/97, or 480.000 cents), just perfect fourth (4/3), and

equal perfect fourth ( 512 ]2[ , approximately 1024/767, or 500.000 cents) with a falsity of +8.672,

ß9.373, and ß11.328 cents at the lower end of the scale; while the meantone perfect fourth

(4/3× 48081 , approximately 103/77, or 503.422 cents), just perfect fourth (4/3), and equal perfect fourth

( 512 ]2[ ) with a falsity of +2.380, +7.781, and +5.826 cents at the higher end.86 The sléndro grave or

small fourth is the identical interval found in the five-tone equal temperament, and is produced by the

ratio 1: 25 )2( =1:1.319508. The sléndro acute or large fifth and grave or small fourth may also be

referred to as the 5-et grave or small fourth and 5-et acute or large fifth, or alternatively as the quintal

equal subfourth and quintal equal superfifth.87

84 Daniélou, Tableau Comparatif des Intervalles Musicaux 119-25.

85 Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 455.

86 Daniélou, Tableau Comparatif des Intervalles Musicaux 118-24.

87 Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 455.

Theory Versus Performance Practice 47

Table 15. Stage two of tuning process – perfect fourths

NUMBER NOTES (POSITION) RANGE (CENTS) AVERAGE (CENTS) STANDARD DEVIATION

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

C3 and F3

G3 and C4

C4 and F4(2)

C4 and F4(1)

G4 and C5

C5 and F5(3)

C5 and F5(2)

G5 and C6

C6 and F6

494.973 � 503.495

483.049 � 494.186

494.592 � 505.826

499.200 � 514.208

488.672 � 498.615

486.694 � 490.450

498.648 � 501.249

490.616 � 495.220

492.435 � 507.593

498.418

490.463

501.034

508.238

494.807

487.968

500.319

493.385

502.293

4.488880

6.420814

5.795747

7.960749

5.365009

2.149959

1.450061

2.440269

8.544883

Simultaneous sonorities based on stage two of the tuning process for the three tunings reveal tempered

perfect fourths with a mean value of 501.373 cents, a range between 488.349 and 518.700 cents, and a

standard deviation of 8.744854. The primary tempered fourths are represented by F4 (2nd position) and

BÝ4, Aî4 and Dî4, BÝ4 and EÝ5, Eî4 and Aî4, and the mean values of 494.926, 509.466, 497.494, and

495.487 cents respectively. The range of these fourths being between 488.743 and 518.700 indicates

an inclination to again (as in the primary perfect fourths) represent the sléndro grave or small fourth

( 25 ]2[ ), just perfect fourth (4/3), and equal perfect fourth ( 512 ]2[ ), but this time with a falsity of +8.098,

ß9.302, and ß11.257 cents at the lower end of the scale, while the acute or large fourth (27/20, or

519.551 cents), just perfect fourth (4/3), and equal perfect fourth ( 512 ]2[ ) with a falsity of ß0.852,

+20.655, and +18.700 cents at the higher end.

Table 16. Stage two of tuning process – tempered perfect fourths

NUMBER NOTES (POSITION) RANGE (CENTS) AVERAGE (CENTS) STANDARD DEVIATION

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

F3 and BÝ3

Aî3 and Dî4

BÝ3 and EÝ4

Eî4 and Aî4

F4(2) and BÝ4

Aî4 and Dî5

BÝ4 and EÝ5

F5(2) and BÝ5

BÝ5 and EÝ6

488.349 � 495.385

508.362 � 512.257

501.771 � 512.161

491.928 � 498.275

488.743 � 501.094

495.451 � 518.700

496.279 � 499.681

495.746 � 499.267

505.825 � 518.158

490.802

509.679

505.353

495.487

494.926

509.466

497.494

498.011

511.136

3.973019

2.232698

5.898947

3.243106

6.175730

12.340378

1.897984

1.965164

6.342067

48 Theory Versus Performance Practice

An analysis of simultaneous sonorities based on stage two of the tuning process for the three tunings on

the other hand reveal neutral thirds with a mean value of 350.057 cents, a range between 338.758 and

355.954 cents, and a standard deviation of 6.396299. The primary neutral third is represented by F4

(2nd position) and Aî4, and the mean value of 353.752 cents. The range of these thirds being between

350.940 and 355.954 indicates an inclination to represent seven equal quarter-tones (approximately

60/49, or 350.00 cents), the just major third (5/4, or 386.314 cents), and equal major third ( 3 2 ,

approximately 63/50, or 400.000 cents)88 with a falsity of +0.940, ß35.374, and ß49.060 cents at the

lower end of the scale; while an extreme grave or small major third (27/22, or 354.547 cents), just major

third (5/4), and equal major third ( 3 2 ) with a falsity of +1.407, ß30.359, and ß44.046 cents at the

higher end. Seven equal quarter-tones is the identical interval found in the twenty-four-tone equally-

tempered division of the octave, and is produced by the ratio 1: 724 )2( =1:224054.

Table 17. Stage two of tuning process – neutral thirds

NUMBER NOTES (POSITION) RANGE (CENTS) AVERAGE (CENTS) STANDARD DEVIATION

ÐÑ

ÐÒ

F3 and Aî3

F4(2) and Aî4

338.758 � 353.539

350.940 � 355.954

346.361

353.752

7.399249

2.562172

Analysis of Variance The fourth table presents an ‘analysis of variance’ with regards to the three tunings, and therefore

provides a platform for the evaluation of the tuning of the seventy-two strings of the santñr for dastgàh-e

segàh (on F) utilizing average and standard deviation criterion.89 In this case standard deviation has been

derived from cents data, and frequencies normalized to A=440Hz, although in direct relation to ratio

data, which has been obtained via the mathematical equation of 1200ïcentsÏ 24 2 =ratio.

88 Daniélou, Tableau Comparatif des Intervalles Musicaux 90-102.

89 “A computational procedure frequently used to analyze the data from an experimental study employs a

statistical procedure known as the analysis of variance. For a single-factor experiment, this procedure uses a hypothesis

test concerning equality of treatment means to determine if the factor has a statistically significant effect on the response

variable. For experimental designs involving multiple factors, a test for the significance of each individual factor as well as

interaction effects caused by one or more factors acting jointly can be made. A variety of numerical measures are used to

summarize data. The proportion, or percentage, of data values in each category is the primary numerical measure for

qualitative data. The mean, median, mode, percentiles, range, variance, and standard deviation are the most commonly

used numerical measures for quantitative data. The mean, often called the average, is computed by adding all the data

values for a variable and dividing the sum by the number of data values, while standard deviation is a measure of the

variability (dispersion or spread) of any set of numerical values about their arithmetic mean. It is specifically defined as the

square root of the arithmetic mean of the squared deviations.” For a further discussion, see Encyclopaedia Britannica,

“Statistics,” Encyclopaedia Britannica 2001, deluxe ed., CD-ROM, Chicago: Encyclopaedia Britannica, 2001.

Theory Versus Performance Practice 49

Table 18. Tuning of the santñr for dastgàh-e segàh (on F) – analysis of variance 1st position

STRING SET NOTE FREQUENCY (HERTZ) RATIO (DECIMAL) AVERAGE (CENTS) STANDARD DEVIATION

ÐÑ ÐÒ ÐÓ ÐÔ ÐÕ ÐÖ Ð× ÐØ ÐÙ

C3

F3

G3

Aî3

BÝ3

C4

Dî4

EÝ4

F4

130.368

173.862

196.163

212.369

230.848

260.408

285.069

309.099

349.261

1.993204

1.329089

1.499570

1.623461

1.764720

1.990689

1.089605

1.181457

1.334963

1194.107

492.526

701.459

838.887

983.327

1191.922

148.566

288.680

500.160

7.713846

5.999891

7.191309

1.889998

2.620925

9.337949

0.512523

8.496235

1.380069

2nd position

STRING SET NOTE FREQUENCY (HERTZ) RATIO (DECIMAL) AVERAGE (CENTS) STANDARD DEVIATION

ÑÐ ÑÑ ÑÒ ÑÓ ÑÔ ÑÕ ÑÖ Ñ× ÑØ

Eî4

F4

G4

Aî4

BÝ4

C5

Dî5

EÝ5

F5

320.470

347.810

393.173

426.662

462.912

523.251

572.642

617.019

698.585

1.224917

1.329419

1.502808

1.630811

1.769368

1.000000

1.094404

1.179203

1.335086

351.221

492.956

705.193

846.708

987.881

0.000

156.174

285.375

500.319

1.248094

4.817387

5.365009

2.660578

1.737114

0.000000

10.954514

0.273364

1.450061

3rd position

STRING SET NOTE FREQUENCY (HERTZ) RATIO (DECIMAL) AVERAGE (CENTS) STANDARD DEVIATION

ÑÙ ÒÐ ÒÑ ÒÒ ÒÓ ÒÔ ÒÕ ÒÖ Ò×

Eî5

F5

G5

AÝ5

BÝ5

C6

Dî6

EÝ6

F6

640.217

693.619

786.006

823.584

924.807

1048.400

1138.557

1242.435

1397.011

1.223537

1.325595

1.502158

1.573975

1.767425

1.001813

1.087964

1.187227

1.334933

349.270

487.968

704.444

785.295

985.979

1197.828

145.957

297.115

500.121

5.354195

2.149959

9.338180

2.027411

0.245081

6.963106

2.771441

6.147445

1.610794

50 Theory Versus Performance Practice

Utilizing the mean ratios from the ‘analysis of variance’ data, and considering solely the base pitch of

second position C5, as well as all primary derivative pitches (second position F4, G4, B"4, Aù4, Dù5, E"5,

Eù4, and third position A"5), it is possible to arrive at the ‘tuning characteristics’ personified by the three

tunings, and therefore what may be stated as being a ‘performance practice’ tuning obtained via the

intervallic analysis of Persian performer Qmars Piraglu’s instrument, which was tuned to dastgàh-e segàh by

ear, and therefore not artificially influenced by tuning devices in order to adhere to strict theoretical

schemes. Piraglu’s tuning methodology involved the obtainment of a fundamental C note calibrated

approximately to A=440Hz, which would then serve as the reference for the generation of all required

perfect fifths and fourths, as well as tempered fourths, with problem intervals such as A koron and A flat

left to the discretion of the ear and the perceived musicality of performed extracts from associated gušes

(individual pieces which make up the repertoire of a particular dastgàh).

Table 19. Dastgàh-e segàh (on F) – tuning characteristics Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE

NUMBER

NOTE APPROXIMATE INTERVAL FREQUENCY

(HERTZ)

RATIO

(DECIMAL)

AVERAGE

(CENTS)

ÐÑ ÐÒ ÐÓ ÐÔ ÐÕ ÐÖ Ð× ÐØ ÐÙ

ÐÑ

C4

D4

EÝ4

Eî4

F4

G4

Aî4

BÝ4

Bî4

C5

unison

just major tone (9th harmonic)

Pythagorean minor third, or trihemitone

neutral third

just and Pythagorean perfect fourth

just and Pythagorean perfect fifth (3rd harmonic)

grave or small major sixth

Pythagorean minor seventh

grave or small major seventh

octave

261.626

295.748

309.753

320.939

348.206

393.594

430.750

464.127

482.120

523.251

1.000000

1.130424

1.183957

1.226709

1.330933

1.504416

1.646439

1.774012

1.842785

2.000000

0.000

212.237

292.339

353.752

494.926

707.044

863.219

992.419

1058.265

1200.000

The abovementioned process consequently produces intervals approximating the just major tone (9/8),

Pythagorean minor third (32/27), neutral third (11/9), just perfect fourth (4/3), just perfect fifth (3/2), grave

or small major sixth (400/243), Pythagorean minor seventh (16/9), and grave or small major seventh

(50/27). The findings proclaim the obvious Pythagorean connection with the resulting ‘mean’

measurements for the major second (equal to 212.237 cents, with a range between and 208.821 and

214.193 cents, and a standard deviation of 2.968984); minor third (equal to 292.339 cents, with a range

between 288.626 and 299.361 cents, and a standard deviation of 6.084603); perfect fourth (equal to

494.926 cents, with a range between 488.743 and 501.094 cents, and a standard deviation of

Theory Versus Performance Practice 51

6.175730); perfect fifth (equal to 707.044 cents, with a range between 702.864 and 712.313 cents,

and a standard deviation of 4.817387); and minor seventh (equal to 992.419 cents, with a range

between 988.424 and 997.374 cents, and a standard deviation of 4.551339). The research interestingly

also suggests a distinction between the neutral orientation of microtonal inflections such as Eù4 (equal to

353.752 cents, with a range between 350.940 and 355.954 cents, and a standard deviation of

2.562172); Aù5 (equal to 863.219 cents, with a range between 846.391 and 874.654 cents, and a

standard deviation of 14.883326); and Bù4 (equal to 1058.265 cents, with a range between 1052.666

and 1064.027 cents, and a standard deviation of 5.682304); which reveal a ‘grave or small major’

intervallic orientation.

Tuning System Comparison

A tuning system comparison table incorporating data from the twenty-four equally-tempered quarter-

tone scale of Vaziri, twenty-two-note Pythagorean scale of Barkešli, and Farhat’s theory of flexible

intervals, or of the five primary intervals of performance practice; as well as the performance practice

tuning of Piraglu, reveals a close link between the latter two. Farhat and Piraglu share similarities with all

intervals (a falsity of +8.237, ß1.661, ß10.248, ß3.074, +5.044, +1.219, ß3.581, and ß7.735 cents

on each count) but for the neutral third (Eî4), which has a closer association with the equally-tempered

quarter-tone of Vaziri (a falsity of +3.752, as opposed to ß10.248). Interestingly, at 384.360 cents,

Barkešli’s neutral third represents the Pythagorean diminished fourth (8192/6561), which is 34.360,

24.360, and 30.608 cents larger than the neutral thirds of Vaziri, Farhat, and Piraglu.

Table 20. Dastgàh-e segàh (on F) – tuning system comparison

DEGREE

NUMBER

NOTE ALI NAQI VAZIRI

(CENTS)

MEHDI BARKEÊLI

(CENTS)

HORMOZ FARHAT

(CENTS)

QMARS PIRAGLU

(CENTS)

ÐÑ ÐÒ ÐÓ ÐÔ ÐÕ ÐÖ Ð× ÐØ ÐÙ

ÐÑ

C4

D4

EÝ4

Eî4

F4

G4

Aî4

BÝ4

Bî4

C5

0.000

200.000

300.000

350.000

500.000

700.000

850.000

1000.000

1050.000

1200.000

0.000

203.910

294.135

384.360

498.045

701.955

882.405

996.090

1086.315

1200.000

0.000

204.000

294.000

364.000

498.000

702.000

862.000

996.000

1066.000

1200.000

0.000

212.237

292.339

353.752

494.926

707.044

863.219

992.419

1058.265

1200.000

52 Theory Versus Performance Practice

Utilizing Farhat’s theory of the division of the whole-tone and nomenclature for Persian intervals, it is then

possible to conduct an analysis of all minor second (D and E"), small neutral tone (Eù and F, Bù and C),

large neutral tone (D and Eù, G and Aù), and major second (C and D, E" and F, F and G, B" and C)

intervals (encountered within the framework established via the base and primary derivative pitches in the

performance practice tuning of Piraglu); and consequently produce a comparison table outlining range,

as well as the average for each interval in the two supportive propositions.

Table 21. Dastgàh-e segàh (on F) – Hormoz Farhat’s and Qmars Piraglu’s division of the whole-tone

DEGREE

NUMBER

INTERVAL HORMOZ FARHAT

RANGE (CENTS)

HORMOZ FARHAT

AVERAGE (CENTS)

QMARS PIRAGLU

RANGE (CENTS)

QMARS PIRAGLU

AVERAGE (CENTS)

ÐÑ ÐÒ ÐÓ ÐÔ ÐÕ

unison

minor second

small neutral tone

large neutral tone

major second

0.000

90.000

125.000 � 145.000

150.000 � 170.000

204.000

0.000

90.000

135.000

160.000

204.000

0.000

74.433 � 85.664

135.973 � 147.334

136.747 � 162.655

200.117 � 214.193

0.000

80.102

141.454

148.845

208.631

Performance Practice and Tuning

In conclusion, it must be stated that Farhat’s theory of flexible intervals or of the five primary intervals of

performance practice certainly holds true in the final analysis, and especially in view of the fact that

although general assumptions may be reached with regards to tuning practice, there is no doubt that

standard deviation data is so conflicting in some instances (the three tunings producing an overall

standard deviation average of 3.258156, with a range between 0.245081 and 10.954514; while octaves

between 0.187035 and 13.715187; perfect fifths between 3.980471 and 9.338180; perfect fourths

between 1.450061 and 7.960749; tempered fourths between 1.897984 and 12.340378; and neutral

thirds between 2.562172 and 7.399249) that the results cannot be stated as being the axiom. Ayers’s

own tuning research into the performance practice of Persian music – utilizing “thirteen improvised

recorded examples from the mode áahàrgàh and two recorded examples from the mode êhñr” –

declare the following proposition:

“The analysis of actual performance practice in Persian music raises the issue of accuracy in acoustic

performance of any microtonal music. If pitch variation in a mode of as much as 30 cents, which is a sixth

of a tone, can be acceptable, then what place do theoretical systems have in this music? If we are

composing computer music, then we don’t have to depend on performers and we can tune the

Theory Versus Performance Practice 53

computer as accurately as we wish. If we chose a precise tuning, will it sound realistic if the pitches never

vary? Will perfect music sound natural?”90

N. A. Jairazbhoy, and A. W. Stone, in their studies of intonation in present-day North Indian classical music

found that, “It would appear that within each performance the intonation does vary, and that a variation

of as much as fifteen cents, in seconds and thirds at least, could easily pass unnoticed. This is particularly

noticeable in the series of notes taken from the upper register which suggests that there may be a

tendency towards sharpening intervals in this register.” With regards to the notion of the universal

systemization of a theoretical tuning, the following conclusions are reached:

“With this divergence between musicians (the maximum divergence noticed in the interval of the third was

between Pannalal Gosh (439 cents) and Ustàd Umaro Khan (375 cents), a difference of 64 cents, or more

than a quarter-tone), it would appear that intonation is a matter of personal choice, perhaps influenced by

the teacher’s intonation, but not bound to it, and that any intonation within certain limits (perhaps within 25

or 30 cents in either side of the tempered intonation) can be acceptable. Under these circumstances it

would seem pointless to consider applying the ancient 22 æruti system, or for that matter, any system of

exact intonation to North Indian classical music.”91

In support of these findings, in Intervals, Scales and Temperaments: An Introduction to the Study of

Musical Intonation Boyle makes the appropriate observation that “sounds should be pitched according to

the dictates of the ear. For this certain notes in the scale must be free to move about a comma

(approximately 21.506 cents), which is perfectly possible on all but keyboard instruments.” It is further

relevant to note that according to foreword contributor Kenneth Van Barthold, in the piano tuning

practice of the latter part of the twentieth century, octaves began to be often “stretched for added

brilliance,” and therefore “theoretically accurate equal temperament” has never existed in piano tuning,

with stretched octaves often induced in the highest and lowest octaves.92 Theoretically accurate octaves

or the ratio 2/1 is defined in musical acoustics by a “beat-free condition between all the partials of the

upper musical tone and the even-numbered partials of the lower tone.” Stretched, as well as

compressed octaves a common occurrence in Piraglu’s tuning of the santñr.

90 Ayers, “Exploring Microtonal Tunings: A Kaleidoscope of Extended Just Tunings and their Compositional

Applications,” 197-98.

91 N. A. Jairazbhoy, and A. W. Stone, “Intonation in Present-Day North Indian Classical Music,” Bulletin of the School

of Oriental and African Studies, University of London 26.1 (1963): 130-31.

92 Lloyd, and Boyle, Intervals, Scales and Temperaments: An Introduction to the Study of Musical Intonation ix.

54 Theory Versus Performance Practice

The Piano Tuner’s Octave and Inharmonicity The ‘piano tuner’s octave’ may be is stated as being an octave with a falsity of around three cents, or in

other words, when (for example) “the fundamental component of C5 on a good piano is set about three

cents higher than twice the fundamental component of C4.”93

Table 22. Octaves with a falsity of 3 cents

C4 (PARTIAL) FREQUENCY (HERTZ) C5 (PARTIAL) FREQUENCY (HERTZ) BEATS (HERTZ)

1

2

3

4

5

6

7

8

261.626

523.251

784.877

1046.502

1308.128

1569.753

1831.379

2093.005

– – – –

1

– – – –

2

– – – –

3

– – – –

4

– – – –

524.159

– – – –

1048.317

– – – –

1572.476

– – – –

2096.635

– – – –

0.908

– – – –

1.815

– – – –

2.723

– – – –

3.630

In a chapter concerned with tunings and temperaments Boyle adds to the discussion of theoretically

accurate octaves with the following comments:

“The tuner, who tunes to eliminate beats between the fundamental of the upper note and the second

partial tone of the lower note, will naturally stretch the octave physically, and the musical ear, not

surprisingly, seems to prefer them that way. This is simply a case of the ear asserting its right to judge –

within the limits of choice allowed to it – when such an instrument is mostly nearly in tune with itself

musically, and provides a good example of the danger of leaving out the ear and regarding any particular

musical interval as being defined primarily and absolutely by a mathematical ratio.”

The present discussion coincides with the natural phenomenon of inharmonicity, which is a factor

especially affecting plucked and struck strings (along with other musical sounds with a short decay), and

one that displaces the upper partials of a vibrating string with the harmonic series.94 The upwards slope

of inharmonicity on the pianoforte is encountered at around middle C in both directions. A reduction of

93 Benade, Fundamentals of Musical Acoustics 319.

94 “In a grand piano, the tension in each string is over 100 pounds, creating a total force on the frame of between

40,000 and 60,000 pounds. A large variation in tension between the lower and the higher strings could lead to warping

of the piano frame, so that, in order to apply even tension throughout, the higher strings are shorter and smaller in

diameter while the bass strings are constructed of heavy wire wound with additional thin wire. This construction makes

the wires stiff, causing the overtones to be higher in frequency than the ideal harmonics and leading to the slight

inharmonicity that plays an important part in the characteristic piano tone.” For a further discussion, see Encyclopaedia

Britannica, “Sound,” Encyclopaedia Britannica 2001, deluxe ed., CD-ROM, Chicago: Encyclopaedia Britannica, 2001.

Theory Versus Performance Practice 55

inharmonicity in a string may be brought about via an increase of physical length, and a decrease of

diameter, which is the reasoning behind the double and triple stringing (unison twos and threes, or

bichords and trichords) of the pianoforte. Physical length is therefore synonymous with flexibility, and

directly responsible for the consequential decrease in inharmonicity, and hence, a more well-defined

musical tone.95 The Tuning of Unisons A survey incorporating both musically trained and untrained participants conducted in 1959 by Roger Kirk

of the Baldwin Piano Company revealed the general preference for one or two cents deviation among

the strings of a triple-string unison of a piano. Although the upper threshold could have been lessened

for the musically trained, unisons with as much as eight cents deviation were accorded with wide

acceptance. Unisons with a falsity of two cents are described by Arthur H. Benade as “reasonably

smooth”, while ones with a falsity of eight cents as a “rather brighter sound, but is not yet the sort of

jangle one gets with a spread of 15 to 20 cents.” It should be pointed out that in general performance

practice it can be extremely difficult or sometimes even impossible to tune unisons to a “true zero-beat

condition,” and one must also consider the important fact that theoretical unisons will produce reduced

decay times.

Table 23. Unisons with a falsity of 2 cents

C4 (PARTIAL) FREQUENCY (HERTZ) C4 (PARTIAL) FREQUENCY (HERTZ) BEATS (HERTZ)

1

2

3

4

5

6

7

8

261.626

523.251

784.877

1046.502

1308.128

1569.753

1831.379

2093.005

1

2

3

4

5

6

7

8

261.928

523.856

785.784

1047.712

1309.640

1571.568

1833.496

2095.424

0.302

0.605

0.907

1.210

1.512

1.814

2.117

2.419

95 “This reduction is directly proportional to the fourth power of the length and to the square of the diameter,”

explains Lloyd and Boyle. “Thus, increasing the length by a 51 th, i.e. from 1 to 5

6 , reduces the inharmonicity by

( 56 ) 4 =2.07, just over two times, i.e. more than halves its previous value. Again, decreasing the diameter by a 2

1 , i.e. from

1 to 21 , reduces its inharmonicity by ( 2

1 ) 2 = 41 , i.e. quarters its previous values.” For a further discussion, see Lloyd, and

Boyle, Intervals, Scales and Temperaments: An Introduction to the Study of Musical Intonation 166-67.

56 Theory Versus Performance Practice

Table 24. Unisons with a falsity of 8 cents

C4 (PARTIAL) FREQUENCY (HERTZ) C4 (PARTIAL) FREQUENCY (HERTZ) BEATS (HERTZ)

1

2

3

4

5

6

7

8

261.626

523.251

784.877

1046.502

1308.128

1569.753

1831.379

2093.005

1

2

3

4

5

6

7

8

262.837

525.675

788.512

1051.349

1314.187

1577.024

1840.496

2102.699

1.212

2.424

3.635

4.847

6.059

7.271

9.117

9.694

The Persian santär, essentially 72 strings organized into 18 sets of quadruple-string unisons, is an

interesting case for comparison. In order to access deviation among the strings of a quadruple-string

unison of a santär, analysis of a separate performance practice tuning by Piraglu restricted to the primary

pitches of the second position strings is conducted. The results reveal unisons with a mean falsity of

4.442 cents, which is a pronouncement that coincides with a median perspective of Benade’s

“reasonably smooth” and “rather brighter” unisons with falsities of two and eight cents respectively.96

Table 25. Unisons with a falsity of 4.442 cents

C4 (PARTIAL) FREQUENCY (HERTZ) C4 (PARTIAL) FREQUENCY (HERTZ) BEATS (HERTZ)

1

2

3

4

5

6

7

8

261.626

523.251

784.877

1046.502

1308.128

1569.753

1831.379

2093.005

1

2

3

4

5

6

7

8

262.298

524.595

786.893

1049.191

1311.489

1573.786

1836.084

2098.382

0.672

1.344

2.016

2.689

3.361

4.033

4.705

5.377

The ‘unison’ experiment consists of the spectrum analysis of second position pitches Eî4, F4, G4, Aî4,

BÝ4, C5, Dî5, and EÝ5, which have all been tuned in relationship to the first string of each set. In other

words, in stage one of the tuning process of the santñr for dastgàh-e segàh (on F), second position F4 is

acquired by tuning the first ‘outside’ F4 string to the first ‘outside’ C5 string (generating a perfect fifth). The

tuning of F4 is then concluded with the generation of unisons by the three other strings of the set (2nd,

3rd, and 4th strings of the quadruple-string unison of the santär) in relationship to the first string. The

96 Benade, Fundamentals of Musical Acoustics 334-36.

Theory Versus Performance Practice 57

analysis of the eight pitches surveyed conclude the characterization of unisons with a mean value of

4.442 cents, a range between ß9.381 and +9.130 cents, and a standard deviation of 2.416688. In

consideration of beats, the results present a mean value of 1.201Hz, a range between ß1.280 and

+1.283Hz, and a standard deviation of 0.749103. Climate and Tuning Specific climatic conditions no doubt play a significant part in tuning, and therefore the important factor

of relative humidity must be taken into account.97 Piraglu makes a point of stating that he has found the

climatic conditions of Melbourne, Australia, in comparison to Tehran, Iran, as “unsatisfactory” for the tuning

of the santär. The questions therefore arise as to what are the desirable climatic conditions for the tuning

of an instrument, and how can official meteorological statistical data for climatic dissimilarities between

the two regions support Piraglu’s observations? The climate of Tehran, Iran (latitude: 35.41°N, longitude:

51.25°E, elevation: 1191m) features average minimum and maximum temperatures of 10.3°C and 22.8°C;

record minimum and maximum temperatures ranging between ß21.0°C and 43.0°C (averaging ß1.0°C

and 31.9°C); an average monthly total precipitation of 20.8mm; and average relative humidity ranging

between 59.1% in the morning to 53.3% in the afternoon (averaging 56.2%, with a total range of 5.8%).

Melbourne, Australia on the other hand (latitude: ß37.49°S, longitude: 144.58°E, elevation: 35m) features

average minimum and maximum temperatures of 9.9°C and 19.8°C; record minimum and maximum

temperatures ranging between ß3.0°C and 46.0°C (averaging 1.1°C and 34.8°C); an average monthly

total precipitation of 54.2mm; and average relative humidity ranging between 68.7% in the morning to

55.8% in the afternoon (averaging 62.2%, with a total range of 12.9%).98

The tuning of a pianoforte for example, can be affected severely by significant changes in relative

humidity over a one- or two-day period, with perpetual and excessive precipitation detrimental to the

intonation of the instrument. The ideal level of humidity is 42%, or a level within the range of 35% and

97 According to the Encyclopaedia Britannica humidity is “the amount of water vapour in the air. It is the most

variable characteristic of the atmosphere and constitutes a major factor in climate and weather. Atmospheric water

vapour is an important factor in weather for several reasons. It regulates air temperature by absorbing thermal radiation

both from the Sun and the Earth. Moreover, the higher the vapour content of the atmosphere, the more latent energy is

available for the generation of storms. In addition, water vapour is the ultimate source of all forms of condensation and

precipitation.” The article making the further point of importance that “care must be taken to distinguish between the

relative humidity of the air and its moisture content or density, known as absolute humidity.” For a further discussion, see

Encyclopaedia Britannica, “Humidity,” Encyclopaedia Britannica 2001, deluxe ed., CD-ROM (Chicago: Encyclopaedia

Britannica, 2001) n. pag.

98 “BBC Weather Centre: World Weather,” BBC Home Page, 6 Apr. 2006, British Broadcasting Corporation, 6 Apr.

2006, <http://212.58.224.86/weather/>

58 Theory Versus Performance Practice

Table 26. Tuning of the santñr for dastgàh-e segàh (on F) – relationship of unisons to first strings

PITCH STRING FREQUENCY

(HERTZ)

RATIO

(DECIMAL)

BEATS

(HERTZ)

FALSITY

(CENTS)

Eî4

F4

G4

Aî4

BÝ4

C5

Dî5

EÝ5

Ñ Ò Ó Ô Ñ Ò Ó Ô Ñ Ò Ó Ô Ñ Ò Ó Ô Ñ Ò Ó Ô Ñ Ò Ó Ô Ñ Ò Ó Ô Ñ Ò Ó Ô

322.490

323.260

322.860

321.740

351.100

351.600

351.830

350.360

396.020

395.740

394.630

393.880

428.820

429.530

429.410

430.750

466.700

467.430

464.670

464.760

529.180

527.870

527.300

528.530

572.060

574.090

572.470

573.470

622.200

625.490

623.430

621.190

1.000000

1.002388

1.001147

0.997674

1.000000

1.001424

1.002079

0.997892

1.000000

0.999293

0.996490

0.994596

1.000000

1.001656

1.001376

1.004501

1.000000

1.001564

0.995650

0.995843

1.000000

0.997524

0.996447

0.998772

1.000000

1.003549

1.000717

1.002465

1.000000

1.005288

1.001977

0.998377

0.000

0.770

0.370

0.750

0.000

0.500

0.730

0.740

0.000

0.280

1.390

2.140

0.000

0.710

0.590

1.930

0.000

0.730

2.030

1.940

0.000

1.310

1.880

0.650

0.000

2.030

0.410

1.410

0.000

3.290

1.230

1.010

+0.000

+4.129

+1.985

ß4.031

+0.000

+2.464

+3.596

ß3.653

+0.000

ß1.224

ß6.087

ß9.381

+0.000

+2.864

+2.380

+7.774

+0.000

+2.706

ß7.547

ß7.211

+0.000

ß4.291

ß6.161

ß2.128

+0.000

+6.133

+1.240

+4.262

+0.000

+9.130

+3.419

ß2.813

Theory Versus Performance Practice 59

55%. High levels of humidity, 60% and over, produce an expansion of the soundboard, greater string

tension, and therefore an increase in overall pitch. “Sticking keys, sluggish action, and rusting strings/tuning

pins are other consequences of continued high humidity,” points out Martha Beth Lewis. Low levels of

humidity, 34% and under, on the other hand produces a contraction of the soundboard, lesser string

tension, and therefore a decrease in overall pitch. “Other effects of low humidity include rattling (loose)

keys, slipping tuning pins, and cracks in the soundboard,” explains Lewis. Instruments constructed of a

soundboard of reduced density and deficient of the structural support provided by the iron plate of the

pianoforte are even more predisposed to the adverse effects of humidity.99

Meteorological statistical data reveal Melbourne as having no climactic period with the ideal level

of relative humidity within the range of 35% and 55%, while Tehran, in striking contrast, a period of seven

months between April and October. High levels of humidity, 60% and over, are prevalent in Melbourne

for ten months of the year, between February and November, while in Tehran, only for five months of the

year, between November and March. Neither region suffers from low levels of humidity of 34% and

under. The average monthly total precipitation of Melbourne exceeds that of Tehran by 33.4mm, over

two and a half times the level, and is therefore a significant divergent factor. All these factors substantiate

Piraglu’s claims about the “ideal” conditions for the tuning of the santñr in Tehran. Gušes of Dastgàh-e Segàh A prominent radif associated with Mñsà Marñfi (1889-1965) has been utilized in the organization of

modal material for the work, which subscribes to a collection of the following twenty-four gušes for

dastgàh-e segàh:

i. Mogadameh ii. Daràmad-e Avva

iii. Daràmad-e Dovvom iv. Daràmad-e Sevvom

v. Piš Zangñleh vi. Zangñleh

vii. Zangeh Šotor viii. Zàbol

ix. Zàbol (Qesmat-e Dovvom) x. Zangñleh

xi. Panjeh Mñye xii. Àvàz-e Mñye

xiii. Forñd-e Mñye xiv. Bagiye-e Zàbol

xv. Hesàr xvi. Hesàr (Qesmat-e Dovvom)

xvii. Nagmeh xviii. Hesàr (Qesmat-e Sevvom)

xix. Kerešmeh xx. Forñd-e Hesàr

99 Martha Beth Lewis, “Tuning Your Piano: Why Pianos Go Out of Tune,” Martha Beth Lewis’ Home Page, 1999, 8

Apr. 2006, <http://www.serve.com/marbeth/tune_piano.html>.

60 Theory Versus Performance Practice

xxi. Hozzàn xxii. Pas Hesàr

xxiii. Moarbad xxiv. Moxàlef

Àzàdeh for santñr and tape will incorporate six of the most prominent elements of the radif of the

dastgàh-e segàh belonging to the school of Marñfi. Pitch material from the main gušes, which include

guše-ye zàbol, mñye, moxàlef and maqlub, as well as the daràmad and forñd have been accessed and

categorized (according to Piraglu) to be then utilized in adherence to Western contemporary

compositional practices, and therefore resulting in a work that has no intention of representing Persian

classical music tradition.100 The selection of the six guše-ye of dastgàh-e segàh are represented within the

compositional framework as individual demarcations (dividing the work methodically), with composer

improvisation (sequentially exploring each guše) utilized to generate all the notated material of the work.

Ex. 6. Six of the most prominent elements of the radif of the dastgàh-e segàh First tetrachord i. Daràmad-e segàh

ii. Guêe-ye zabçl

iii. Guêe-ye mñye

Second tetrachord iv. Guêe-ye moxàlef

100 Piraglu, “The Persian Music Tradition (In Iran) under Qajar and Pahlavi Dynasties,” 41-43.

Theory Versus Performance Practice 61

v. Guêe-ye maqlub

vi. Segàh forñd

A structural scheme based on ‘golden mean’ or ‘golden section’ proportions has been incorporated in

the linear plan. The aesthetic notion of these proportions being a technique directly borrowed from art

and architecture, with its conceptual basis stating that “if the proportion of ‘ab’ to ‘bc’ is the same as the

proportion of ‘bc’ to the whole line, then ‘ac’ is segmented according to the golden mean.” The ratio

represented by this ideology is approximately 1:1.618 (or the relationship acbc

bcab � ), and manifests itself

in the work at guêe-ye moxàlef with its introduction of melodic material based on the second tetrachord

of dastgàh-e segàh (720 seconds Ï .618 = 444.96 seconds, or approximately 7:25).101

Table 27. Structural scheme

NUMBER EVENT CLOCK COUNTER

(SECONDS)

DURATION GOLDEN

MEAN

1

2

3

4

5

6

7

8

9

10

11

12

13

14

daràmad-e segàh (first tetrachord)

taknavàzi-e santñr

guêe-ye zabçl

taknavàzi-e santñr

guêe-ye mñye

taknavàzi-e santñr

guêe-ye moxàlef (second tetrachord)

taknavàzi-e santñr

guêe-ye maqlub

taknavàzi-e santñr

forñd-e segàh

taknavàzi-e santñr

fade out

end

0:00

2:50

4:35

5:40

6:20

7:00

7:25

9:10

10:15

10:55

11:20

11:45

12:00

13:00

0

170

275

340

380

420

445

550

615

655

680

705

720

780

2:50

1:45

1:05

0:40

0:40

0:25

1:45

1:05

0:40

0:25

0:25

0:15

1:00

0:00

– – – –

3 (3-1)

2 (7-1)

5 (5-3)

4 (7-3)

6 (7-5)

1 (13-1)

8 (9-7)

7 (13-7)

10 (11-9)

9 (13-9)

11 (13-11)

– – – –

– – – –

101 Stefan Kostka, Materials and Techniques of Twentieth-Century Music, 2nd ed. (Upper Saddle River, NJ: Prentice-

Hall, 1999) 150-51.

62 Theory Versus Performance Practice

Sampling of the Santñr and Vocals The following stage of the compositional process involves the assembly of the tape element of the work,

which begins with the transferral of the complete set of santñr samples (the twenty-seven individual

samples collected during the first recording session with Piraglu) into the memory of an Akai S3000XL

Midi Stereo Digital Sampler. This will provide a platform for the retuning (to a hundredth of a cent) of

each individual sample in the edit sample ‘pitch offset’ parameter window of the Akai S3000XL (to firstly

adhere to equal temperament), in order to be then readjusted (in cent increments) in the edit program

‘tune’ parameter window (enabling the capture of the frequency ratios of Barkešli’s twenty-two-note

division of the octave). The frequencies (in hertz), obtained via the spectrum analyzing module of

Adobe Audition 2.0, once converted into cents will provide all the necessary data required to offset

pitch. The alternative tuning system will service the accompanying sampled santñr – one of the two

principal tape elements of the work – and the ‘call and response’ component that will effectuate the

‘theoretical’ and ‘performance practice’ tuning comparison proposed by Àzàdeh for santñr and tape.

Performances of the ‘sampled santñr’ notated phrases of the work have been recorded on a midi

sequencer (Roland MC-500 Mark II Micro Composer), utilizing the Akai S3000XL as a sound source, with

the product finally digitally transferred to Sony Sound Forge 8.0 via optical cable (routed through the

Fostex D-160 Digital Multitrack Recorder). A modified ‘white noise’ preset of the Waves X-Noise 5.2 DX

plug-in (“an audio plug-in that intelligently learns from a section of noise, and then applies a broadband

noise reduction to eliminate background noise from any source”) is utilized to remove unwanted white

noise (threshold: ß20dB, reduction: 70%, [dynamics] attack: 30ms, release: 220ms; [high shelf]

frequency: 1415Hz, gain: +0.0dB).

Table 28. Mehdi Barkešli’s twenty-two-note division of the octave tuning matrix ‘key of F’ (Program 01)

NOTE C – – Dî E" Eî F – – G A" Aî B" – –

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +00 ß20 ß06 ß16 ß02 +00 +02 ß08 ß18 ß04 +00

Stage two of the assembly requires the recording of the recitation of poetry by Piraglu (an original poem

entitled Àzàdeh), as well as the individual syllables constituting the Farsi (Modern Persian) vocalizations of

Àzàdeh,102 which include À, zà, and deh – the three syllables sung at two distinct pitches (348.834Hz

102 “The official language of Iran is Iranian or Modern Persian (Iranian/Indo-European), which is written in Arabic

script.” For a further discussion, see Akira Nakanishi, Writing Systems of the World: Alphabets, Syllabaries, Pictograms

(Rutland, VT: Charles E. Tuttle Co., 1980) 30.

Theory Versus Performance Practice 63

and 174.417Hz, or F4 and F3). Àzàdeh includes the following four lines of text, which are presented

below in Farsi (Arabic and Roman script), together with the English translation:

Kas nadànad darde bi darmàne mà,

Nobody knows the pain inside me,

Jore Leili dar dele por êñre mà.

The pain that emanates, being away from my beloved Àzàdeh.

Hazar kardim ze haráe xalgo donyà

Because my soul is with her, I am oblivious to everybody and everything

Kajàst Àzàdeh in mahpeykare mà?

Where is my beloved Àzàdeh, who reflects the beauty of moonlight?

Àzàdeh, Qmars Piraglu

Àzàdeh, Qmars Piraglu

All the samples are recorded on the Fostex D-160 Digital Multitrack Recorder, and then processed within

Sony Sound Forge 8.0 utilizing various audio plug-ins: Waves L3 Multimaximizer 5.2 (“a multi-band peak

limiting audio plug-in”)103 is utilized for general compression (left and right input: +0.0dB, threshold:

ß2.3dB, out ceiling: ß0.3dB, release: 10.00ms); dB-L Audioware Mastering Limiter 1.05 (“an intelligent

loudness maximizer and brickwall limiter audio plug-in with dithering”),104 for general limiting (gain: 3.0dB,

release: 50ms, output: ß0.3dB, attack: hard, dither: off); while Sony Noise Reduction 2.0 (“a collection of

four professional-level sound restoration audio plug-ins: vinyl restoration, noise reduction, clipped peak

103 Waves: The World’s Leading Developer of Audio Signal Processing Software, 2006, Waves Incorporated, 15

Dec. 2006, <http://www.waves.com/>.

104 dB Audioware: Professional Audio Software, 2006, dB Audioware Limited, 15 Dec. 2006, <http://www.db-

audioware.com/>.

64 Theory Versus Performance Practice

restoration, and click and crackle removal”),105 to remove unwanted tape hiss (reduction type: mode 3,

reduce noise by: 30.0dB, noise bias: +0.0dB, attack speed: 90ms, release speed: 50ms, windowing FFT

size: 1024, windowing overlap: 67%).

The two sets of three syllables are then arranged within the digital domain as a stereo configuration

sounding a low F3-set panned left, left-right, and left-right-left; followed by a high F4-set panned right,

right-left, and right-left-right – the time points selected in accordance to the ‘golden mean’ structural

scheme and therefore acting as structural markers. An Antares Auto-tune 4.31 DX plug-in (“an audio

plug-in that corrects intonation problems in vocals or solo instruments”)106 is utilized to tune the samples

(input type: low male voice, key: F, scale: chromatic, remove: F! and E), while a Spin Audio 3D Delays

1.1 DX plug-in (“a multi-tap dimensional space delay audio plug-in that provides six independent stereo

delay lines which can be freely positioned in 3D sound field”),107 to generate simultaneous quarter-note

and eighth-note triplet delays (delay 1 [mode: ms, tempo: 120bpm, time: 375ms, feedback: 55%, filter:

off, taps: ß6.0dB]; delay 2 [mode: ms, tempo: 120bpm, time: 125ms, feedback: 65%, filter: off, taps:

ß6.0dB]; I/O [in: ß3.0dB, dry: +0.0dB, wet: ß4.0dB]).

The singular 174.417Hz À component is further developed as an F pedal point or drone in the

form of a 500ms and 250ms rhythmic pulse (at Ê=120 equal to simultaneous crotchets and quavers). A

Waves Super Tap 5.0 DX plug-in (“an audio plug-in with six taps with up to six seconds of mono or true

stereo delay, as well as independent Q10-style filtering, rotation [stereo panning], gain, and precision

time control”)108 is utilized to generate a thirteen-minute rhythmic pulse ([tempo: 120bmp, ms: 500,

modulation: off; gain: ß3.6dB]; delay 1 [gain: +0.0dB, rotation: ß45°, delay: 250ms, eq: off]; delay 2

[gain +0.0dB, rotation: ß45°, delay: 500ms, eq: off]; feedback [gain: 99%, rotation: 0°, delay: 500ms]),

while a PSP Audioware Nitro 1.0.2 DX plug-in (“a multimode filter audio plug-in with many filter types

derived from analog prototypes in addition to other useful processing blocks such as phaser, bit-

crusher/downsampler, waveshaper, and interpolated delay blocks”),109 to generate ‘state variable low-

pass’ filter sweeps throughout the sample (op 1 [frequency: 553Hz, resonance: 29.40%, level: +6.0dB];

op 2 [frequency: 790Hz, resonance: 68.97%, level: +6.0dB]; op 3 [attenuation: +0.0dB, level:

maximum]; op 4 [attenuation: +0.0dB, level: maximum]; I/O [in: +0.0dB, mix: 25.00%, out: +1.0dB]).

105 Sony Media Software: Home for Vegas, Sound Forge and Acid, 2006, Sony Corporation of America, 15 Dec.

2006, <http://www.sonymediasoftware.com/>.

106 Antares Audio Technologies, 2006, Antares Audio Technologies, 15 Dec. 2006,

<http://www.antarestech.com/>.

107 Spin Audio Software, 2006, Spin Audio Software, 15 Dec. 2006, <http://www.spinaudio.com/>.

108 Waves: The World’s Leading Developer of Audio Signal Processing Software, 2006, Waves Incorporated, 15

Dec. 2006, <http://www.waves.com/>.

109 PSP Audioware: Audio Processors and Effects Plug-ins, 2006, PSP Audioware, 15 Dec. 2006,

<http://www.pspaudioware.com/>.

Theory Versus Performance Practice 65

A secondary thirteen-minute rhythmic pulse is then created from the sequential layering of a one-

second sample (in order to duplicate sonic parameters generated by the Waves Super Tap 5.0 DX plug-

in) of the primary rhythmic pulse, which is then subjected to thirteen digital fade-ins and therefore every

‘golden mean’ point of the structural scheme (a 500ms fade-out additionally executed to smooth out the

junctions). The abovementioned PSP Audioware Nitro 1.0.2 DX plug-in setting with a slight modification

of the effect level (mix: 75.00%) is also utilized to generate ‘state variable low-pass’ filter sweeps

throughout the sample. The adoption of dissimilar effect levels (25.00% in the primary and 75.00% in

secondary rhythmic pulses), and the fact that the superimposition of the two rhythmic pulses essentially

generates a series of gradual digital cross-fades, adds up to an unbroken rhythmic pulse of continually

transforming timbral colour.

The sequential layering of a one-second sample of the secondary thirteen-minute rhythmic pulse is

then employed to create a tertiary thirteen-minute rhythmic pulse. The PSP Audioware Nitro 1.0.2 DX

plug-in is utilized to generate ‘stereo width balance’ spatial nuances (stereo field modifications) in the

sample (op 1 [width: 100.00%, balance: left/100%, level: maximum]; op 2 [width: 100.00%, balance:

right/100%, level: maximum]; op 3 [feedback: 50.00%, depth: 94.04%, level: maximum]; op 4 [pan left:

left/26.6%, pan right: right/24.6%, level: +0.0dB]; I/O [in: +2.0dB, mix: 100.00%, out: ß8.0dB]). The

resulting .wav file (similarly to the secondary rhythmic pulse) is then also subjected to thirteen digital fade-

ins. The next step involves a DSound Stomp’n Fx DN-SG1 Noise Gate DX plug-in (“a dynamics effects

processor audio plug-in whose function is to remove unwanted audio material below a certain

threshold”),110 which is utilized to suppress the sustaining quality of the sample (tone level: ß32dB,

attack: 170ms, release: 1880ms), while the Spin Audio 3D Delays 1.1 DX plug-in, to generate

simultaneous ‘high cut’ filtered multiple-time delays (delay 1 [mode: ms, tempo: 120bpm, time: 1250ms,

feedback: 3%]; delay 2 [mode: ms, tempo: 120bpm, time: 1000ms, feedback: 3%]; filter 1 and 2

[routing: out, type: low pass, frequency: 400Hz, gain: +0.0dB, q/filter slope steepness: 1.0]; delay 3

[mode: ms, tempo: 120bpm, time: 1500ms, feedback: 3%]; delay 4 [mode: ms, tempo: 120bpm, time:

1750ms, feedback: 3%]; filter 3 and 4 [routing: out, type: low pass, frequency: 200Hz, gain: +0.0dB,

q/filter slope steepness: 1.0]; taps: ß1.9dB; I/O [in: ß1.0dB, dry: +0.0dB, wet: +0.0dB]).

The final recordings are further subjected to digital sound processing modifications (namely

equalization and reverberation) at the final stages of mixing and mastering, where all the tracks (including

the principal solo santñr recording [recorded separately on the Fostex D-160 Digital Multitrack Recorder],

as well as the accompanying ‘sampled’ santñr, primary pulse, secondary pulse, tertiary pulse, vocal

samples, and recitation of poetry, which constitute the tape element of the work) are conclusively

assembled. A separate mix omitting the principal solo santñr is also produced in order to reproduce the

work in a live performance context.

110 DSound, 2005, DSound, 15 Dec. 2006, <http://www.dsound1.com/>.

66 Theory Versus Performance Practice

2. The Equally-Tempered Archetype: Exposiciones for Sampled Microtonal Schoenhut Toy Piano

Equal Temperaments

According to American composer and theorist Harry Partch,111 there are two distinct classes of equal

temperaments, with the first including “those which divide the already equal tone into further equal

parts,” while the second including “those which compress one of the Pythagorean cycles into the 2/1 to

obtain such divisions as nineteen equal degrees and fifty-three equal degrees.”112 The results of the

former being third-tones (as in eighteen-tone equal temperament), quarter-tones (as in twenty-four-tone

equal temperament), fifth-tones (as in thirty-tone equal temperament), sixth-tones (as in thirty-six-tone

equal temperament), eighth-tones (as in forty-eight-tone equal temperament), twelfth-tones (as in

seventy-two-tone equal temperament), and sixteenth-tones (as in ninety-six-tone equal temperament);

while of the latter, namely nineteen-tone, thirty-one-tone, forty-three-tone, and fifty-three-tone equal

temperaments. The superimposition of a second twelve-tone equally-tempered division of the octave a

sixth of a tone, or 33.333 cents higher, as in thirty-six-tone equal temperament, is termed by Partch as a

polypythagoreanism, due to its correlation to the Pythagorean concept of the juxtaposition of perfect

fifths.113 The general accepted abbreviation for equal temperament, or tuning system of “logarithmically

equal intervals” is ‘ET’, although some theorists subscribe to the lesser ambiguity of ‘EDO’, or equally-

divided octave, which better defines the probability, or improbability of the ‘extended’ perimeter. Dan

Streams makes the following observations: “Usually, but not always, equal temperaments assume octave-

equivalence, of which the usual 12-edo is the most obvious example. For many theorists the preferred

abbreviation or these types of temperaments is EDO, for which some other theorists substitute ED2; both

of these specify that it is the 2:1 ratio which is to be equally divided.”114

111 “Visionary composer, theorist, and creator of musical instruments, Harry Partch (1901-1974) was a leading figure

in the development of an indigenously American contemporary music. A pioneer in his explorations of new instruments

and new tunings, Partch created multimedia theatre works that combine sight and sound in a compelling synthesis. He is

acknowledged as a major inspiration to postwar experimental composers as diverse as Gyõrgy Ligeti, Lou Harrison, Philip

Glass, and Laurie Anderson, and his book Genesis of a Music, first published in 1949, is now considered a classic.” For a

further discussion, see Bob Gilmore, Harry Partch: A Biography (New Haven, CT: Yale U. Press, 1998) n. pag.

112 Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 425. 113 Read, 20th-Century Microtonal Notation 13-145.

114 Dan Stearns, “Some Thoughts on an Alternative Definition of Equal Temperament,” Kronosonic, 2006, The

International Society for Creative Guitar and String Music, 1 Aug. 2006,

<http://kronoson.station185.com/kronosonic/forum/index>.

Studies of Microtonal Equal Temperaments Mexican composer Julián Carrillo Trujillo (1875-1965) utilized quarter-tones, eighth-tones, and sixteenth-

tones in his compositions; publishing his theories under the heading of Sonido Trece or the ‘Thirteenth

Sound’ in a series of writings published in Mexico and the United States between 1927 and 1957 – the

number thirteen merely representing a system beyond that of the traditional twelve tones, and essentially

ninety-six-tone equal temperament ( 96 2 ). The pitch resources of such a system provide for 780 distinct

intervals as opposed to the 12 of twelve-tone equal temperament. “The reason for stopping with this

intervallic division was not dictated by theoretical considerations, but by a purely physical one: Carrillo

felt that the human ear could not distinguish clearly beyond 1/16th of a tone,” explains Gerald R.

Benjamin. Carrillo also introduced new or adapted instruments to produce quarter-, eighth-, and

sixteenth-tones, and devised a numerical notation system based on “relative number notation.” The

quarter-tone or “quarters of tone” system is represented thus: 0, 1/4, 2/4, 3/4, 5/4, 6/4, 7/4, 8/4, 9/4,

10/4, 11/4, 12/4, 13/4, 14/4, 15/4, 16/4, 17/4, 18/4, 19/4, 20/4, 21/4, 22/4, 23/4, and 0.115

Italian composer Ferruccio Busoni (1866-1957) on the other hand, subscribed to the theorem

pronouncing the third-tone as a product of the division of the octave into six equal tones, and the further

threefold subdivision of each equal tone, which results in an eighteen-tone scale and therefore eighteen-

tone equal temperament ( 18 2 ). In his 1911 publication of Sketch of a New Aesthetic of Music Busoni

states, “Tripartite tones are wholly independent intervals with a pronounced character, and not to be

confounded with ill-tuned semitones. They form a refinement in chromatics based, as at present

appears, on the whole-tone scale.” Other tripartite systems investigated by Busoni include sixth-tones, or

the threefold subdivision of each equal semitone, which results in a thirty-six-tone scale and therefore

thirty-six-tone equal temperament ( 36 2 ). Busoni devised a notation system for sixth-tones consisting of a

six-line staff designating “open noteheads” to the spaces and “solid noteheads” to the lines, with each

representing the tripartite divisions of diatonic and chromatic semitones respectively.116

If one were to entertain the notion that the “most important characteristic sought in a temperament

is its ability, in principle, to support both traditional tonality and atonality with intervals of greater purity

than are found in twelve-tone equal temperament,” Ramon Fuller’s A Study of Microtonal Equal

Temperaments presents a set of sound guidelines. According to Fuller, octaves classify as the most

important interval, because of their ability “to maintain audibly stable musical structures”, and therefore

‘pure’ or ‘just’ 2/1 ratios are of the outmost importance. Second in line come fifths, or 3/2s (the third

harmonic), “because of their role in musical structure and the ear’s sensitivity to mistuning”, which should

be represented within a temperament with a size ranging from 699.500 cents to 704.400 cents, or a just 115 Gerald R. Benjamin, “Julian Carrillo and ‘Sonido Trece’ (Dedicated to the Memory of Nabor Carrillo),” Anuario 3

(1967): 33-68. 116 Read, 20th-Century Microtonal Notation 95-96.

68 The Equally-Tempered Archetype

perfect fifth (3/2) with a falsity of around two cents; the ideology representing a tempered cycle of fifths

as opposed to a “mathematically exact” one, and therefore providing a link between the perfect fifth and

the octave. Major and minor thirds (resembling the just major third, with a ratio of 5/4, or 386.314; and

just minor third, with a ratio of 6/5, or 315.641 cents) become the next consideration. It is interesting to

note that although twelve-tone equal temperament qualifies on the first two counts (representing the just

perfect octave [2/1] correctly, while the just perfect fifth [3/2] with a falsity of ß1.955 cents), the

common temperament features major and minor thirds with a falsity of +13.686 and ß15.641 cents

respectively; the mean of the falsity equal to the value of 9.776 cents, and the maximum amount of error,

17.596 cents (represented by the equal minor seventh).

Table 29. Deviation of basic equally-tempered intervals from just intonation

COMPARATIVE TABLE JUST INTONATION EQUAL TEMPERAMENT

INTERVAL RATIO CENTS RATIO CENTS FALSITY (CENTS)

unison

diatonic semitone

major tone

minor third

major third

perfect fourth

tritone

perfect fifth

minor sixth

major sixth

minor seventh

major seventh

octave

1/1

16/15

9/8

6/5

5/4

4/3

45/32

3/2

8/5

5/3

9/5

15/8

2/1

0.000

111.731

203.910

315.641

386.314

498.045

590.224

701.955

813.686

884.359

1017.596

1088.269

1200.000

1.000000

1.059463

1.122462

1.189207

1.259921

1.334840

1.414214

1.498307

1.587401

1.681793

1.781797

1.887749

2.000000

0.000

100.000

200.000

300.000

400.000

500.000

600.000

700.000

800.000

900.000

1000.000

1100.000

1200.000

+0.000

ß11.731

ß3.910

ß15.641

+13.686

+1.955

+9.776

ß1.955

ß13.686

+15.641

ß17.596

+11.731

+0.000

Fuller also recommends that major thirds “should be no more than seven cents sharp in a good

microtonal temperament,” adding that “to accommodate the ear’s bias, we will require a major third to

be no more than four cents flat, for a total acceptable range from about 382.000 cents and 393.000

cents;” while allocating the rule for minor thirds that they be “no more than eight cents flat or four cents

sharp, for a total acceptable range from 308.000 cents to 320.000 cents.” In conclusion, Fuller’s eight

best temperaments (selected from all equal temperaments from 1 to 144) in ascending order include the

following: fifty-three-tone, sixty-five-tone, eighty-seven-tone, ninety-nine-tone, one hundred and six-tone,

one hundred and eighteen-tone, one hundred and thirty-tone, and one hundred and forty-tone equal

temperaments (53-et, 65-et, 87-et, 99-et, 106-et, 118-et, 130-et, and 140-et).

The Equally-Tempered Archetype 69

Table 30. Fuller’s eight best equal temperaments

TEMPERAMENT MINOR THIRD

PC NUMBER

MINOR THIRD

(CENTS)

MAJOR THIRD

PC NUMBER

MAJOR THIRD

(CENTS)

PERFECT FIFTH

PC NUMBER

PERFECT FIFTH

(CENTS)

53-et

65-et

87-et

99-et

106-et

118-et

130-et

140-et

14

17

23

26

28

31

34

37

316.981

313.846

317.241

315.152

316.981

315.254

313.846

317.143

17

21

28

32

34

38

42

45

384.906

387.692

386.207

387.879

384.906

386.441

387.692

385.714

31

38

51

58

62

69

76

82

701.887

701.538

703.448

703.030

701.887

701.695

701.538

702.857

In The Structure of Recognizable Diatonic Tunings Easley Blackwood117 presents the concept of equal

temperaments and ‘recognizable diatonic tunings’ – “those in which the perfect fifths ultimately form a

closed circle” – proposing that “in order for any array of notes to contain recognizable diatonic scales, it

is both necessary and sufficient that the array should contain seven adjacent intervals that are the same

size, and are perfect fifths within the range of recognizability.” The theorem in effect pronouncing that for

a tuning to be capable of generating recognizable diatonic scales, its intervallic boundaries must adhere

to the formula: ava 53

74 << , or 685.714 << v 720.000. The criteria for a “perfect fifth within the range

of recognizability” therefore stipulates that the interval be not smaller than four sevenths of an octave, and

not larger than three fifths of an octave, hence within the range of +18.045 cents and ß16.241 cents

from a just perfect fifth (3/2). Tunings accommodating Blackwood’s ideals include 12, 17, 19, 22, 24, 26,

27, 29, 31, 33, 34, and 36 or more equal divisions of the octave.118 Dutch musicologist Rudolf Rasch

subscribes to the modification of the formula: ava 53

74 �� (‘less than’ replaced by ‘less than or equal

to’), which has the capacity to indicate “which equal temperaments have no recognizable v, as a by-

product of showing which have more than one.”119 117 American composer-theorist Easley Blackwood, a significant proponent of the equally-tempered modus

operandi, subscribes to the notion that “extended microtonal systems based on equal-temperament tunings are as valid

acoustically and musically, and as technically challenging as fractional divisions of the octave adhering to just or mean-tone

intonational principles.” For a further discussion, see Read, 20th-Century Microtonal Notation 5.

118 Easley Blackwood, The Structure of Recognizable Diatonic Tunings (Princeton, NJ: Princeton U. Press, 1985) 221-

54.

119 Paul Rapoport, “The Structural Relationships of Fifths and Thirds in Equal Temperaments,” Journal of Music Theory

37.2 (Autumn, 1993): 359.

70 The Equally-Tempered Archetype

Nicolas Mercator’s Fifty-Three-Tone Equally-Tempered Division of the Octave According to Fuller, fifty-three-tone equal temperament ( 53 2 ) is “the most nearly ideal of all

temperaments for working with pure intervals and just scales,” noting that due to the fact that it has a

prime structure, there are no “embedded subtemperaments,” and “any one of the intervals of T53 can

be arranged in a cycle that will generate the complete set of T53 pitch-classes.”120 Nicolas Mercator

(1620-87) is acknowledged for discovering “that if the octave is divided into 53 equal intervals, 31 of

them give a very perfect fifth and 17 a very good major third.”121 Daniel James Wolf offers the following

criticism of fifty-three-tone equal temperament:

“While 53tet does provide excellent approximations of 5-limit intervals, it is problematic in at least two

ways. For one, the temperament is awkward, if not unsuitable, for the performance of existing repertoire.

If Western classical triadic tonality can be heard as premised upon having the best major third present in

the tuning system equivalent to the (octave-equivalent) sum of four consecutive perfect fifths – a properly

equally present in each of the major tuning or temperament environments used in common practice

counterpoint and harmony (meantone, well-temperaments, 12tet) – then this premise is unfulfilled by

53tet. In 53tet, which might be thought of a scale of 53 modestly tempered syntonic commas, the best

major third remains one scale step distant from the sum of four perfect fifths; as a consequence,

realizations of existing repertoire may well tend to ‘drift’ in pitch from an initial tonic by the approximate

comma interval of 1/53 octave. But, perhaps more critically in this speculative context, given the

extravagant resources required to implement 53tet in notation or instruments, it does not offer significantly

better and consistent representations of intervals beyond the 5-limit.”122

A contrasting ideology of the equally-tempered paradigm is presented by Dirk de Klerk in his study of

equal temperaments (subscribing to thirty-four and forty-six equal divisions of the octave), which

presents the following analytical commentary:

“If we ignore the seventh harmonic, the tunings with 34 and 46 divisions in the octave give results that are

feasible as in the Mercator temperament (53-et). They are also a good deal more perfect than our 12-

semitone system and also better than those of Wesley Woolhouse (19-et), Christian Huygens (31-et), and

Von Janko (41-et). There is no point preferring Mercator’s system to 34 and 46 and Janko’s to 34. If we

also include the seventh harmonic it appears that it is as feasible in the temperaments of Janko and

Mercator and in 46 as it is in that of Huygens (and Fokker), whereas in the cases of 46 and Mercator the

fifth and the minor third are better.”123

120 Ramon Fuller, “A Study of Microtonal Equal Temperaments,” Journal of Music Theory 35.1/2 (Spring-Autumn,

1991): 212-20. 121 Dirk de Klerk, “Equal Temperament,” Acta Musicologica 51.1 (Jan.-Jun., 1979): 140. 122 Daniel James Wolf, “Alternative Tunings, Alternative Tonalities,” Contemporary Music Review 22.1-2 (2003): 4. 123 Klerk, “Equal Temperament,” 150.

The Equally-Tempered Archetype 71

Tabl

e 31

. N

icol

as M

erca

tor’s

fifty

-thre

e-to

ne e

qua

lly-te

mp

ered

div

ision

of t

he o

ctav

e Re

lativ

e Pi

tch:

A4=

440H

z / C

4 (m

iddl

e C)

=26

1.62

5565

4Hz

DE

GRE

E N

UMBE

R N

OTE

IN

TERV

AL

FACT

OR

RATI

O

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

TUN

ING

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÑÓ

ÑÔ

ÑÕ

ÑÖ

Ñ×

ÑØ

ÑÙ

ÒÐ

ÒÑ

ÒÒ

ÒÓ

ÒÔ

ÒÕ

ÒÖ

C C¢

C£

CÚ

D¹

D¸

DÝ

D³

D²

D D¢

D£

DÚ

E¸

EÝ

E³

E² E E¢

E£

F³

F² F F¢

F£

FÚ

uniso

n 53

-et s

ynto

nic

com

ma

53-e

t gre

at d

iesis

53

-et g

rave

or s

mal

l jus

t chr

omat

ic s

emito

ne, o

r min

or h

alf-t

one

53-e

t Pyt

hago

rean

lim

ma

53-e

t jus

t dia

toni

c se

mito

ne, o

r maj

or h

alf-t

one

53-e

t gre

at li

mm

a, a

cute

or l

arge

hal

f-ton

e 53

-et g

rave

or s

mal

l ton

e 53

-et j

ust m

inor

tone

53

-et j

ust m

ajor

tone

53

-et a

cute

or l

arge

tone

53

-et s

uper

maj

or s

econ

d

53-e

t aug

men

ted

seco

nd

53-e

t Pyt

hago

rean

min

or th

ird, o

r trih

emito

ne

53-e

t jus

t min

or th

ird

53-e

t neu

tral t

hird

53

-et g

rave

or s

mal

l maj

or th

ird

53-e

t jus

t maj

or th

ird

53-e

t Pyt

hago

rean

maj

or th

ird, o

r dito

ne

53-e

t acu

te o

r lar

ge m

ajor

third

53

-et s

ubfo

urth

53

-et g

rave

or s

mal

l fou

rth

53-e

t jus

t and

Pyt

hago

rean

per

fect

four

th

53-e

t acu

te o

r lar

ge fo

urth

53

-et s

uper

four

th

53-e

t gra

ve o

r sm

all a

ugm

ente

d fo

urth

1/1

532

253

)2

(

353

)2

(

453

)2

(

553

)2

(

653

)2

(

753

)2

(

853

)2

(

953

)2

(

1053

)2

(

1153

)2

(

1253

)2

(

1353

)2

(

1453

)2

(

1553

)2

(

1653

)2

(

1753

)2

(

1853

)2

(

1953

)2

(

2053

)2

(

2153

)2

(

2253

)2

(

2353

)2

(

2453

)2

(

2553

)2

(

1.00

0000

1.

0131

64

1.02

6502

1.

0400

15

1.05

3705

1.

0675

77

1.08

1630

1.

0958

69

1.11

0295

1.

1249

11

1.13

9720

1.

1547

23

1.16

9924

1.

1853

25

1.20

0929

1.

2167

38

1.23

2756

1.

2489

84

1.26

5426

1.

2820

84

1.29

8961

1.

3160

61

1.33

3386

1.

3509

39

1.36

8723

1.

3867

41

261.

626

265.

070

268.

559

272.

094

275.

676

279.

305

282.

897

286.

707

290.

482

294.

306

298.

180

302.

105

306.

082

310.

111

314.

194

318.

330

322.

520

326.

766

331.

068

335.

426

339.

841

344.

315

348.

848

353.

440

358.

093

362.

807

0.00

0 22

.642

45

.283

67

.925

90

.566

11

3.20

8 13

5.84

9 15

8.49

1 18

1.13

2 20

3.77

4 22

6.41

5 24

9.05

7 27

1.69

8 29

4.34

0 31

6.98

1 33

9.62

3 36

2.26

4 38

4.90

6 40

7.54

7 43

0.18

9 45

2.83

0 47

5.47

2 49

8.11

3 52

0.75

5 54

3.39

6 56

6.03

8

+00

+

23

+45

ß

32

ß09

+

13

+36

ß

42

ß19

+

04

+26

+

49

ß28

ß

06

+17

+

40

ß38

ß

15

+08

+

30

ß47

ß

25

ß02

+

21

+43

ß

34

72 The Equally-Tempered Archetype

DEG

REE

NUM

BER

NO

TE

INTE

RVA

L FA

CTO

R RA

TIO

(D

ECIM

AL)

FR

EQUE

NCY

(H

ERTZ

) CE

NTS

TU

NIN

G

Ò×

ÒØ

ÒÙ

ÓÐ

ÓÑ

ÓÒ

ÓÓ

ÓÔ

ÓÕ

ÓÖ

Ó×

ÓØ

ÓÙ

ÔÐ

ÔÑ

ÔÒ

ÔÓ

ÔÔ

ÔÕ

ÔÖ

Ô×

ÔØ

ÔÙ

ÕÐ

ÕÑ

ÕÒ

ÕÓ ÐÑ

F¤ F¥

GÝ

G³

G²

G G¢

G£

GÚ

A¸

AÝ

A³

A²

A A¢

A£

AÚ

B¹

B¸

BÝ

B³

B² B B¢

B£

C³

C² C

53-e

t jus

t trit

one,

or a

ugm

ente

d fo

urth

53

-et a

cute

or l

arge

trito

ne, o

r aug

men

ted

four

th

53-e

t acu

te o

r lar

ge d

imin

ished

fifth

53

-et s

ubfif

th

53-e

t gra

ve o

r sm

all f

ifth

53-e

t jus

t and

Pyt

hago

rean

per

fect

fifth

53

-et a

cute

or l

arge

fifth

53

-et s

uper

fifth

53

-et a

ugm

ente

d fi

fth

53-e

t Pyt

hago

rean

min

or s

ixth

53

-et j

ust m

inor

six

th

53-e

t neu

tral s

ixth

53

-et g

rave

or s

mal

l maj

or s

ixth

53

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ust m

ajor

six

th

53-e

t Pyt

hago

rean

maj

or s

ixth

53

-et a

cute

or l

arge

maj

or s

ixth

53

-et a

ugm

ente

d s

ixth

53

-et g

rave

or s

mal

l min

or s

even

th

53-e

t Pyt

hago

rean

min

or s

even

th

53-e

t acu

te o

r lar

ge m

inor

sev

enth

53

-et n

eutra

l sev

enth

53

-et g

rave

or s

mal

l maj

or s

even

th

53-e

t jus

t dia

toni

c m

ajor

sev

enth

53

-et P

ytha

gore

an m

ajor

sev

enth

53

-et a

cute

or l

arge

maj

or s

even

th

53-e

t sub

octa

ve

53-e

t gra

ve o

r sm

all o

ctav

e

octa

ve

2653

)2

(

2753

)2

(

2853

)2

(

2953

)2

(

3053

)2

(

3153

)2

(

3253

)2

(

3353

)2

(

3453

)2

(

3553

)2

(

3653

)2

(

3753

)2

(

3853

)2

(

3953

)2

(

4053

)2

(

4153

)2

(

4253

)2

(

4353

)2

(

4453

)2

(

4553

)2

(

4653

)2

(

4753

)2

(

4853

)2

(

4953

)2

(

5053

)2

(

5153

)2

(

5253

)2

(

2/

1

1.40

4996

1.

4234

92

1.44

2231

1.

4612

16

1.48

0452

1.

4999

41

1.51

9686

1.

5396

92

1.55

9960

1.

5804

96

1.60

1302

1.

6223

82

1.64

3739

1.

6653

77

1.68

7301

1.

7095

12

1.73

2017

1.

7548

17

1.77

7918

1.

8013

23

1.82

5036

1.

8490

61

1.87

3402

1.

8980

64

1.92

3050

1.

9483

65

1.97

4014

2.00

0000

367.

583

372.

422

377.

324

382.

292

387.

324

392.

423

397.

589

402.

823

408.

126

413.

498

418.

942

424.

457

430.

044

435.

705

441.

441

447.

252

453.

140

459.

105

465.

149

471.

272

477.

476

483.

762

490.

130

496.

582

503.

119

509.

742

516.

452

52

3.25

1

588.

679

611.

321

633.

962

656.

604

679.

245

701.

887

724.

528

747.

170

769.

811

792.

453

815.

094

837.

736

860.

377

883.

019

905.

660

928.

302

950.

943

973.

585

996.

226

1018

.868

10

41.5

09

1064

.151

10

86.7

92

1109

.434

11

32.0

75

1154

.717

11

77.3

58

12

00.0

00

ß11

+

11

+34

ß

43

ß21

+

02

+25

+

47

ß30

ß

08

+15

+

42

ß40

ß

17

+06

+

28

ß49

ß

26

ß04

+

19

+42

ß

36

ß13

+

09

+32

ß

45

ß23

+00

The Equally-Tempered Archetype 73

Pietro Aron’s Quarter-Comma Meantone Tempered Division of the Octave

Before the advent of twelve-tone equal temperament in the West, Pythagorean intonation was the

predominant doctrine (“presumably from the Dark Ages to the 1200s”), although musicians eventually

became dissatisfied with the extreme sharpness of the Pythagorean thirds (the just major third [5/4] with a

falsity of +21.506 cents), and began to experiment by altering the fifths with an “indefinite amount of

diminution called temperament,” or participate.124 Historically, the first theoretical account of

temperament in the West (or quasi-Pythagorean intonation) comes during the Renaissance from Italian

music theorist and composer Franchinus Gaffurius’s (1451-1522) Practica musica, which was published in

Milan, in 1496. Meantone temperament, or the division of the just major third (5/4) into two mean tones

(equal to the meantone major tone, 9/8× 28180 , or 193.157 cents), may be attributed to Italian music

theorist and composer Pietro Aron (1489-1555), who in his 1523 treatise entitled Toscanello in musica

described a method for tempering the fifths that would in time serve as the foundation for numerous

systems of tempered intonation.125 It should be noted that as Mark Lindley rightly points out, “When

meantone systems gradually went out of fashion on keyboard instruments after c. 1650, they yielded, not

to equal temperament, but to a type of irregular system.” Irregular systems were prevalent between

1680 and c. 1800, and include ‘well temperaments’ prescribed by composers Andreas Werckmeister

(1645-1706), Jean-Philippe Rameau (1683-1764), Francesco Antonio Vallotti (1697-1780), and Muzio

Clementi (1752-1832); as well as by scientists and mathematicians Jacopo Francesco Riccati (1676-

1754), Johann Heinrich Lambert (1728-77), and Thomas Young (1773-1829). In A Venerable

Temperament Rediscovered, Douglas Leedy provides an insightful account of the rationale behind the

half-millennium domination of meantone temperament.126

“In the history of Western music, the keyboard temperament with the longest run was meantone, which

was the nearly universal standard from 1350 or so until it was almost totally eclipsed by twelve-tone equal

temperament in the first half of the nineteenth century. Meantone and the development (and ultimate

success) of keyboard instruments are inextricably intertwined: because its sonorities are close to those of

just tuning, meantone was able to emulate the sweetness of post-Pythagorean, pure triadic intonation

124 “In the visual arts, the mid- to late-15th century was a time of mathematically governed precision and elegance.

Geometrically planned perspective was introduced into painting and intarsia, and the first great masters (Brunelleschi,

Alberti, Francesco di Giorgio Martini) were architects who made remarkably sensitive use of very simple ratios in their

buildings. It was at this time that musicians, evidently alert to the distinctive sound of the nearly pure thirds in a quasi-

Pythagorean system, began to temper the fifths among the naturals so as to have unequivocally consonant thirds there as

well.” For a further discussion, see Mark Lindley, Mathematical Models of Musical Scales: A New Approach (Bonn: Verlag

für Systematische Musikwissenschaft, 1993) 134-38.

125 Barbour, Tuning and Temperament: A Historical Survey 25-28.

126 Lindley, Mathematical Models of Musical Scales: A New Approach 134-52.

74 The Equally-Tempered Archetype

characteristics of vocal music from around the time of Dunstable – the earliest pieces were in fact

transcriptions of vocal works.”

The author then continues the discussion with this evocative description of Meantone’s intervallic

properties:

“What unfortunately cannot be conveyed in words is the warm and serene beauty of meantone’s

consonant harmonies, its arresting dissonances, or the vivid colour, kaleidophonic variety, and expressive

strength of its melodic intervals. These need to be heard and savoured over time, for time is necessary to

accustom oneself to meantone’s richness and to becomes attuned to its subtlety and shading - much as

one would need time to become accustomed to the replacement of a diet of uniform blandness with the

variety of flavour, colour, and piquancy afforded by a Lucullan cuisine.”127

Pietro Aron’s quarter-comma meantone tempered division of the octave is based on the premise that the

just major tone (9/8) and just minor tone (10/9), or the sum of the two intervals (the just major third [5/4])

is reduced to the mean of these two intervals via the cumulative diminution of just perfect fifths (3/2s) by

the amount of a quarter of a syntonic comma. The syntonic comma, or comma of Didymus,128 is

represented by the ratio 81/80, or 21.506 cents (equal to the subtraction of 10/9 from 9/8, or the

difference between the two tones), and a quarter of this measurement is equal to 48081 , 1:1.003110, or

5.377 cents.129 81/80, and its reciprocal, 80/81, may also be theoretically referred to as the acute, or

comma, and grave, or hypocomma.130 The consequence of this process (the diminution of just perfect

fifths) also naturally leads to the augmentation of just perfect fourths (4/3s) by the amount of a quarter of

a syntonic comma, and ultimately to ‘true’ just major thirds (5/4s). Lloyd and Boyle offer the following

explanation: “The true major third (ratio 5/4) is the sum of a major and minor tone. Basic mean-tone

temperament therefore made all the whole tones the mean of these two intervals, half a comma smaller

than the major tone, half a comma larger than the minor tone, leaving each of the diatonic semitones (EF

and BC) too sharp by quarter comma.”131

In order to generate a meantone scale, it is first necessary to construct a Pythagorean scale with

twenty-seven distinct pitches, which are the necessary intervals to facilitate the general modulations of

127 Douglas Leedy, “A Venerable Temperament Rediscovered,” Perspectives of New Music 29.2 (Summer, 1991):

202-03.

128 “Didymus, academic philosopher of Nero’s time (ruled 54-68 A.D.) gave his name,” to the syntonic comma

(81/80), “the difference between two between-degree relationships – 9/8 and 10/9 – in his diatonic tetrachord.” For a

further discussion, see Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 368.

129 Lloyd, and Boyle, Intervals, Scales and Temperaments: An Introduction to the Study of Musical Intonation 168.

130 Alexander J. Ellis, “On the Conditions, Extent, and Realization of a Perfect Musical Scale on Instruments wit Fixed

Tones,” Proceedings of the Royal Society of London 13 (1863-64): 95.

131 Lloyd, and Boyle, Intervals, Scales and Temperaments: An Introduction to the Study of Musical Intonation 168.

The Equally-Tempered Archetype 75

Western tonal music.132 The fifteen intervals generated by an ascending series of fifths, or the pitches C, G,

D, A, E, B, F!, C!, G!, D!, A!, E!, B!, F#, C#, and G# (beginning with G (3/2) are then cumulatively reduced

by the amount of a quarter of a syntonic comma, or by ß¼, ß½, ß¾, ß1, ß1¼, ß1½, ß1¾, ß2,

ß2¼, ß2½, ß2¾, ß3, ß3¼, ß3½, and ß3¾ of a comma in each case; while the other eleven

intervals, generated by a descending series, or C, F, B", E", A", D", G", C", F", B$, E$, and A$, are

cumulatively increased by the amount of a quarter of a syntonic comma, or by +¼, +½, +¾, +1,

+1¼, +1½, +1¾, +2, +2¼, +2½, and +2¾ of a comma in each case. The abovementioned pattern

demonstrates that by a subtraction of a syntonic comma (81/64×80/81=5/4), the Pythagorean major

third (81/64) will be suitably reduced to a just major third (5/4). It may be therefore summarized that the

calculation of the factor for each meantone interval firstly involves the calculation of the reciprocal, or

multiplicative inverse of a series of one-quarter increments (¼, ½, ¾, 1, 1¼, 1½, 1¾, 2, 2¼, 2½, 2¾, 3,

3¼, 3½, and 3¾ equal to 4, 2, 1.333333, 1, 0.8, 0.666666, 0.571429, 0.5, 0.444444, 0.4, 0.363636,

0.307692, 0.285714, and 0.266666), which is then be utilized to divide the syntonic comma (81/80)

into the appropriate quarter-comma additions or subtractions that are cumulatively applied to each

Pythagorean ratio. For example: the meantone perfect fifth is equal to the first ascending Pythagorean fifth

(3/2) minus one quarter of a comma, so 3/2 must be multiplied by the fourth root of 80/81 (four being

the reciprocal of 0.25, or ¼ï1=4), which is mathematically expressed as the factor 3/2× 48180 , or the

decimal ratio 1.495349. The meantone perfect fifth (3/2× 48180 ) at 696.578 cents therefore represents

the just perfect fifth (3/2) tempered by the subtraction of a quarter of a syntonic comma (81/80), or

1:3/2× 48180 =1:1.495349 (equivalent to 696.578+5.377=701.955); while the meantone perfect fourth

(4/3× 48081 ) at 503.422 cents represents the just perfect fourth (4/3) tempered by the addition of a

quarter of a syntonic comma, or 1:4/3× 48081 =1:1.337481 (equivalent to 503.422ß5.377=498.045

cents).

The system reduced to twelve notes to the octave (being the tuning convention applied to

standard keyboard instruments) produces the following series of pitches: C, C!, D, E", E', F, F!, G, G!, A,

B", B', and C, or eight ascending fifths and three descending fifths modified by the quarter-comma

meantone process. The selection of sharps and flats represent “the chromatically altered notes used in

modal music,” and results in F!, G!, and C!, as just major thirds (5/4s) above D, E, and A respectively;

while B", and E", as just major thirds (5/4s) below D and G. “The result of tuning the black notes in this

way was to produce ‘wolves’ in the keys E" major and E major, and in all the major keys more remote,

while the minor keys of C and E were faulty,” notes Lloyd and Boyle. Major keys represented well within

this temperament include C, G, D, A, F, and B"; as well as minor keys G, D, and A,133 and hence “any keys

132 Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 434.

133 Lloyd, and Boyle, Intervals, Scales and Temperaments: An Introduction to the Study of Musical Intonation 102-

68.

76 The Equally-Tempered Archetype

beyond the six which can be got with only three sharps and two flats are very seriously out of tune, and

full of wolves.”134 The so-called ‘wolves’ are in reference to the A" major (G!, C, and E") and F minor

triads (F, G!, and C), which are severely compromised in twelve-tone meantone temperament by the

replacement of a major third, perfect fifth, and minor third with diminished fourth, diminished sixth, and

augmented second intervals. The major third, perfect fifth, and minor third each at 427.373, 737.637,

and 269.205 cents therefore represent just intervals with a falsity of +41.059, +35.683, and ß46.436

cents on each count. “That is why we find all the early music written in simple keys – they were the only

ones available on keyboard instruments,” notes N. Lindsay Norden.135 The ‘wolf’ fifth (or meantone

diminished sixth) at 737.637 cents, approximates the interval of a superfifth (192/125, or 743.014 cents),

which is significantly higher than an acute or large fifth (243/160, or 723.014 cents), and way beyond any

recognizable form of Fuller’s or Blackwood’s definition of a perfect fifth with a 720.000 cent upper limit.

The interval further represents the eleventh descending Pythagorean fifth (or Pythagorean diminished sixth)

tempered by an addition of two and three-quarter commas, or 262144/177147× 0.3636368081 =1.531237.

“The horrible effect was familiarly compared to the howling of ‘wolves’,” notes Helmholtz. “Similarly for B,

D!, and F!, it was necessary to use B, E", and F!, E" being a great diesis (128/125, or 41.059 cents) too

sharp, with similar excruciating effects.”136

It is interesting to note that according to Edward Dunne, and Mark McConnell, “the syntonic

comma is actually a much greater problem in Western music than the Pythagorean comma. To encounter

a Pythagorean comma, a piece would have to modulate through all twelve keys of the circle of fifth, but

only a few modulations bring you to the syntonic comma.” Also, the fact that the syntonic comma

(81/80) represents an equation involving three primes (81/80=1, or 2ô.5=3ô), just like 2x=3y (tripling

fifths and doubling octaves), which “will never be a scale in which all the fifths, or a complete set of fifths

and thirds, are correct,” the mathematical verity becomes apparent that “any method of constructing a

twelve-tone scale by rational numbers is doomed to inconsistency.”137

134 James Swinburne, “The Ideal Scale: Its AEtiology, Lysis and SequelAE,” Proceedings of the Musical Association,

63rd sess. (1936-1937): 39-64.

135 N. Lindsay Norden, “A New Theory of Untempered Music: A Few Important Features with Special Reference to

‘A Capella’ Music,” The Musical Quarterly 22.2 (Apr., 1936): 221.

136 Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 434-55.

137 Edward Dunne, and Mark McConnell, “Pianos and Continued Fractions,” Mathematics Magazine 72.2 (Apr.,

1999): 107-09.

The Equally-Tempered Archetype 77

Tabl

e 32

. Pi

etro

Aro

n’s

qua

rter-

com

ma

mea

nton

e te

mp

ered

div

ision

of t

he o

ctav

e Re

lativ

e Pi

tch:

A4=

440H

z / C

4 (m

iddl

e C)

=26

1.62

5565

4Hz

DEG

REE

NUM

BER

NO

TE

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

TUN

ING

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÑÓ

ÑÔ

ÑÕ

C CÚ

DÝ

CÛ

D EÞ

DÚ

EÝ E FÝ

EÚ F FÚ

GÝ

FÛ

uniso

n

mea

nton

e ch

rom

atic

sem

itone

, or m

inor

hal

f-ton

e (A

) V

II4

3 1ß

mea

nton

e m

inor

sec

ond

(D)

V41

1�

mea

nton

e d

oubl

e au

gmen

ted

oct

ave

(A)

XIV

213

ß

mea

nton

e m

ajor

tone

(A)

II21

ß

mea

nton

e d

imin

ished

third

(D)

X21

2�

mea

nton

e au

gmen

ted

sec

ond

(A)

IX41

2ß

mea

nton

e m

inor

third

(D)

III4

33�

just

maj

or th

ird (A

) IV

1ß (

5th

harm

onic

)

mea

nton

e d

imin

ished

four

th (D

) V

III2�

mea

nton

e au

gmen

ted

third

(A) X

I4

32

ß

mea

nton

e p

erfe

ct fo

urth

(D)

I41

�

mea

nton

e tri

tone

, or a

ugm

ente

d fo

urth

(A) V

I21

1ß

mea

nton

e d

imin

ished

fifth

(D) V

I21

1�

mea

nton

e d

oubl

e au

gmen

ted

four

th (A

) XI

II41

3ß

1/1

2187

/204

8×0.

5714

298180

256/

243×

0.8

8081

4782

969/

4194

304×

0.28

5714

8180

9/8×

28180

6553

6/59

049×

0.4

8081

1968

3/16

384×

0.44

4444

8180

32/2

7×1.

3333

338081

81/6

4×80

/81,

or 5

/4

8192

/656

1×0.

58081

, or 3

2/25

1771

47/1

3107

2×0.

3636

368180

4/3×

48081

729/

512×

0.66

6667

8180

1024

/729

×0.

6666

678081

1594

323/

1048

576×

0.30

7692

8180

1.00

0000

1.04

4907

1.06

9984

1.09

1830

1.11

8034

1.14

4867

1.16

8241

1.19

6279

1.25

0000

1.28

0000

1.30

6133

1.33

7481

1.39

7542

1.43

1084

1.46

0302

261.

626

273.

374

279.

935

285.

651

292.

506

299.

526

305.

642

312.

977

327.

032

334.

881

341.

718

349.

919

365.

633

374.

408

382.

052

0.00

0

76.0

49

117.

108

152.

098

193.

157

234.

216

269.

206

310.

265

386.

314

427.

373

462.

363

503.

422

579.

471

620.

529

655.

536

+00

ß24

+17

ß48

ß07

+34

ß31

+10

ß14

+27

ß38

+03

ß21

+21

ß44

78 The Equally-Tempered Archetype

DEG

REE

NUM

BER

NO

TE

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

TUN

ING

ÑÖ

Ñ×

ÑØ

ÑÙ

ÒÐ

ÒÑ

ÒÒ

ÒÓ

ÒÔ

ÒÕ

ÒÖ

Ò× ÐÑ

G AÞ

GÚ

AÝ

GÛ

A BÞ

AÚ

BÝ B CÝ

BÚ C

mea

nton

e p

erfe

ct fi

fth (A

) I

41ß

mea

nton

e d

imin

ished

six

th (D

) XI

43

2�

mea

nton

e au

gmen

ted

fifth

(A)

VIII

2ß

(25

th h

arm

onic

)

just

min

or s

ixth

(D)

IV1�

mea

nton

e d

oubl

e au

gmen

ted

fifth

(A) X

V4

33

ß

mea

nton

e m

ajor

six

th (A

) III

43

ß

mea

nton

e d

imin

ished

sev

enth

(D)

IX41

2�

mea

nton

e au

gmen

ted

six

th (A

) X21

2ß

mea

nton

e m

inor

sev

enth

(D)

II21

�

mea

nton

e m

ajor

sev

enth

(A) V

411ß

mea

nton

e d

imin

ished

oct

ave

(D) V

II4

3 1�

mea

nton

e au

gmen

ted

sev

enth

(A)

XII

3ß

octa

ve

3/2×

48180

2621

44/1

7714

7×0.

3636

368081

6561

/409

6×0.

58180

, or 2

5/16

128/

81×

81/8

0, o

r 8/5

1434

8907

/838

8608

×0.

2666

678180

27/1

6×1.

3333

338180

3276

8/19

683×

0.44

4444

8081

5904

9/32

768×

0.4

8180

16/9

×2

8081

243/

128×

0.8

8180

4096

/218

7×0.

5714

298081

5314

41/5

2428

8×0.

3333

338180

2/1

1.49

5349

1.53

1237

1.56

2500

1.60

0000

1.63

2667

1.67

1851

1.71

1975

1.74

6928

1.78

8854

1.86

9186

1.91

4046

1.95

3125

2.00

0000

391.

221

400.

611

408.

790

418.

601

427.

147

437.

399

447.

896

457.

041

468.

010

489.

027

500.

763

510.

987

523.

251

696.

578

737.

637

772.

627

813.

686

848.

676

889.

735

930.

794

965.

784

1006

.843

1082

.892

1123

.951

1158

.941

1200

.000

ß03

+38

ß27

+14

ß51

ß10

+31

ß34

+07

ß17

+24

ß41

+00

The Equally-Tempered Archetype 79

Joseph Sauveur’s Forty-Three-Tone Equally-Tempered Division of the Octave There are a variety of other meantone temperaments that require special mention, and namely the 2/7

comma temperament of Italian music theorist and composer Gioseffo Zarlino (1517-90), 1/3 comma

temperament of Spanish music theorist and organist Francisco de Salinas (1513-90), 1/5 comma

temperament of Dutch organist Abraham Verheijen (fl. 1600), 2/9 comma temperament of Italian music

theorist Lemme Rossi (1602-73), and 1/6 comma temperament of German organ builder and instrument

maker Gottfried Silbermann (1683-1753). Additionally, 1/11 comma meantone temperament serves as an

excellent approximation of twelve-tone equal temperament. 1/11of a syntonic comma (81/80) is equal to

118081 , 1:1.001130, or 1.955 cents, and results in a fifth and fourth equal to 1:3/2× 11

8180 =1:1.498307, or

700.000 cents, and 1:4/3× 118081 =1:1.334840, or 500.000 cents respectively.

Fifth-comma meantone temperament may be expressed as the mathematical equation that

produces the “first ratio for the fifth” as the “fifth root of 15:2,” or 52

15 . The fifth of a syntonic comma

(81/80) is equal to 58081 , 1:1.002488, or 4.301 cents. The fifth-comma meantone perfect fifth

(3/2× 58180 ) at 697.654 cents represents the just perfect fifth (3/2) tempered by the subtraction of a fifth

of a syntonic comma (81/80), or 1:3/2× 58180 =1:1.496278 (equivalent to 697.654+4.301=701.955);

while the fifth-comma meantone perfect fourth (4/3× 58081 ) at 502.346 cents represents the just perfect

fourth (4/3) tempered by the addition of a fifth of a syntonic comma (81/80), or

1:4/3× 58081 =1:1.336650 (equivalent to 502.346ß4.301=498.045 cents). The temperament (which

incidentally approximates the forty-three-tone equally-tempered division of the octave [ 43 2 ]) features

the “equal distortion of the fifths and the major thirds (equal to 390.615 cents); the former being one-fifth

comma flat, while the latter, sharp by the same amount.”138

French acoustician Joseph Sauveur (1653-1716) proposed a forty-three-tone equally-tempered

division of the octave (which closely approximates fifth-comma meantone temperament) in 1701, and

consequently devised a highly complex notation system based on seven diatonic notes and syllables.

Read explains: “Ut to si, were divided into six parts, which he termed mérides. Furthermore, each

mérides was divided into seven eptamérides and these in turn into ten decamérides each, resulting in an

octave comprised of 43 mérides, 301 eptamérides, and 3010 decamérides.”139 The fact that the

common logarithm of two is equal to 0.301030, means that the decamérides “unit of musical measure”

may be derived directly from this source. The 43-et perfect fifth ( 2543 ]2[ ) measures 697.674 cents; the

43-et perfect fourth ( 1843 ]2[ ), 502.326 cents; while the 43-et major third ( 1443 ]2[ ), 390.698 cents.

These measurements represent fifth-comma meantone temperament intervals with a falsity of +0.020,

ß0.020, and +0.083 cents on each count.140 138 Barbour, Tuning and Temperament: A Historical Survey 31-83.

139 Read, 20th-Century Microtonal Notation 133-34.

140 Barbour, Tuning and Temperament: A Historical Survey 122.

80 The Equally-Tempered Archetype

Tabl

e 33

. Jo

sep

h Sa

uveu

r’s fo

rty-th

ree-

tone

eq

ually

-tem

per

ed d

ivisi

on o

f the

oct

ave

Rela

tive

Pitc

h: A

4=44

0Hz

/ C4 (

mid

dle

C)=

261.

6255

654H

z

DEG

REE

NUM

BER

NO

TE

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

TUN

ING

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÑÓ

ÑÔ

ÑÕ

ÑÖ

Ñ×

ÑØ

ÑÙ

ÒÐ

ÒÑ

C D$

B#

C!

D"

E&

C#

D E$

C%

D!

E"

F$

D#

E F"

G&

E! F G$

E#

uniso

n

43-e

t dim

inish

ed s

econ

d

43-e

t dou

ble

augm

ente

d s

even

th

43-e

t chr

omat

ic s

emito

ne, o

r min

or h

alf-t

one

43-e

t min

or s

econ

d

43-e

t dou

ble

dim

inish

ed th

ird

43-e

t dou

ble

augm

ente

d o

ctav

e

43-e

t maj

or to

ne

43-e

t dim

inish

ed th

ird

43-e

t trip

le a

ugm

ente

d o

ctav

e

43-e

t aug

men

ted

seco

nd

43-e

t min

or th

ird

43-e

t dou

ble

dim

inish

ed fo

urth

43-e

t dou

ble

augm

ente

d s

econ

d

43-e

t maj

or th

ird

43-e

t dim

inish

ed fo

urth

43-e

t trip

le d

imin

ished

fifth

43-e

t aug

men

ted

third

43-e

t per

fect

four

th

43-e

t dou

ble

dim

inish

ed fi

fth

43-e

t dou

ble

augm

ente

d th

ird

1/1

432

243

)2

(

343

)2

(

443

)2

(

543

)2

(

643

)2

(

743

)2

(

843

)2

(

943

)2

(

1043

)2

(

1143

)2

(

1243

)2

(

1343

)2

(

1443

)2

(

1543

)2

(

1643

)2

(

1743

)2

(

1843

)2

(

1943

)2

(

2043

)2

(

1.00

0000

1.01

6250

1.03

2765

1.04

9547

1.06

6603

1.08

3936

1.10

1550

1.11

9450

1.13

7642

1.15

6129

1.17

4916

1.19

4009

1.21

3412

1.23

3131

1.25

3169

1.27

3534

1.29

4229

1.31

5261

1.33

6634

1.35

8355

1.38

0429

261.

626

265.

877

270.

198

274.

588

279.

051

283.

585

288.

194

292.

877

297.

636

302.

473

307.

388

312.

383

317.

460

322.

618

327.

861

333.

189

338.

603

344.

106

349.

698

355.

380

361.

155

0.00

0

27.9

07

55.8

14

83.7

21

111.

628

139.

535

167.

442

195.

349

223.

256

251.

163

279.

070

306.

977

334.

884

362.

791

390.

698

418.

605

446.

512

474.

419

502.

326

530.

233

558.

140

+00

+28

ß44

ß16

+12

+40

ß33

ß05

+23

ß49

ß21

+07

+35

ß37

ß09

+19

+47

ß26

+02

+30

ß42

The Equally-Tempered Archetype 81

DEG

REE

NUM

BER

NO

TE

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

TUN

ING

ÒÒ

ÒÓ

ÒÔ

ÒÕ

ÒÖ

Ò×

ÒØ

ÒÙ

ÓÐ

ÓÑ

ÓÒ

ÓÓ

ÓÔ

ÓÕ

ÓÖ

Ó×

ÓØ

ÓÙ

ÔÐ

ÔÑ

ÔÒ

ÔÓ ÐÑ

F!

G"

A&

F #

G

A$

F %

G!

A"

B &

G# A

B $

C &

A! B "

C $

A# B C "

D &

B ! C

43-e

t trit

one,

or a

ugm

ente

d fo

urth

43-e

t dim

inish

ed fi

fth

43-e

t dou

ble

dim

inish

ed s

ixth

43-e

t dou

ble

augm

ente

d fo

urth

43-e

t per

fect

fifth

43-e

t dim

inish

ed s

ixth

43-e

t trip

le a

ugm

ente

d fo

urth

43-e

t aug

men

ted

fifth

43-e

t min

or s

ixth

43-e

t dou

ble

dim

inish

ed s

even

th

43-e

t dou

ble

augm

ente

d fi

fth

43-e

t maj

or s

ixth

43-e

t dim

inish

ed s

even

th

43-e

t trip

le d

imin

ished

oct

ave

43-e

t aug

men

ted

six

th

43-e

t min

or s

even

th

43-e

t dou

ble

dim

inish

ed o

ctav

e

43-e

t dou

ble

augm

ente

d s

ixth

43-e

t maj

or s

even

th

43-e

t dim

inish

ed o

ctav

e

43-e

t dou

ble

dim

inish

ed s

econ

d

43-e

t aug

men

ted

sev

enth

octa

ve

2143

)2

(

2243

)2

(

2343

)2

(

2443

)2

(

2543

)2

(

2643

)2

(

2743

)2

(

2843

)2

(

2943

)2

(

3043

)2

(

3143

)2

(

3243

)2

(

3343

)2

(

3443

)2

(

3543

)2

(

3643

)2

(

3743

)2

(

3843

)2

(

3943

)2

(

4043

)2

(

4143

)2

(

4243

)2

(

2/1

1.40

2861

1.42

5658

1.44

8825

1.47

2369

1.49

6296

1.52

0611

1.54

5321

1.57

0433

1.59

5953

1.62

1888

1.64

8244

1.67

5029

1.70

2249

1.72

9911

1.75

8022

1.78

6591

1.81

5624

1.84

5128

1.87

5112

1.90

5583

1.93

6549

1.96

8019

2.00

0000

367.

024

372.

989

379.

050

385.

209

391.

469

397.

831

404.

296

410.

866

417.

542

424.

327

431.

223

438.

230

445.

352

452.

589

459.

944

467.

418

475.

014

482.

733

490.

577

498.

549

506.

651

514.

884

523.

251

586.

047

613.

953

641.

860

669.

767

697.

674

725.

581

753.

488

781.

395

809.

302

837.

209

865.

116

893.

023

920.

930

948.

837

976.

744

1004

.651

1032

.558

1060

.465

1088

.372

1116

.279

1144

.186

1172

.093

1200

.000

ß14

+14

+42

ß30

ß02

+26

ß47

ß19

+09

+37

ß35

ß07

+21

+49

ß23

+05

+33

ß40

ß12

+16

+44

ß28

+00

82 The Equally-Tempered Archetype

Origins of Equal Temperament Equal temperament, or the division of the octave into equal parts may be accredited to Prince Chu Tsai-

yü, “a sixth-generation descendant of Hung Hsi, the fourth emperor of the Ming dynasty,” who in his 1584

treatise entitled A New Account of the Science of the Pitch Pipes presented the string lengths for twelve-

tone equal temperament.141 According to Kuttner:

“Lü Hsüeh Hsin Shuo (A New Account of the Science of the Pitch Pipes) contains a complete nine-digit

monochord of equal temperament with the octave ratio 10:5 for what appears to be a calculation of string

lengths; for the lengths of pitch-pipes there monochords based on the octave ratios 100:50 and 90:45,

with four decimals, as well as tabulations of pipe diameters and circumferences on the base 100:50 with

two decimals.”

The roots of two for the string lengths in a monochord are then published in Chu Tsai-yü’s 1595-96

follow-up treatise entitled Lü Lü Ching I, and although correct to nine places, have been calculated with

the absence of logarithms. It should be noted that although “the twelfth root of two is numerically

implied as quantitative definition of the semitone in equal temperament,” in Tsai-yü’s second theoretical

work, “it is never stated explicitly as a mathematical expression.”142 Partch offers the following

commentary on the nature of Prince Chu Tsai-yü’s accomplishment:

“Of the prince’s accomplishment a contemporary modern theorist reminds us that ‘the computation would

have to begin, for certain tones, with numbers containing 108 zeros, of which the 12th root would have to

be extracted, as (Marin) Mersenne did, by taking the square root twice and then the cube root. This

lengthy and laborious procedure was followed without error.’”143

The nine-digit string lengths of Tsai-yü’s monochord depicted in the following table are not a result of

rounded off values of a ten-digit calculation, and therefore a reduction to lower terms, but the true values

of a calculation based on the hundredth millionth.144

141 Stuart Isacoff, Temperament: How Music Became a Battleground for the Great Minds of Western Civilization

(New York: Vintage, 2003) 163-66. 142 Chu Tsai-yü’s 1595-96 treatise entitled Lü Lü Ching I “contains an enormously detailed mathematical investigation

of all conceivable parameters involved in the definition of pitches in equal temperament tuning, including string lengths

and pitch-pipe dimensions, such as tube lengths, inner and outer diameters, circular surface areas, bore and volume of

pipes through three octave ranges in 9- and 10-digit tabulations,” states Fritz A. Kuttner. For a further discussion, see Fritz

A. Kuttner, “Prince Chu Tsai-Yu’s Life and Work: A Re-Evaluation of His Contribution to Equal Temperament Theory,”

Ethnomusicology 19.2 (May, 1975): 166-67.

143 Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 380-81.

144 Barbour, Tuning and Temperament: A Historical Survey 77-78.

The Equally-Tempered Archetype 83

Table 34. Chu Tsai-yü’s monochord

DEGREE

NUMBER

NOTE LENGTHS RATIO

(DECIMAL)

CENTS FALSITY

(CENTS)

ÐÑ ÐÒ ÐÓ ÐÔ ÐÕ ÐÖ Ð× ÐØ ÐÙ ÑÐ ÑÑ ÑÒ

ÐÑ

C4

C!4

D4

D!4

E4

F4

F!4

G4

G!4

A4

A!4

B4

C5

1,000,000,000

943,874,312

890,898,718

840,896,415

793,700,525

749,153,538

707,106,781

667,419,927

629,960,524

594,603,557

561,231,024

529,731,547

500,000,000

1.000000

1.059463

1.122462

1.189207

1.259921

1.334840

1.414214

1.498307

1.587401

1.681793

1.781797

1.887749

2.000000

0.000

100.000

200.000

300.000

400.000

500.000

600.000

700.000

800.000

900.000

1000.000

1100.000

1200.000

+0.000

+0.000

+0.000

+0.000

+0.000

+0.000

+0.000

+0.000

+0.000

+0.000

+0.000

+0.000

+0.000

The mathematical solution to the dilemma of the cycle of fifths, or the irresolvable succession of just

perfect fifths or 3/2s rests on the unequivocal formula, equivalent to the frequency ratio 1.059463094, or

the numerical approximation of the proportion 749:500 (derived from 750:500, which is equal to the

ratio 3/2, and refined in chapter one of Lü Lü Ching I to the proportion 749.153.538:500.000.000). The

explanation for the infinite nature of a 3/2 series is that octaves (or the ratio 2/1) are based on multiples of

two, while fifths (or the ratio 3/2), on multiples of three; and because two and three are prime numbers

(divisible only by themselves and one), presented is the mathematical verity that only powers of identical

prime numbers can be equal.145 Calculus illustrates that a series of twelve 3/2s yields the following values

in cents: 0.000, 113.685, 203.910, 317.595, 407.820, 521.505, 611.730, 701.955, 815.640, 905.865,

1019.550, 1109.775, and 1223.460. The intonational anomaly generated by the twelfth ascending fifth,

which has a ratio of 531441/524288 (1:1.013643) and measures 23.460 cents, is generally referred to as

the Pythagorean comma. In other words, “the sum of twelve 3/2s (3ñò/2ñù) is greater than seven octaves

(2÷/1ñ) by a Pythagorean comma.”146 In striking contrast, a series of twelve 749:500s yields: 0.000,

97.516, 199.290, 296.806, 398.581, 496.097, 597.871, 699.645, 797.161, 898.935, 996.451,

1098.226, and 1195.742; while a series of twelve 749.153.538:500.000.000s yields twelve equal

semitones.

145 Isacoff, Temperament: How Music Became a Battleground for the Great Minds of Western Civilization 40-170.

146 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 34.

84 The Equally-Tempered Archetype

In the West, French monk, mathematician and physicist Marin Mersenne (1588-1648) is not only

acknowledged for “the discovery of the overtone series in the natural sounds of the trumpet,” but also

for being “the first Westerner to give the correct mathematical solution for equal temperament.”147

Mersenne’s most accurate geometrical approximations are characterised by Barbour as follows:

“By the familiar Euclidean method he found the mean proportional between a line and its double,

subtracted the original line from the mean, and then subtracted this difference from the doubled line. The

length thus found was the larger of the desired means – that is, the string length for the major third.”

In mathematical terms, Mersenne’s solution is depicted via the equation (3ß 2 2 ):2, which is equal to the

string length of 0.7928930, or the decimal ratio 1.261204 for the major third. The division of the major

third into four equal semitones or “mean proportionals” produce a semitone equal to 100.440 cents in

this region, while the subsequent division of the remaining major sixth (the distance between the major

third and the octave) into eight equal semitones produce a dissimilar semitone equal to 99.780 cents.

The mean falsity of the semitones in Mersenne’s geometrical approximation for equal temperament,

therefore have a value of 0.881 cents, with a maximum error of 1.762 cents.

Table 35. Marin Mersenne’s geometrical approximation for equal temperament

DEGREE

NUMBER

NOTE LENGTHS RATIO CENTS FALSITY

(CENTS)

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÐÑ

C4

C!4

D4

D!4

E4

F4

F!4

G4

G!4

A4

A!4

B4

C5

1.000000

0.943634

0.890446

0.844670

0.792893

0.748487

0.706567

0.666996

0.629640

0.594377

0.561088

0.529664

0.500000

1.000000

1.059733

1.123033

1.190115

1.261204

1.336029

1.415293

1.499260

1.588209

1.682435

1.782251

1.887989

2.000000

0.000

100.440

200.881

301.321

401.762

501.542

601.321

701.101

800.881

900.661

1000.440

1100.220

1200.000

+0.000

+0.440

+0.881

+1.321

+1.762

+1.542

+1.321

+1.101

+0.881

+0.661

+0.440

+0.220

+0.000

147 Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 381-82.

The Equally-Tempered Archetype 85

It is then, circa 1596, that Flemish mathematician and inventor Simon Stevin (1548-1620) recognizes that

the solution to equal temperament lies in finding “eleven mean proportional parts between two and

one,”148 and hence produces “an essay containing the mathematical formulation of equal temperament as 12 2 for the first time in Western musical theory.” The essay – rediscovered and edited in 1884 by Dutch

mathematician Dr. David Bierens de Haan (1822-95), to be then published for the very first time –

presents the calculation of a monochord defining “twelve semitone values, correct to four decimal

places, as the 12 successive powers of the twelfth root of two.”149 The method adopted by Stevin

involves the calculation of degree numbers 7, 4, and 5 (or F!, D!, and E) – each subordinate to cubic

and quartic levels of mathematical complexity – with the next step requiring the strategy of proportion, or

“the rule of three”, and therefore the division of the fifth degree (7937) by the fourth (8408) to produce

the second degree (9440). “This method is much easier than to extract the roots for each individual

note, which runs into difficulties with the roots of the prime numbers, as for the notes 2, 6, 8, and 12 (C!,

F, G, and B), where the 12th root itself must be extracted,” explains Barbour. The method delivers a

mean falsity of 0.199 cents, with a maximum error of 0.391 cents.

Table 36. Simon Stevin’s monochord

DEGREE

NUMBER

NOTE LENGTHS RATIO CENTS FALSITY

(CENTS)

ÐÑ ÐÒ ÐÓ ÐÔ ÐÕ ÐÖ Ð× ÐØ ÐÙ ÑÐ ÑÑ ÑÒ

ÐÑ

C4

C!4

D4

D!4

E4

F4

F!4

G4

G!4

A4

A!4

B4

C5

10000

9440

8911

8408

7937

7493

7071

6675

6301

5945

5612

5298

5000

1.000000

1.059322

1.122209

1.189343

1.259922

1.334579

1.414227

1.498127

1.587050

1.682086

1.781896

1.887505

2.000000

0.000

99.769

199.609

300.199

400.001

499.662

600.017

699.792

799.617

900.302

1000.096

1099.776

1200.000

+0.000

ß0.231

ß0.391

+0.199

+0.001

ß0.338

+0.017

ß0.208

ß0.383

+0.302

+0.096

ß0.224

+0.000

148 Barbour, Tuning and Temperament: A Historical Survey 54-55.

149 Kuttner, “Prince Chu Tsai-Yu’s Life and Work: A Re-Evaluation of His Contribution to Equal Temperament Theory,”

Ethnomusicology 167-68.

86 The Equally-Tempered Archetype

It must be noted that in spite of the geometrical and numerical approximations of Mersenne, Stevin, and

other European theorists of the sixteenth and seventeenth centuries, it is not until 1630 that the first

printed numerical solution to equal temperament based on the theory of logarithmic computation

appears; German mathematician Johann Faulhaber (1580-1635) establishing lengths derived from the

division of a monochord with a length of the required 20,000 units in order to generate a series

legitimately expressing twelve equally-tempered semitones. The mean falsity of 0.100 cents (with a

maximum error of 0.181 cents) depicted in Faulhaber’s monochord is a direct result of the expected

error obtained via the utilization of logarithmic tables.150

Table 37. Johann Faulhaber’s monochord

DEGREE

NUMBER

NOTE LENGTHS RATIO CENTS FALSITY

(CENTS)

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÐÑ

C4

C!4

D4

D!4

E4

F4

F!4

G4

G!4

A4

A!4

B4

C5

20000

18877

17817

16817

15874

14982

14141

13347

12598

11891

11224

10594

10000

1.000000

1.059490

1.122523

1.189273

1.259922

1.334935

1.414327

1.498464

1.587554

1.681944

1.781896

1.887861

2.000000

0.000

100.045

200.095

300.096

400.001

500.124

600.139

700.181

800.166

900.156

1000.096

1100.103

1200.000

+0.000

+0.045

+0.095

+0.096

+0.001

+0.124

+0.139

+0.181

+0.166

+0.156

+0.096

+0.103

+0.000

The Twelve-Tone Equally-Tempered Division of the Octave The twelve-tone equally-tempered division of the octave is the division of the octave into twelve equal

intervals, technically referred to as tempered half-tones; the frequency ratio of each semitone therefore

mathematically representing the twelfth root of two, or in different terms, the distance between any two

tones representing twelve times the logarithm on the base of two of the frequency ratio.151 The ratio of

150 Barbour, Tuning and Temperament: A Historical Survey 54-78.

151 Olson, Music, Physics and Engineering 46-47.

The Equally-Tempered Archetype 87

the equally-tempered semitone may be expressed in mathematical terms as 1: 12 2 =1:1.059463094,152

or the fraction approximations: 1024/967 (2ñð/967) and 512/483 (2ù×3×7×23). The correct

measurements for the string lengths (based on 2,000,000 units) of the equally-tempered monochord are

depicted in the following table.153

Table 38. The equally-tempered monochord

DEGREE

NUMBER

NOTE LENGTHS RATIO CENTS FALSITY

(CENTS)

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÐÑ

C4

C!4

D4

D!4

E4

F4

F!4

G4

G!4

A4

A!4

B4

C5

2000000

1887749

1781797

1681793

1587401

1498307

1414214

1334840

1259921

1189207

1122462

1059463

1000000

1.000000

1.059463

1.122462

1.189207

1.259921

1.334840

1.414214

1.498307

1.587401

1.681793

1.781797

1.887749

2.000000

0.000

100.045

200.095

300.096

400.001

500.124

600.139

700.181

800.166

900.156

1000.096

1100.103

1200.000

+0.000

+0.000

+0.000

+0.000

+0.000

+0.000

+0.000

+0.000

+0.000

+0.000

+0.000

+0.000

+0.000

The methodology involved in the construction of a twelve-tone equally-tempered scale requires the

establishment of twelve degrees via the equation f= 12 2 ; the equation producing the figure

1.059463094 (the frequency ratio of one tempered semitone), which when multiplied by the powers of

2, 3, 4, etc., generates the ratios for the remainder of the scale degrees. The multiplication by the power

of 9 ( 912 ]2[ ) generates the frequency ratio of A=440Hz (1.68179283), or equal major sixth ( 34 ]2[ ),

while the mathematical equation of 440ï1.68179283 presents the relative frequency of middle C

(261.6255654Hz), or 1/1. Once data is available for all frequency ratios it simply becomes a matter of

the multiplication of every ratio by 261.6255654Hz (the frequency of middle C) in order to obtain the

frequency values in hertz (Hz) of every scale degree. The mathematical formula for cents is then utilized

to generate the cent values of every scale degree, although unnecessary in equal temperament because

of the obvious outcomes.

152 Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 431.

153 Daniélou, Tableau Comparatif des Intervalles Musicaux 28.

88 The Equally-Tempered Archetype

The acoustical realities of equal temperament are that although equally-tempered fifths approximate just

perfect fifths, the system delivers major thirds and minor thirds tempered by seven and eight times as

much as fifths respectively. “Lustrous and calm in their pure form, they were now slightly rough and

somewhat bland,” notes Stuart Isacoff with regards to the intonational transformation of just major thirds

(5/4s) into equal major thirds ( 3 2 ).154 The root position twelve-tone equally-tempered major triad is

represented by the ratio 1.000000:1.259921:1.498307, and the pitches C, E¢, and G; and presents the

just major third (5/4) and just perfect fifth (3/2) with a falsity of ß13.686 and +1.955 cents on each

count. G4 (391.995Hz), or the equal perfect fifth ( 712 ]2[ ), presents 0.886 beats between the third

harmonic of C4 (784.877Hz) and the second harmonic of G4 (783.991Hz), and 1.772 beats between

the sixth harmonic of C4 (1569.753Hz) and the fourth harmonic of G4 (1567.982Hz); while E¢4

(329.628Hz), or the equal major third ( 3 2 ), 10.382 beats between the fifth harmonic of C4 (1308.128Hz) and the fourth harmonic of E¢4 (1318.510Hz).

Table 39. The beating characteristics of the twelve-tone equally-tempered major triad

C4

(PARTIAL)

FREQUENCY

(HERTZ)

E¢4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

G4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

5

6

– – – –

7

– – – –

8

261.626

– – – –

– – – –

523.251

– – – –

784.877

– – – –

1046.502

– – – –

1308.128

1569.753

– – – –

1831.379

– – – –

2093.005

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

– – – –

5

– – – –

6

– – – –

– – – –

329.628

– – – –

– – – –

659.255

– – – –

988.883

– – – –

– – – –

1318.510

– – – –

1648.138

– – – –

1977.765

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

10.382

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

4

– – – –

– – – –

5

– – – –

– – – –

– – – –

391.995

– – – –

– – – –

783.991

– – – –

– – – –

1175.986

– – – –

1567.982

– – – –

– – – –

1959.977

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

0.886

– – – –

– – – –

– – – –

– – – –

1.772

– – – –

– – – –

– – – –

– – – –

The root position twelve-tone equally-tempered minor triad is represented by the ratio 1.000000:

1.189207:1.498307, and the pitches C, E¸, and G; and presents the just minor third (6/5) and just

perfect fifth (3/2) with a falsity of ß15.641 and +1.955 cents on each count. G4, or the equal perfect

154 Isacoff, Temperament: How Music Became a Battleground for the Great Minds of Western Civilization 118.

The Equally-Tempered Archetype 89

fifth ( 712 ]2[ ), presents 0.886 beats between the third harmonic of C4 and the second harmonic of G4,

and 1.772 beats between the sixth harmonic of C4 and the fourth harmonic of G4; while E¸4

(311.127Hz), or the equal minor third ( 4 2 ), 14.118 beats between the sixth harmonic of G4 and the

fifth harmonic of EÝ4 (1555.635Hz).

Table 40. The beating characteristics of the twelve-tone equally-tempered minor triad

C4

(PARTIAL)

FREQUENCY

(HERTZ)

E¸4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

G4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

– – – –

5

6

7

– – – –

– – – –

8

261.626

– – – –

– – – –

523.251

– – – –

784.877

– – – –

1046.502

– – – –

– – – –

1308.128

1569.753

1831.379

– – – –

– – – –

2093.005

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

– – – –

5

– – – –

6

– – – –

– – – –

– – – –

311.127

– – – –

– – – –

622.254

– – – –

933.381

– – – –

– – – –

1244.508

– – – –

1555.635

– – – –

1866.762

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

14.118

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

– – – –

4

– – – –

– – – –

5

– – – –

– – – –

– – – –

391.995

– – – –

– – – –

783.991

– – – –

– – – –

1175.986

– – – –

– – – –

1567.982

– – – –

– – – –

1959.977

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

0.886

– – – –

– – – –

– – – –– –

– –

– – – –

1.772

– – – –

– – – –

– – – –

– – – –

The following series of tables depict the beating characteristics of the thirds, sixths, fifths, and fourths of

twelve-tone equal temperament with reference to the tuning scheme of the pianoforte. The ‘tuning

beats’ column represents the approximation of beats per second that a piano tuner must consider. The

tuning methodology begins with the tuning of the first fifth, or F3 and C4 (middle C), with 0.591 (or 0.6)

beats; and is followed by the tuning of the first fourth, G3 and C4, with 0.886 (or 0.9) beats. “Leave the F

slightly sharp, so that it sounds as a very slow wow – wow trailing off to nothing,” explains Ian McCombie.

“This interval should beat just over one beat in two seconds. It is the only note in the tempered scale

which is sharp. The G is left slightly flat, and should beat about beat per second flat.” The process

continues, to include a selection of thirds, sixths, fifths, and fourths, as well as relevant octave (2/1)

relationships.155

155 Ian McCombie, The Piano Handbook (London: David & Charles, 1980) 89-95.

90 The Equally-Tempered Archetype

Table 41. Piano tuning – twelve-tone equal temperament (thirds)

NUMBER LOWER NOTE 5TH PARTIAL (HERTZ) UPPER NOTE 4TH PARTIAL (HERTZ) BEATS (HERTZ) TUNING BEATS

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

F3

FÚ3

G3

GÚ3

A3

AÚ3

B3

C4

873.071

924.986

979.989

1038.262

1100.000

1165.409

1234.708

1308.128

A3

AÚ3

B3

C4

CÚ4

D4

DÚ4

E4

880.000

932.328

987.767

1046.502

1108.731

1174.659

1244.508

1318.510

6.929

7.341

7.778

8.241

8.731

9.250

9.800

10.382

7.0

7.5

8.0

8.5

9.0

9.5

10.0

10.5 Table 42. Piano tuning – twelve-tone equal temperament (sixths)

NUMBER LOWER NOTE 5TH PARTIAL (HERTZ) UPPER NOTE 3RD PARTIAL (HERTZ) BEATS (HERTZ) TUNING BEATS

ÐÑ

ÐÒ

ÐÓ

ÐÔ

F3

FÚ3

G3

GÚ3

873.071

924.986

979.989

1038.262

D4

DÚ4

E4

F4

880.994

933.381

988.883

1047.685

7.924

8.395

8.894

9.423

8.0

8.5

9.0

9.5 Table 43. Piano tuning – twelve-tone equal temperament (fifths)

NUMBER LOWER NOTE 3RD PARTIAL (HERTZ) UPPER NOTE 2ND PARTIAL (HERTZ) BEATS (HERTZ) TUNING BEATS

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

F3

G3

A3

FÚ3

GÚ3

AÚ3

523.842

587.993

660.000

554.992

622.957

699.246

C4

D4

E4

CÚ4

DÚ4

F4

523.251

587.330

659.255

554.365

622.254

698.456

0.591

0.664

0.745

0.626

0.703

0.789

0.6

0.7

0.8

0.6

0.7

0.8 Table 44. Piano tuning – twelve-tone equal temperament (fourths)

NUMBER LOWER NOTE 4TH PARTIAL (HERTZ) UPPER NOTE 3RD PARTIAL (HERTZ) BEATS (HERTZ) TUNING BEATS

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

G3

A3

B3

FÚ3

GÚ3

AÚ3

783.991

880.000

987.767

739.989

830.609

932.328

C4

D4

E4

B3

CÚ4

DÚ4

784.877

880.994

988.883

740.825

831.548

933.381

0.886

0.994

1.116

0.836

0.938

1.053

0.9

1.0

1.1

0.8

1.0

1.0

The Equally-Tempered Archetype 91

Exposiciones for Sampled Microtonal Schoenhut Toy Piano Exposiciones for Sampled Microtonal Schoenhut Toy Piano is an ‘acousmatic’156 work that attempts to

explore the equally-tempered sound world within the context of a sampled microtonal Schoenhut

model 6625, 25-key toy piano and a complex polyrhythmic scheme. All equal temperaments between

one and twenty-four – essentially functioning as tuning modulations – as well as all polyrhythms (divisible

only by 1 and including their inversions) between the ranges of 2 and 15 are presented. In other words,

polyrhythmic ratios 3:2Ê (2:3Ê�), 5ç:2Ê (2É:5Ê�), 4:3Ê (3:4�), 5:3Ê (3:5�), and so on – fifty-seven polyrhythmic

sets in total, with the last set represented by 15:14Ê (14:15Ê) – alongside two complementary scales

(Indonesian pélog and sléndro forms with primary and secondary scale tones, as well as primary and

secondary auxiliary tones) shaped via microtonal inflections produced by sequential tuning modulations

featuring the first twenty-four equally-tempered divisions of the octave.

The work was especially composed for ‘The Extensible Toy Piano Project’ – a toy piano festival and

symposium hosted by Clark University, Department of Visual and Performing Arts, in Worcester,

Massachusetts, USA between Friday and Saturday, November 4 and 5, 2005.

“It has a deceptively simple mechanism – plastic hammers hitting steel rods. Yet, the toy piano produces a

rich and quirky sound palette. John Cage brought the instrument from a treasured plaything to a bona fide

musical instrument with his Suite for Toy Piano (1948). Our aim is to bring the instrument into the 21st

Century.” 157

A Brief History of the Toy Piano The history of the toy piano (Fr. piano jouet; Ger. Spielzeugklavier; It. pianino-giocattolo) begins as

Troiger’s ‘Stahlklavier’ (Dessau, 1792) and Franz Schuster’s ‘Adiaphonon’ (Vienna, c. 1818), to be later

developed in Philadelphia, USA, in 1872, where German immigrant Albert Schoenhut ultimately conceives

156 According to Francis Dhomont, “The term Acousmatic Music (or Art) designates works that have been

composed for loudspeakers, to be heard in the home – on radio or on CD/tape – or in concert, through the use of

equipment (digital or analog) that allows the projection of sound in 3-dimensional space.” For a further discussion see,

Francis Dhomont, “Acousmatic Update,” Contact! 8.2 Spring, 1995, CEC – Communauté Électroacoustique Canadienne /

Canadian Electroacoustic Community, 27 Jan. 2006, <http://cec.concordia.ca/contact/contact82Dhom.html>. A further

description of the term ‘acousmatic’ is offered in an interview with François Bayle by Sandra Desantos: “In acousmatic

music, one may recognize the sound sources, but one also notices that they are out of their usual context. In the

acousmatic approach, the listener is expected to reconstruct an explanation for a series of sound events, even if this

explanation is provisional.” For a further discussion see, Sandra Desantos, “Acousmatic Morphology: An Interview with

François Bayle,” Computer Music Journal 21.3 (Fall, 1997): 17.

157 The Extensible Toy Piano Project, ed. David Claman and Matt Malsky, 1 Jan. 2005, Clark U., Worcester, MA, 21

August 2005, <http://www.clarku.edu/xtp/xtp.html>.

92 The Equally-Tempered Archetype

the child’s toy that in time will also capture the imagination of the modern composer. The instrument is

usually made out of wood or plastic, and is dimensionally less than fifty centimetres in width, with a range

between two diatonic and three chromatic octaves. It has a simple sounding mechanism (similar to that

of the full-sized keyboard glockenspiel) consisting of plastic hammers operated via a keyboard, which

strike fixed metal plates or steel rods. Traditionally, toy pianos were modelled on uprights, but following

the 1950s grand piano varieties were commonplace.

Fig. 3. Matt Malsky, Schoenhut Model 6625: 25-Key Toy Piano.158

Contemporary works that have incorporated the toy piano include John Cage’s Suite for Toy Piano

(1948) and George Crumb’s Ancient Voices of Children (1970), as well as other works by Renaud

Gagneux, Mauricio Kagel, Louis Roquin, Zygmunt Krauze, and Leonid Aleksandrovich, among many.

Internationally acclaimed concert pianist Margaret Leng Tan made her debut on the toy piano in 1993 at

New York’s Lincoln Centre, and went on to introduce the model 6625, 25-key Schoenhut Traditional

Spinet to Carnegie Hall in 1997; also releasing a compact disc entitled The Art of the Toy Piano in that

same year – a collection of works by Stephen Montague, John Lennon and Paul McCartney, Toby

Twining, Jed Distler, Philip Glass, David Lang, Julia Wolfe, Ludwig van Beethoven, Guy Klucevsek, Raphael

Mostel, and Erik Satie.159

158 Matt Malsky, “Schoenhut Model 6625: 25-Key Toy Piano,” Feb. 2005, The Extensible Toy Piano Project, ed.

David Claman and Matt Malsky, 1 Mar. 2005, Clark U., Worcester, MA, 21 Aug. 2005,

<http://www.clarku.edu/xtp/photo.html>. 159 Hugh Davies, “Toy Piano,” The New Grove Dictionary of Music and Musicians, ed. Stanley Sadie and John Tyrrell,

2nd ed., vol. 12 (London: Macmillan Reference, 2001) 615.

The Equally-Tempered Archetype 93

The Schoenhut Toy Piano Sample The Schoenhut model 6625, 25-key toy piano samples (recorded in dead studio space [96kHz/24bit]

by engineer John Shirley at Clark University, Department of Visual and Performing Arts, in Worcester, MA,

USA, utilizing two Neumann TLM 103s [positioned front-L and back-R] and a Nuendo recorder) include

three sets of twenty-five (two-octave chromatic span) forte ( � ), mezzo forte ( � ), and piano ( � )

samples, as well as one corresponding set of keyboard release clicks. All these sounds multi-sampled on

an Akai S3000XL Midi Stereo Digital Sampler – tuned firstly to standard A=440Hz twelve-tone equal

temperament, modified within thirty velocity cross-faded patches, and then operated via a midi

sequencer; a gong-like detuned middle C (octave down) sounding the tonal centre, as well as marking

the downbeat, while another severely gated alternative providing the rhythmic pulse. The final Schoenhut

Toy Piano sample may therefore be characterized as comprising of six basic sample elements (three

dynamic ranges, keyboard clicks, gong, and counter).160 The musical rationale of the gong and counter

has a close association with that of the gong and kempli in the gamelan music of Indonesia. Michael

Tenzer explains the significance of the gong in gamelan music:

“The most important function of the gongs in gamelan music is to mark structural points in a composition.

The number of gongs employed for this purpose depends on the ensemble. Gong kebyar uses one or

two gong ageng (the largest and deepest) for the beginnings and endings of melodies and other strong

accents. If two, they are used in alternation and never together. Other divisions are the responsibility of

the medium-sized kempur, the small, chiming kemong, and the nearly ever-present kempli, whose clear, dry

sound taps out the steady beat on which all musicians depend when the rhythms get tricky. Other types

of gamelans make use of some of these gongs, plus others like the tiny kelenang, the flat-bossed kajar and

bebende, or a resonant version of the kempli called tawa-tawa.”161

The elements then undergoing further processing within a digital audio editor PC software environment

(Sony Sound Forge 8.0). A modified ‘white noise’ preset of the Waves X-Noise 5.2 plug-in is utilized to

remove unwanted noise from all the samples (threshold: ß20.0dB, reduction: 70%; [dynamics] attack:

30ms, release: 220ms; [high shelf] frequency: 1415Hz, gain: +0.0dB), while the DSound Stomp’n Fx DN-

160 The Schoenhut model 6625, 25-key toy piano features are as follows: “25-key two-octave designer spinet; age

range: 3 and up; chromatically tuned with lovely chime like notes produced by little hammers striking precision ground,

German steel music rods; play-by-colour teaching method makes playing simple and fun; removable colour strip fits

behind the keys to guide small fingers from chord to chord; helps to build your child’s confidence and develop basic

playing skills; comes with a songbook which contains a collection of familiar tunes; available in mahogany, white, or red

finish; dimensions: 19¾” highÏ10¼” deepÏ17” wide; matching bench: 9¼” highÏ6” deepÏ10” wide; weighs 17 lbs.”

For a further discussion, see Schoenhut Toy Piano Company: Over 130 Years, 2005, Schoenhut Toy Piano Company, 21

Aug. 2005, <http://www.toypiano.com/6625.htm>.

161 Michael Tenzer, Balinese Music (Singapore: Periplus Editions, 1991) 37.

94 The Equally-Tempered Archetype

SG1 Noise Gate DX plug-in, to gate any remaining unwanted sonic material from the keyboard clicks and

counter samples (tone level: ß45dB, attack: 0.0ms, release: 185ms). Sound Diffusion The optimum method of performance for Exposiciones for Sampled Microtonal Schoenhut Toy Piano is

7-channel sound diffusion, with a scheme designed around the speaker system available in the

performance space of the Razzo Recital Hall at Clark University, Department of Visual and Performing Arts,

in Worcester, MA, USA, although the work may be performed utilizing a simple two-channel system (CD

player). The system incorporates seven EAW speakers in various dimensions: three on-stage large soffit-

mounted speakers (left front = 1, centre front = 2, right front = 3); plus four other smaller ‘left and right

side sets’ soffit-mounted speakers (right side = 4, left side = 7); as well as left and right rear sets (left rear

= 6, right rear = 5). Sound diffusion organized in such a manner as to sonically delineate the various

elements of the toy piano samples: toy piano (1 = 75.0%, 3 = 100.0%, 4 = 12.5%, 7 = 50.0%); key

clicks (1 = 100.0%, 3 = 75.0%, 4 = 50.0%, 7 = 12.5%); gong (2 = 100.0%); downbeat counter (4 =

50.0%, 5 = 100.0%, 6 = 100.0%, 7 = 50.0%); and upbeat counter (1 = 50.0%, 3 = 50.0%, 4 =

100.0%, 5 = 50.0%, 6 = 50.0%, 7 = 100.0%).

Fig. 4. Sound diffusion matrix (Razzo Recital Hall, Clark University, Worcester, MA, USA)

The Equally-Tempered Archetype 95

Polyrhythmic Theory The New Harvard Dictionary of Music defines a polyrhythm as being “the simultaneous use of two or

more rhythms that are not readily perceived as deriving from one another or as simple manifestations of

the same meter; sometimes also cross-rhythm.” The most common examples of this phenomena is

illustrated via the juxtaposing of triple and duple subdivisions of the beat, or alternatively of 3/4 and 6/8

meters. In classical music, the simple polyrhythmic technique is termed ‘hemiola’.162 According to Kevin

Barrett, a better understanding of polyrhythms can be achieved via their conceptualization as rhythmic

ratios, and therefore “three notes of equal value over a pulse of two notes” interpreted as a “ratio of

three notes to two notes, or 3:2;” or the inverted alternative of “two notes of equal value over a pulse of

three notes” interpreted as a “ratio of two notes to three notes, or 2:3.” Barrett establishes the criteria for

a true polyrhythm as being when the two rhythms in question do not have a common divisor other than

one, and therefore excluding ratios such as 4:2 and 2:4.163

American guitarist Steve Vai provides a thorough discussion on the technique required to perform

complex polyrhythmic ratios in Little Black Dots: Tempo Metal (an online article discussing his 1978

transcriptions for American composer, guitarist, singer, film director, and satirist, Frank Zappa [1940-

93]).164 In the article, 5ç:2Ê (transcribed as five quavers in the time of two crotchets) is utilized as an

example to demonstrate that the “first number (5) shows the number of beats to be superimposed over

the space provided,” while the “second number (2) designates the number of beats upon which the first

number is to be superimposed.” It is understood therefore, that by finding a common denominator for

the two (in this case the number ten, and therefore sixteenth notes), and “subdividing and putting five

units of measurements on both sides of the beat, you can now see that the second beat will fall on the

upstroke of the third eighth note of the quintuplet.” The following example demonstrates how the 5:2Ê polyrhythmic subdivision, which has the common denominator of ten, is subdivided into five groups of

semiquavers in order to delineate crotchet pulses.

162 Randel, ed., The New Harvard Dictionary of Music 646.

163 Kevin Barrett, “Understanding Polyrhythms,” Funkster’s Groove Theory, 23 Mar. 2004, 18 Jan. 2006,

<http://www.funkdrums.com/polyrhythm.html>. 164 “With more than 60 albums to his credit, composer, arranger, guitarist, and bandleader Frank Zappa

demonstrated a mastery of pop idioms ranging from jazz to rock of every conceivable variety, penned electronic and

orchestral works, parlayed controversial satire, and testified in Congress against censorship. As astute an entrepreneur as

he was a musician, he was impatient with any division between popular and high art; he combined scatological humour

with political wit, required of his players (Little Feat founder Lowell George, guitarists Adrian Belew and Steve Vai, and

drummer Terry Bozzio among them) an intimidating skill, and displayed consistent innovation in instrumental and studio

technology.” For a further discussion, see Patricia Romanowski, Holly George-Warren, and Jon Pareles, ed., The New

Rolling Stone Encyclopedia of Rock & Roll (New York: Fireside, 1995) 1111.

96 The Equally-Tempered Archetype

Ex. 7. 5:2Ê Polyrhythmic subdivision

The second example demonstrates how the 7ç:3Ê polyrhythmic subdivision (transcribed as seven

quavers in the time of three crotches), which has the common denominator of twenty-one, is subdivided

into seven groups of semiquaver triplets in order to delineate crotchet pulses.165

Ex. 8. 7ç:3Ê Polyrhythmic subdivision

The organizational framework of the ‘polyrhythm 1-16 set’ scheme utilized in the work is based on the

proposition of including all the fifty-seven polyrhythmic sets available between the ranges of two and

fifteen, and therefore one-hundred-and-fourteen unique polyrhythms. All the polyrhythms have been

organized in ascending order of complexity, as well as ‘golden mean’ arch form defined polyrhythmic

density, with equal temperaments one to four allocated one polyrhythmic set each (3:2Ê [2:3Ê]; 5ç:2Ê

[2É:5Ê]; 4:3Ê� [3:4�]; 5:3Ê� [3:5�]); five to eight, two (7ç:2Ê [2É�:7Ê�] and 5:4Ê [4:5Ê]; 7ç:3Ê [3É�:7Ê] and 9è:2Ê

[2È:9Ê]; 8ç:3Ê [3É:8Ê] and 7:4Ê [4:7Ê]; 6:5Ê [5:6Ê] and 7:5Ê�� [5:7Ê]; nine to twelve, three (11è:2Ê [2È:11Ê],

10ç:3Ê [3É:10Ê], and 9ç:4Ê [4É:9Ê]; 8:5Ê [5:8Ê], 7:6Ê [6:7Ê], and 11ç:3Ê [3É:11Ê]; 9:5Ê [5:9Ê], 13è:2Ê

[2È:13Ê], and 11ç:4Ê [4É:11Ê]; 8:7Ê [7:8Ê], 13è:3Ê [3È:13Ê], and 11ç:5Ê [5É:11Ê]); thirteen to sixteen, four

(9:7Ê [7:9Ê], 15è:2Ê [2È:15Ê], 14è:3Ê [3È:14Ê], and 13ç:4Ê [4É:13Ê]; 12ç:5Ê [5É:12Ê], 11:6Ê [6:11Ê], 10:7Ê

[7:10Ê], and 9:8Ê [8:9Ê]; 13ç:5Ê [5É:13Ê], 11:7Ê [7:11Ê], 15ç:4Ê [4É:15Ê], and 14ç:5Ê [5É:14Ê]; 13ç:6Ê

[6É:13Ê], 12:7Ê [7:12Ê], 11:8Ê [8:11Ê], and 10:9Ê [9:10Ê]); seventeen to nineteen, three (13:7Ê [7:13Ê],

11:9Ê [9:11Ê], and 13:8Ê [8:13Ê]; 11:10Ê [10:11Ê], 15ç:7Ê [7É:15Ê], and 13:9Ê [9:13Ê]; 15:8Ê [8:15Ê], 14:9Ê

[9:14Ê], and 13:10Ê [10:13Ê]); twenty to twenty-two, two (12:11Ê [11:12Ê] and 13:11Ê [11:13Ê]; 14:11Ê

[11:14Ê] and 13:12Ê [12:13Ê]; 15:11Ê [11:15Ê] and 14:13Ê [13:14Ê]); while the final two, twenty-three to

twenty-four, one each (15:13Ê [13:15Ê]; 15:14Ê [14:15Ê]). A further aspect of the tripartite organizational

framework is the lower limit imposed to the units of time: 2, or half note (1-et to 9-et); 4, or quarter note

(10-et to 17-et); and 8, or eighth note (18-et to 24-et).

165 Steve Vai, “Little Black Dots: Tempo Metal,” The Official Steve Vai Website, 1983, 18 Jan. 2006,

<http://www.vai.com/LittleBlackDots/tempomental.html>

The Equally-Tempered Archetype 97

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�) (5

:7�)

(3É:1

0Ê)

(6:7

Ê) (2

È:13Ê

)

(3È:1

3Ê)

(2È:1

5Ê)

(6:1

1Ê)

(7:1

1Ê)

(7:1

2Ê)

(9:1

1Ê)

(7É:1

5Ê)

(9:1

4Ê)

(11:

13Ê)

(12:

13Ê)

(13:

14Ê)

9ç:4

Ê 11

ç:3Ê

11ç:

4Ê

11ç:

5Ê

14è:

3Ê

10:7

Ê 15

ç:4Ê

11:8

Ê 13

:8Ê

13:9

Ê 13

:10Ê

(4É:9

Ê) (3

É:11Ê

)

(4É:1

1Ê)

(5É:1

1Ê)

(3È:1

4Ê)

(7:1

0Ê)

(4É:1

5Ê)

(8:1

1Ê)

(8:1

3Ê)

(9:1

3Ê)

(10:

13Ê)

13ç:

4�

9:8Ê

14ç:

5Ê

10:9

Ê

(4É:1

3Ê)

(8:9

Ê) (5

É:14Ê

)

(9:1

0Ê)

98 The Equally-Tempered Archetype

Alain Daniélou’s Scale of Proportions The intervallic structure of the Indonesian pélog and sléndro pentatonic scales utilized in the work have

been defined by the ratios of just intonation, or the ‘scale of proportions’ (the harmonic division of the

octave), as presented by Daniélou in Music and the Power of Sound: The Influence of Tuning and Interval

on Consciousness. The ‘harmonic division of the octave’ is based on the harmonic series, and is a

division of the octave into fifty-three distinct intervals. It is a scale of just intonation, where the intervals

are called ‘pure’ or ‘just’ because there are no beats between the notes or their harmonics. The quarter-

tone (three-quarter-tone) is a result of the further division of the disjunctions of this scale (just diatonic

semitone, or major half-tone [16/15]), which altogether presents a total of sixty-six unique intervals (the

octave included). In Indian musical theory this system is referred to as the ‘sixty-six ærutis’ – a theoretical

extension to the system of the ‘twenty-two ærutis’ of contemporary performance practice.166 It must be

further noted that the ‘scale of proportions’ (the modal or harmonic division of the octave) is based on

the ‘modal’ musical model, and therefore in principal serves a selection of intervals with primary

relationships to a fixed tonic. Daniélou explains the significance of the ‘modal’ musical application of

intervals:

“Indian deæå music is essentially modal, which means that the intervals on which the musical structure is built

are calculated in relation to a permanent tonic. This does not mean that the relations between sounds

other than the tonic are not considered, but that each note will be established first according to its relation

to the fixed tonic and not, as in the case of the cycle of fifths, by any permutation of the basic note. The

modal structure can thus be compared to the proportional division of a straight line rather than to the

periodic movement of a spiral. According to the symbolism of numbers, the proportional divisions are

connected with certain ideas, forms, and emotions.”

With regards to the nature of harmonic science, Daniélou adds the following:

“The object of harmonic science is to classify these proportions according to their symbolism and the

feelings, images, or symbols they express. Only on this basis can modes be logically constructed and their

expression precisely defined. All the notes obtained in the harmonic system are distinct from those of the

cyclic system, which is based on different data. Yet though the notes are theoretically distinct and their

sequence follows completely different rules, in practice they lead to a similar division of the octave into

fifty-three intervals.”167

166 “Æruti means ‘to hear’ or ‘that which is heard’. Musically, it points to the interval, between notes, which can be

just perceived auditorily. Musically viable pitches in an octave are literally infinite: this was recognized and explicitly stated

so in our ancient texts. But for practical reasons twenty-two have been enumerated and distributed within the span Sa to

Sa’ (C to C’). Further, like the seven notes, they have been given names and divided into five classes based, purportedly,

on their aesthetic connotations.” For a further discussion, see B. Chaitanya Deva, Indian Music (New Delhi: Indian Council

for Cultural Relations, 1974) 29-30.

167 Daniélou, Music and the Power of Sound: The Influence of Tuning and Interval on Consciousness 91-145.

The Equally-Tempered Archetype 99

Tabl

e 45

. A

lain

Dan

iélo

u’s

scal

e of

pro

por

tions

(th

e ha

rmon

ic d

ivisi

on o

f the

oct

ave)

Re

lativ

e Pi

tch:

A4=

440H

z / C

4 (m

iddl

e C)

=26

1.62

5565

4Hz

DE

GRE

E N

UMBE

R N

OTE

IN

TERV

AL

RATI

O

(FRA

CTIO

N)

RATI

O

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

TUN

ING

ÐÑ

ÐÒ

ÐÓ ÐÔ

ÐÕ

ÐÖ

Ð× ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ ÑÓ

ÑÔ

ÑÕ ÑÖ

Ñ×

ÑØ

ÑÙ

ÒÐ ÒÑ

ÒÒ

ÒÓ

ÒÔ

ÒÕ ÒÖ

C C¢

C£

CË /

Dì

CÚ

D¹

D¸

DÝ

CÍ /

Dí

D³

D²

D D¢

D£

DË /

Eì

DÚ

E¸

EÝ

DÍ /

Eí

E³

E² E E¢

E£

EË /

Fí

F³

F² F F¢

F£

FË / G

ì FÚ

uniso

n sy

nton

ic c

omm

a gr

eat d

iesis

G

reek

enh

arm

onic

or s

eptim

al q

uarte

r-ton

e gr

ave

or s

mal

l jus

t chr

omat

ic s

emito

ne, o

r min

or h

alf-t

one

Pyth

agor

ean

limm

a ju

st d

iato

nic

sem

itone

, or m

ajor

hal

f-ton

e gr

eat l

imm

a, a

cute

or l

arge

hal

f-ton

e th

ree-

qua

rter-t

one

grav

e or

sm

all t

one

just

min

or to

ne

just

maj

or to

ne (

9th

harm

onic

) ac

ute

or la

rge

tone

su

per

maj

or s

econ

d

five

qua

rter-t

ones

au

gmen

ted

seco

nd (7

5th

harm

onic

) Py

thag

orea

n m

inor

third

, or t

rihem

itone

ju

st m

inor

third

se

ven

qua

rter-t

ones

ne

utra

l thi

rd

grav

e or

sm

all m

ajor

third

ju

st m

ajor

third

(5t

h ha

rmon

ic)

Pyth

agor

ean

maj

or th

ird, o

r dito

ne (8

1st h

arm

onic

) ac

ute

or la

rge

maj

or th

ird

nine

qua

rter-t

ones

su

bfou

rth

grav

e or

sm

all f

ourth

ju

st a

nd P

ytha

gore

an p

erfe

ct fo

urth

ac

ute

or la

rge

four

th

sup

erfo

urth

el

even

qua

rter-t

ones

gr

ave

or s

mal

l aug

men

ted

four

th

1/1

81/8

0 12

8/12

5 31

/30

25/2

4 25

6/24

3 16

/15

27/2

5 13

5/12

4 80

0/72

9 10

/9

9/8

256/

225

5904

9/51

300

93/8

0 75

/64

32/2

7 6/

5 75

/62

8000

/656

1 10

0/81

5/

4 81

/64

32/2

5 31

/24

125/

96

320/

243

4/3

27/2

0 51

2/37

5 62

/45

25/1

8

1.00

0000

1.

0125

00

1.02

4000

1.

0333

33

1.04

1667

1.

0534

98

1.06

6667

1.

0800

00

1.08

8710

1.

0973

94

1.11

1111

1.

1250

00

1.13

7778

1.

1510

53

1.16

2500

1.

1718

75

1.18

5185

1.

2000

00

1.20

9677

1.

2193

26

1.23

4568

1.

2500

00

1.26

5625

1.

2800

00

1.29

1667

1.

3020

83

1.31

6872

1.

3333

33

1.35

0000

1.

3653

33

1.37

7778

1.

3888

89

261.

626

264.

896

267.

905

270.

346

272.

527

275.

622

279.

067

282.

556

284.

834

287.

106

290.

695

294.

329

297.

672

301.

145

304.

140

306.

592

310.

075

313.

951

316.

483

319.

007

322.

995

327.

032

331.

120

334.

881

337.

933

340.

658

344.

527

348.

834

353.

195

357.

206

360.

462

363.

369

0.00

0 21

.506

41

.059

56

.767

70

.672

90

.225

11

1.73

1 13

3.23

8 14

7.14

3 16

0.89

7 18

2.40

4 20

3.91

0 22

3.46

3 24

3.54

5 26

0.67

7 27

4.58

2 29

4.13

5 31

5.64

1 32

9.54

7 34

3.30

1 36

4.80

7 38

6.31

4 40

7.82

0 42

7.37

3 44

3.08

1 45

6.98

6 47

6.53

9 49

8.04

5 51

9.55

1 53

9.10

4 55

4.81

2 56

8.71

7

+00

+

22

+41

+

57

ß29

ß

10

+12

+

33

+47

ß

39

ß18

+

04

+23

+

44

+61

ß

25

ß06

+

16

+30

ß

57

ß35

ß

14

+08

+

27

+43

ß

43

ß23

ß

02

+20

+

39

+55

ß

31

100 The Equally-Tempered Archetype

DEG

REE

NUM

BER

NO

TE

INTE

RVA

L RA

TIO

(F

RACT

ION

) RA

TIO

(D

ECIM

AL)

FR

EQUE

NCY

(H

ERTZ

) CE

NTS

TU

NIN

G

Ò×

ÒØ

ÒÙ ÓÐ

ÓÑ

ÓÒ

ÓÓ

ÓÔ ÓÕ

ÓÖ

Ó× ÓØ

ÓÙ

ÔÐ

ÔÑ

ÔÒ ÔÓ

ÔÔ

ÔÕ

ÔÖ Ô×

ÔØ

ÔÙ

ÕÐ

ÕÑ ÕÒ

ÕÓ ÐÑ

F ¤ F¥

GÝ

FÍ /

Gí

G³

G²

G G¢

G£

GË /

Aì

GÚ

A¸

AÝ

GÍ /

Aí

A³

A²

A A¢

A£

AË /

Bì

AÚ

B¹

B¸

BÝ

AÍ /

Bí

B³

B² B B¢

B£

BË /

Cí

C³

C² C

just

trito

ne, o

r aug

men

ted

four

th (

45th

har

mon

ic)

acut

e or

larg

e tri

tone

, or a

ugm

ente

d fo

urth

ac

ute

or la

rge

dim

inish

ed fi

fth

thirt

een

qua

rter-t

ones

su

bfift

h gr

ave

or s

mal

l fift

h ju

st a

nd P

ytha

gore

an p

erfe

ct fi

fth (

3rd

har

mon

ic)

acut

e or

larg

e fif

th

sup

erfif

th

fifte

en q

uarte

r-ton

es

augm

ente

d fi

fth (

25th

har

mon

ic)

Pyth

agor

ean

min

or s

ixth

ju

st m

inor

six

th

seve

ntee

n q

uarte

r-ton

es

neut

ral s

ixth

gr

ave

or s

mal

l maj

or s

ixth

ju

st m

ajor

six

th

Pyth

agor

ean

maj

or s

ixth

(27t

h ha

rmon

ic)

acut

e or

larg

e m

ajor

six

th

nine

teen

qua

rter-t

ones

au

gmen

ted

six

th

grav

e or

sm

all m

inor

sev

enth

Py

thag

orea

n m

inor

sev

enth

ac

ute

or la

rge

min

or s

even

th

twen

ty-o

ne q

uarte

r-ton

es (

29th

har

mon

ic)

neut

ral s

even

th

grav

e or

sm

all m

ajor

sev

enth

ju

st d

iato

nic

maj

or s

even

th (

15th

har

mon

ic)

Pyth

agor

ean

maj

or s

even

th

acut

e or

larg

e m

ajor

sev

enth

tw

enty

-thre

e q

uarte

r-ton

es

subo

ctav

e (1

25th

har

mon

ic)

grav

e or

sm

all o

ctav

e

octa

ve

45/3

2 64

/45

36/2

5 90

/62

375/

256

40/2

7 3/

2 24

3/16

0 19

2/12

5 31

/20

25/1

6 12

8/81

8/

5 50

/31

81/5

0 40

0/24

3 5/

3 27

/16

128/

75

31/1

8 12

5/72

22

5/12

8 16

/9

9/5

29/1

6 40

00/2

187

50/2

7 15

/8

243/

128

48/2

5 60

/31

125/

64

160/

81

2/

1

1.40

6250

1.

4222

22

1.44

0000

1.

4516

13

1.46

4844

1.

4814

81

1.50

0000

1.

5187

50

1.53

6000

1.

5500

00

1.56

2500

1.

5802

47

1.60

0000

1.

6129

03

1.62

0000

1.

6460

91

1.66

6667

1.

6875

00

1.70

6667

1.

7222

22

1.73

6111

1.

7578

13

1.77

7778

1.

8000

00

1.81

2500

1.

8289

89

1.85

1852

1.

8750

00

1.89

8438

1.

9200

00

1.93

5484

1.

9531

25

1.97

5309

2.00

0000

367.

911

372.

090

376.

741

379.

779

383.

241

387.

593

392.

438

397.

344

401.

857

405.

520

408.

790

413.

433

418.

601

421.

977

423.

833

430.

659

436.

043

441.

493

446.

508

450.

577

454.

211

459.

889

465.

112

470.

926

474.

196

478.

510

484.

492

490.

548

496.

680

502.

321

506.

372

510.

987

516.

791

52

3.25

1

590.

224

609.

776

631.

283

645.

188

660.

896

680.

449

701.

955

723.

014

743.

014

758.

722

772.

627

792.

180

813.

686

827.

592

835.

193

862.

852

884.

359

905.

865

925.

418

941.

126

955.

031

976.

537

996.

090

1017

.596

10

29.5

77

1045

.256

10

66.7

62

1088

.269

11

09.7

75

1129

.328

11

43.2

33

1158

.941

11

78.4

94

12

00.0

00

ß10

+

10

+31

+

45

ß39

ß

20

+02

+

23

+43

+

59

ß27

ß

08

+14

+

28

ß75

ß

37

ß16

+

06

+25

+

41

ß45

ß

23

ß04

+

18

+30

ß

55

ß33

ß

12

+10

+

29

+43

ß

41

ß22

+00

The Equally-Tempered Archetype 101

Notation for the Twenty-Four Equal Temperaments

The notation adopted in Exposiciones for Sampled Microtonal Schoenhut Toy Piano (inspired by

Daniélou’s work) is highly illustrative of the affects of each individual equal temperament on the two

pentatonic scales, and their consequential intervallic deviation from just intonation. It is based on

approximations of the harmonic division of the octave, to the closest syntonic comma (81/80), and

Daniélou’s subdivision of the whole-tone, or just major tone (9/8).

Table 46. Alain Daniélou’s subdivision of the whole-tone

NOTE INTERVAL RATIO (FRACTION) CENTS

C

C¢

C£

CË / Dì

CÚ

D¹

D¸

DÝ

CÍ / Dí

D³

D²

D

unison

syntonic comma

great diesis

Greek enharmonic or septimal quarter-tone

grave or small just chromatic semitone

Pythagorean limma

just diatonic semitone

great limma

three-quarter-tone

small tone

just minor tone

just major tone (9th harmonic)

1/1

81/80

128/125

31/30

25/24

256/243

16/15

27/25

135/124

800/729

10/9

9/8

0.000

21.506

41.059

56.767

70.672

90.225

111.731

133.238

147.143

160.897

182.404

203.910

The system of accidentals utilizes twenty-three unique symbols – three identical to conventional sharp

(Ú), flat (Û), and natural (Ö) accidentals (raising, lowering or neutralizing a tone by 25/24, or 70.672 cents),

with an additional four derived from standard quarter-tone notation in twentieth century contemporary

music practice; the latter representative of the division of the ‘unequal’ major half-tone, or just diatonic

semitone (16/15), and not of the ‘equal’ quarter-tone (1:1.029302237, or 50.000 cents) derived from

twenty-four-tone equal temperament ( 24 2 ). The quarter-tone symbols include (Ë) and (í), raising or

lowering a tone by 31/30 (Greek enharmonic or septimal quarter-tone), or 56.767 cents; and (Í) and

(ì), raising or lowering a tone by 135/124 (three-quarter-tone), or 147.143 cents. Additional symbols

include (¢, ¦, ¤, ¨) and (², ¶, ´, ¸), raising or lowering an unaltered, naturalized, sharpened, or flattened

tone by 81/80 (one syntonic comma), or 21.506 cents; and (£, §, ¥, ©) and (³, ·, μ, ¹), raising or

lowering an unaltered, naturalized, sharpened, or flattened tone by 128/125 (approximately two syntonic

commas, or one great diesis), or 41.059 cents. It should be noted that two syntonic commas is equal to

the ratio 6561/6400, and 43.013 cents.

102 The Equally-Tempered Archetype

Paul Rapoport has devised a viable alternative for notating equal temperaments based on the expansion

of Pythagorean notation, limited to the fifth harmonic. Nevertheless, this approach was not adopted in

the work as it would not have provided a platform for comparative analysis of the twenty-four equal

temperaments. In spite of this very fact, the system of notation was extensively explored and in view of

its virtues, deserves a mention.

In simple terms, Rapoport’s system of notation is based on firstly acknowledging the octave, or the

symbol a, and establishing the number of units and size of an individual unit (in 12-et, twelve units equal

to a hundred cents per unit, or 1200÷12=100). The next task then becomes to identify three key

elements: the fifth, or v closest to just (in 12-et equal to seven units, or 700 cents); major third, or t

closed to just (in 12-et equal to four units, or 400 cents); and the Pythagorean major third, or �,

generated via the formula �=4vß2a (in 12-et also equal to four units, or 400 cents). It is then simply a

matter of further establishing the existence of the following intervals, and setting a unit where appropriate:

i. syntonic comma (81/80), or k (�ßt, or 4vßtß2a)

ii. Pythagorean comma (531441/524288), or p (3�ßa, or 12vß7a)

iii. great diesis (128/125, or 41.059 cents), or d (aß3t, or 3kßp)

iv. skhisma (32805/32768, or 1.955 cents), or s (2�+tßa, or pßk) v. diaskhisma (2048/2025, or 19.553 cents), or q (aß[�ß2t] or 2kßp)

vi. Pythagorean and just major tone (9/8), or �w (2vßa)

vii. Pythagorean limma (256/243), or �h (3aß5v)

viii. Pythagorean apotome (2187/2048), or ! (7vß4a)

ix. just minor third (6/5), or jm3 (vßt)

x. Pythagorean minor third, or trihemitone (32/27, or 294.135 cents), or �m3 (v�)

xi. just diatonic semitone (16/15), or jh (aßvßt)

xii. grave or small just chromatic semitone (25/24), or jc (2tßv)

In 12-et, only �w, �h, !, jm3, �m3, jh, and jc present a positive result, with 2, 1, 1, 3, 3, 1, and 1 units

respectively; and therefore the Pythagorean apotome (2187/2048), or !, is selected as the appropriate

notational device. The paper concludes that the “most important comma for notation is k (syntonic

comma), followed in order by d (great diesis), q (diaskhisma), and s (skhisma).” Rapoport’s system of

notation certainly has its merits when dealing with one single equal temperament at a time, although in the

context of tuning modulations, presented is the dilemma of having to decipher seventy-two unique

symbols for chromas such as the syntonic comma (81/80), Pythagorean comma (531441/524288), great

The Equally-Tempered Archetype 103

diesis (128/125), skhisma (32805/32768), and diaskhisma (2048/2025), along with conventional quarter-

sharp and flat, sharp and flat, and three-quarter-sharp and flat symbols.168

The ‘notation for the twenty-four equal temperaments’ table presents all the possible pitch

allocations within the octave, and represents the scheme utilized in the work for the notation of all equal

temperament between one and twenty-four. The system has been adopted for practicality, although it is

able to approximate the intervals to the closest syntonic comma, and therefore useful when making

comparisons between one equal temperament and another. Sléndro and Pélog Scales Sléndro and pélog scales represent the two genera (the feminine and masculine genus) of Javanese

music. Saléndro or sléndro, derived from Æailendra, being the name of the Indian dynasty in regional rule

from the latter part of the eighth century through to the latter part of the tenth century.169 The tunings of

sléndro and pélog scales from East and Central Java are based on two very distinct non-equidistant

pentatonic and heptatonic tuning systems or laras, and accordingly are represented within the Javanese

orchestra or gamelan via different sets of instruments.170 A gamelan orchestra or ensemble is typically

made up of distinct combinations of gongs, metallophones, xylophones, drums, bowed and plucked

strings, bamboo flutes, small cymbals and singers, with the participation of between “three or four

musicians” and “twenty-five instrumentalists and ten to fifteen singers.”171 A complete gamelan consists of

a double set of instruments; each tuned to sléndro and pélog, and called a gamelan gedhé. The

gamelan situated at the Sri Wedhari theatre auditorium in Solo, Java – belonging to the Central Javanese

gamelan ensemble tradition of Karawitan – serves as an illustration of the two tuning systems, although it

168 Paul Rapoport, “The Notation of Equal Temperaments,” Xenharmonikôn: An Informal Journal of Experimental

Music 16 (Autumn, 1995): 61-84.

169 Laurence Picken, “The Music of Far Eastern Asia: 2. Other Countries,” Ancient and Oriental Music, ed. Egon

Wellesz (London: Oxford U. Press, 1957) 166-67. 170 “Sléndro is a five-tone scale that very roughly approaches equal-size intervals. The intervals vary within a given

scale and across orchestras, but the underlying tuning concept can be considered as a rough five-tone equal

temperament,” explains Braun, while “pélog is a seven-tone scale, whose underlying tuning concept is less obvious. One

possibility is that the tuning approaches a nine-tone equal temperament, using both single steps (133 cents) and double

steps (267 cents) from an imagined nine-tone equal temperament scale, which is a particular element of the historical

tradition of Javanese Music.” Martin Braun, “The Gamelan Pélog Scale of Central Java as an Example of a Non-Harmonic

Musical Scale,” Neuroscience of Music, 11 Mar. 2006, Sweden, 10 Apr. 2006,

<http://web.telia.com/~u57011259/pelog_main.htm>.

171 Benjamin Brinner, Knowing Music, Making Music: Javanese Gamelan and the Theory of Musical Competence and

Interaction (Chicago: U. of Chicago Press, 1995) XVII-XX.

104 The Equally-Tempered Archetype

Table 47. Notation for the twenty-four equal temperaments

NOTATION COMPASS OR RANGE

(CENTS)

NOTATION COMPASS OR RANGE

(CENTS)

C

C¢

C£

CË / Dì

CÚ

D¹

D¸

DÝ

CÍ / Dí

D³

D²

D

D¢

D£

DË / Eì

DÚ

E¸

EÝ

DÍ / Eí

E³

E²

E

E¢

E£

EË / Fí

F³

F²

F

F¢

F£

FË / Gì

FÚ

F¤

1189.247 � 10.752

10.753 � 31.282

31.283 � 48.912

48.913 � 63.719

63.720 � 80.448

80.449 � 100.977

100.978 � 122.483

122.484 � 140.189

140.190 � 154.019

154.020 � 171.650

171.651 � 193.156

193.157 � 213.685

213.686 � 233.503

233.504 � 252.110

252.111 � 267.629

267.630 � 284.358

284.359 � 304.887

304.888 � 322.593

322.594 � 336.423

336.424 � 354.053

354.054 � 375.560

375.561 � 397.066

397.067 � 417.595

417.596 � 435.226

435.227 � 450.032

450.033 � 466.761

466.762 � 487.291

487.292 � 508.797

508.798 � 529.327

529.328 � 546.957

546.958 � 561.764

561.765 � 579.470

579.471 � 599.999

F¥

GÝ

FÍ / Gí

G³

G²

G

G¢

G£

GË / Aì

GÚ

A¸

AÝ

GÍ / Aí

A³

A²

A

A¢

A£

AË / Bì

AÚ

B¹

B¸

BÝ

AÍ / Bí

B³

B²

B

B¢

B£

BË / Cí

C³

C²

C

600.000 � 620.528

620.529 � 638.234

638.235 � 653.041

653.042 � 670.671

670.672 � 691.201

691.202 � 712.483

712.484 � 733.013

733.014 � 750.867

750.868 � 765.674

765.675 � 782.403

782.404 � 802.932

802.933 � 820.638

820.639 � 831.391

831.392 � 849.021

849.022 � 873.605

873.606 � 895.111

895.112 � 915.640

915.641 � 933.271

933.272 � 948.077

948.078 � 965.783

965.784 � 986.313

986.314 � 1006.842

1006.843 � 1023.586

1023.587 � 1037.416

1037.417 � 1056.008

1056.009 � 1077.515

1077.516 � 1099.021

1099.022 � 1119.550

1119.551 � 1136.279

1136.280 � 1151.086

1151.087 � 1168.716

1168.717 � 1189.246

1189.247 � 10.752

The Equally-Tempered Archetype 105

must be noted that absolute pitch, as well as fixed theoretically defined tuning systems, should be

considered anomalies, and hence intonation traditions unique to but one gamelan. The roman numerals

above the pitches represent a form of the Central Javanese cipher notation system of Kepatihan, which is

based on the principles of the undotted numeral representative of the central octave, while dots above

and below, delineating the high and low octaves respectively. The names accorded to each pitch are

based on a system of interchangeable sléndro and pélog repertoire. It must be further noted that in spite

of the heptatonic structure of pélog, the tuning system is essentially pentatonic in traditional

compositions, with additional ‘auxiliary’ pitches serving as temporary substitute neighbour tones.172

Ex. 9. Gamelan gedhé sléndro and pélog tunings (Sri Wedhari theatre auditorium, Solo, Central Java)

In order to ascertain the size of the intervals of this particular gamelan gedhé, the following example

represents a transposition to the key of C of the actual tunings.

Ex. 10. Gamelan gedhé sléndro and pélog tunings in the key of C

The following presentation represents a harmonically tuned ‘just intonation’ reinterpretation of the sléndro

and pélog scales; the intervallic delineations facilitating the display of the harmonic characteristics.

Additional auxiliary tones have been added to the scales in order to generate a heptatonic scale

172 Marc Perlman, Unplayed Melodies: Javanese Gamelan and the Genesis of Music Theory (Berkeley: U. of California

Press, 2004) XV-41.

106 The Equally-Tempered Archetype

structure from the two essentially pentatonic forms. The sléndro tuning may be simplified as consisting of

the intervals: major second, perfect fourth, perfect fifth, and minor seventh; and in the key of C consisting

of the pitches: C, D, F, G, and B"; while pélog: minor second, minor third, perfect fifth, and minor sixth;

and in the key of C consisting of the pitches: C, D", E", G and A".

Ex. 11. The harmonic characteristics of the sléndro and pélog scales i. The sléndro scale

ii. The pélog scale

One-Tone Equal Temperament One-tone equal temperament is generated by the factor 1 2 , and produces the intervals of the octave

(with the simple frequency ratio of 2/1) and the unison (1/1), or what David D. Doty describes as “the

‘non-interval’ between two tones with exactly the same frequency.” 1/1, or the unison represents relative

pitch, and not absolute pitch in general terms, although in this particular context, the frequency relative to

A=440Hz (standard pitch), and equal to 261.626Hz. The octave is a twofold multiplication of 1/1,

hence equal to the frequency of 523.251Hz. The octave may also be stated as being the primary interval

The Equally-Tempered Archetype 107

of the prime number two, and “the only interval just intonation and equal temperament have in

common.”173 Sethares proclaims the octave as “the most consonant interval after the unison.” 2/1 and

1/1 are of course represented by the first two partials of the harmonic series.174 “The proportion of the

whole to its half or of the half to the whole is so natural that it is the first to be understood,” states French

composer and theorist Jean-Philippe Rameau; acknowledging Italian music theorist and composer

Gioseffo Zarlino (1517-90) for recognizing the octave as “the mother, the source, and the origin of all

intervals,” because, “by the division of its two terms all other harmonious chords are generated.”175 The

first three bars of the work highlight the singular application of both intervals.

Ex. 12. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 1-3

The following table depicts the tonal resources of one-tone equal temperament, indicating degree,

notation, interval, ratio, frequency, cents, and tuning.

Table 48. The one-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE

NUMBER

NOTE INTERVAL RATIO

(DECIMAL)

FREQUENCY

(HERTZ)

CENTS TUNING

ÐÑ

ÐÑ

C

C

unison

octave

1.000000

2.000000

261.626

523.251

0.000

1200.000

+00

+00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to

represent sléndro and pélog scales.

173 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 36.

174 Sethares, Tuning, Timbre, Spectrum, Scale 1-3. 175 Jean-Philippe Rameau, Treatise on Harmony, trans. Philip Gossett (New York: Dover Publications, 1971) 8-10.

108 The Equally-Tempered Archetype

Ex. 13. 1-tone equal temperament

The following tuning matrix contains the data required to represent the temperament chromatically within

the program memory of the Akai S3000XL.

Table 49. 1-tone equal temperament tuning matrix no. 1 (Program 01)

NOTE C – – – – – – – – – – – – – – – – – – – – – –

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00

Two-Tone Equal Temperament Two-tone equal temperament is generated by the factor 2 2 , and introduces the equal tritone, or the

sum of three equal whole-tones, which according to Richard Bobbitt was “proscribed in early

polyphonic music as diabolus in musica (the devil in music),” and “remains even today a ferment among

the family of intervals due to its peculiar characteristics.”176 Interesting among these characteristics is the

fact that the interval of a tritone does not appear in the first sixteen partials of the harmonic series, with

the just tritone, or augmented fourth (45/32, or 590.224 cents) first identified as the forty-fifth harmonic;

the alternative cyclic tritone (3ô÷/2÷ô, or 591.885 cents), as the forty-seventh ascending fifth in the ‘scale

of fifths’, or cyclic division of the octave; while the simpler Pythagorean tritone (729/512, or 611.730

cents), as the sixth ascending fifth. The equal tritone, or augmented fourth ( 2 2 ) is produced by the

176 According to Hans Tischler, “The diabolus in musica, however often it was exorcised by theorists, was

nevertheless as familiar to musicians as was Satan to the people of the age – and of much later ages, witness Luther and

the belief in witches. The avoidance of the tritone was hardly so generally practiced that it can be used as an unfailing

guide to musica ficta.” For a further discussion, see Hans Tischler, “Musica Ficta in the Thirteenth Century,” Music & Letters

54.1 (Jan., 1973): 48.

The Equally-Tempered Archetype 109

division of the octave into two equal parts (or two tones equal to 600.000 cents each), and

approximated with the frequency ratio 181/128.177 It is absolutely symmetrical, and is therefore non-

invertible.178 The notation (F¥) is reflective of the equal tritone fitting within the range of 600.000 and

620.528 cents, and may therefore be prescribed as approximating the acute or large tritone, or

augmented fourth (64/45, or 609.776) with a falsity of ß9.776 cents. It should nevertheless be noted

that theoretically it additionally represents the just tritone (45/32) with a falsity of +9.776 cents. The

pélog scale form scheme of the composition is able to accommodate the interval to produce a two-

tone symmetrical scale form, as well as a simultaneous sonority with an intervallic value of 600.000 cents.

Ex. 14. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 4

The following table depicts the tonal resources of two-tone equal temperament, indicating degree,

notation, interval, ratio, frequency, cents, and tuning.

Table 50. The two-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE

NUMBER

NOTE INTERVAL RATIO

(DECIMAL)

FREQUENCY

(HERTZ)

CENTS TUNING

ÐÑ

ÐÒ

ÐÑ

C

F¥

C

unison

equal or acute or large tritone, or augmented fourth

octave

1.000000

1.414214

2.000000

261.626

369.994

523.251

0.000

600.000

1200.000

+00

+00

+00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to

represent sléndro and pélog scales.

177 Daniélou, Tableau Comparatif des Intervalles Musicaux 144-45.

178 Richard Bobbitt, “The Physical Basis of Intervallic Quality and its Application to the Problem of Dissonance,”

Journal of Music Theory 3.2 (Nov., 1959): 190-92.

110 The Equally-Tempered Archetype

Ex. 15. 2-tone equal temperament

The following tuning matrix contains the data required to represent the temperament chromatically within

the program memory of the Akai S3000XL.

Table 51. 2-tone equal temperament tuning matrix no. 1 (Program 02)

NOTE C – – – – – – – – – – F¥ – – – – – – – – – –

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00

Three-Tone Equal Temperament Three-tone equal temperament is generated by the factor 3 2 , and introduces the equal major third

(approximately 63/50, or 400.000 cents) and its inversion, the equal minor sixth ( 23 ]2[ , approximately

100/63, or 800.000 cents).179 Relative to the just major thirds (5/4s) and just minor sixths (8/5s, or

813.686 cents) of just intonation, three-tone equal temperament produces major thirds ( 3 2 ) and minor

sixths ( 23 ]2[ ) with a falsity of +13.686 and ß13.686 cents on each count. The tuning modulation,

although unable to generate a recognizable sléndro or pélog scale form, is nevertheless able to deliver

the first theoretical triad, with the enharmonic reinterpretation of the pitches C, E¢, and A¸ generating an

augmented triad (C, E¢, and G¤). The triad is nevertheless not included in the work for the very reason

that formulated aesthetic guideless predicate a clear intention to highlight principal tones (of the sléndro

or pélog scale forms generated), with an attempt to exclude secondary ‘auxiliary’ tones from the melodic

framework as much as possible. Simultaneous sonorities introduced in the work with this temperament

include the dyads C and A¸, and E¢ and C, which represent the interval of an equal minor sixth ( 23 ]2[ ).

179 Daniélou, Tableau Comparatif des Intervalles Musicaux 102-03.

The Equally-Tempered Archetype 111

Ex. 16. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 7

The following table depicts the tonal resources of three-tone equal temperament, indicating degree,

notation, interval, ratio, frequency, cents, and tuning.

Table 52. The three-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE

NUMBER

NOTE INTERVAL RATIO

(DECIMAL)

FREQUENCY

(HERTZ)

CENTS TUNING

ÐÑ

ÐÒ

ÐÓ

ÐÑ

C

E¢

A¸

C

unison

equal or Pythagorean major third, or ditone

equal or Pythagorean minor sixth

octave

1.000000

1.259921

1.587401

2.000000

261.626

329.628

415.305

523.251

0.000

400.000

800.000

1200.000

+00

+00

+00

+00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to

represent sléndro and pélog scales.

Ex. 17. 3-tone equal temperament

The following tuning matrix contains the data required to represent the temperament chromatically within

the program memory of the Akai S3000XL.

112 The Equally-Tempered Archetype

Table 53. 3-tone equal temperament tuning matrix no. 1 (Program 03)

NOTE C – – – – – – E¢ – – – – – – A¸ – – – – – –

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00

Four-Tone Equal Temperament Four-tone equal temperament ( 4 2 ) presents a series of four equal tones 300.000 cents in size

(approximately 44/37),180 and therefore the juxtaposition of four equal minor thirds, or two sets of two-

tone equally-tempered intervals interlocked an equal minor third ( 4 2 ) apart. In comparison with the just

minor third (6/5), the equal minor third ( 4 2 ) has a falsity of ß15.641 cents, while its inversion, the equal

major sixth ( 34 ]2[ ), a falsity of +15.641 cents with the just major sixth (5/3, or 884.359 cents). Within

the combined sléndro and pélog scale form schemes, the temperament presents the pitches of the

equally-tempered diminished seventh tetrad, with the pitches C, E¸, F¥, and A¢ enharmonically

reinterpreted as C, E¸, G¸, and A¢. The only simultaneous sonority introduced in the work with this

temperament is the dyad E¸ and C, which represents the interval of an equal major sixth ( 34 ]2[ ).

Ex. 18. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 9

The following table depicts the tonal resources of four-tone equal temperament, indicating degree,

notation, interval, ratio, frequency, cents, and tuning.

180 Daniélou, Tableau Comparatif des Intervalles Musicaux 78.

The Equally-Tempered Archetype 113

Table 54. The four-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE

NUMBER

NOTE INTERVAL RATIO

(DECIMAL)

FREQUENCY

(HERTZ)

CENTS TUNING

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÑ

C

E¸

F¥

A¢

C

unison

equal or Pythagorean minor third, or trihemitone

equal or acute or large tritone, or augmented fourth

equal or Pythagorean major sixth

octave

1.000000

1.189207

1.414214

1.681793

2.000000

261.626

311.127

369.994

440.000

523.251

0.000

300.000

600.000

900.000

1200.000

+ 00

+ 00

+ 00

+ 00

+ 00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to

represent sléndro and pélog scales.

Ex. 19. 4-tone equal temperament

The following tuning matrix contains the data required to represent the temperament chromatically within

the program memory of the Akai S3000XL.

Table 55. 4-tone equal temperament tuning matrix no. 1 (Program 04)

NOTE C – – – – E¸ – – – – F¥ – – – – A¢ – – – –

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00

114 The Equally-Tempered Archetype

Five-Tone Equal Temperament Five-tone equal temperament is generated by the factor 5 2 , and with its five equal tones 240.00 cents

in size, which represent the 5-et supermajor second (approximately 54/47),181 is not only the first equally-

tempered division of the octave able to produce intervals outside of the domain of twelve-tone equal

temperament, but also the first able to adequately represent at least one of the Javanese scale forms.

The complete sléndro form is outlined via the pitches C, D£, F², G¢, and A!, and subsequently presents

the opportunity to evaluate the common notion proposed by theorists such as Martin Braun, proclaiming

that “the tuning concept (of sléndro) can be considered as a roughly five-tone equal temperament.”182

The sléndro tuning of the gamelan gedhé at the Sri Wedhari theatre auditorium in Solo, Central Java,

analyzed by Marc Perlman presents a platform for comparison with the following series of unequal

intervals: 238.000, 237.000, 252.000, 223.000, and 250.000 cents. The mean of this example (and the

division of the octave into five equal tones) is represented by the figure of 240.000 cents (with a

standard deviation value of 11.683321); proving the theory as somewhat significant, although analysis of

deviation values (ß2.000, ß3.000, +12.000, ß17.000, and +10.000 cents in each case) illustrate a

disproportionate relationship, and make the theoretical position no doubt inconclusive for the very

reasons that Perlman makes clear with the following statement: “There is no absolute pitch, or even a

standardized intonation, for these two laras; each fine gamelan may have its own distinctive realization of

them. The sample tone measurements are therefore only illustrative, not definitive.”183 Simultaneous

sonorities introduced in the work with this temperament include the dyads D£ and A!, F² and C, and C

and G¢, which at 720.000 cents, represents the 5-et acute or large fifth ( 35 ]2[ ). The interval is at the

upper limits of Blackwood’s criteria for a “perfect fifth within the range of recognizability”, and represents

the just perfect fifth (3/2) with a falsity +18.045 cents.184

Ex. 20. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 12

181 Daniélou, Tableau Comparatif des Intervalles Musicaux 64.

182 Braun, “The Gamelan Pélog Scale of Central Java as an Example of a Non-Harmonic Musical Scale,” Neuroscience

of Music n. pag.

183 Perlman, Unplayed Melodies: Javanese Gamelan and the Genesis of Music Theory 41. 184 Blackwood, The Structure of Recognizable Diatonic Tunings 197.

The Equally-Tempered Archetype 115

The following table depicts the tonal resources of five-tone equal temperament, indicating degree,

notation, interval, ratio, frequency, cents, and tuning.

Table 56. The five-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE

NUMBER

NOTE INTERVAL RATIO

(DECIMAL)

FREQUENCY

(HERTZ)

CENTS TUNING

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÑ

C

D£

F²

G¢

AÚ

C

unison

5-et supermajor second

5-et grave or small fourth

5-et acute or large fifth

5-et augmented sixth

octave

1.000000

1.148698

1.319508

1.515717

1.741101

2.000000

261.626

300.529

345.217

396.550

455.517

523.251

0.000

240.000

480.000

720.000

960.000

1200.000

+00

+40

ß20

+20

ß40

+00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to

represent sléndro and pélog scales.

Ex. 21. 5-tone equal temperament

The following tuning matrix contains the data required to represent the temperament chromatically within

the program memory of the Akai S3000XL.

116 The Equally-Tempered Archetype

Table 57. 5-tone equal temperament tuning matrix no. 1 (Program 05)

NOTE C – – D£ – – – – F² – – G¢ – – – – A! – –

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +00 +40 +00 +00 ß20 +00 +20 +00 +00 ß40 +00

Six-Tone Equal Temperament Six-tone equal temperament ( 6 2 ), with its return to intervals within the domain of twelve-tone equal

temperament, presents the essential scalar material to produce the equally-tempered whole-tone

hexatonic scale, featuring six equal whole-tones 200.00 cents in size (approximately 55/49).185 The

temperament may alternatively be expressed as the juxtaposition of three sets of two-tone equally-

tempered intervals, or two sets of three-tone equally-tempered intervals interlocked an equal major tone

( 6 2 ) apart. It must be noted that in comparison with the just major tone (9/8) and the just minor tone

(10/9, or 182.404 cents), the equal major tone ( 6 2 ) has a falsity of +3.910 and ß17.596 cents on

each count. Charles Villiers Stanford makes the following statement in opposition to the very notion of an

equally-tempered whole-tone and the scale derived via its juxtaposition – a series of six equally-

tempered whole-tones encompassing the span of an octave:

“It is physically impossible for a scale of whole-tones to reach a half at the octave (9/8, or 203.910×3 =

611.730 cents); a scale of whole greater tones would arrive at an octave which is too sharp (9/8, or

203.910×6 = 1223.460 cents); and any combination of the greater and lesser tones will be found equally

impossible at the octave. The only way to reach the octave by whole-tones is to make each and every

one of the intervals out of tune as the pianoforte is.”186

“The whole-tone hexatonic scale offers a limited basis for extended musical expression,” notes Vincent

Persichetti, and due to the fact that “when the scale is mirrored there is no change except in register.”

The scale also offers only two possible transpositions (minor seconds apart), with no prospects for

modality and hence the generation of unique modes. Persichetti concludes that “its intervallic make-up

deprives the scale of the fundamental intervals, the perfect fourth and fifth, and of the leading tone,” and

that “a real feeling of tonality, therefore, must be established by harmony outside the whole-tone

category.”187 The temperament also introduces the equal minor seventh ( 56 ]2[ , approximately 98/55, or

185 Daniélou, Tableau Comparatif des Intervalles Musicaux 102.

186 Charles Villiers Stanford, “On Some Recent Tendencies in Composition,” Proceedings of the Musical Association,

47th Sess. (1920): 40. 187 Persichetti, Twentieth-Century Harmony: Creative Aspects and Practice 54-55.

The Equally-Tempered Archetype 117

1000.000 cents),188 which represents the acute or large minor seventh (9/5, or 1017.596 cents) with a

falsity of ß17.596 cents. Simultaneous sonorities appearing in the work include the equal major third

( 3 2 ) and major sixth ( 34 ]2[ ), which are depicted via the Pythagorean approximations of A¸ and C, and

C and A¸.

Ex. 22. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 16

The following table depicts the tonal resources of six-tone equal temperament, indicating degree,

notation, interval, ratio, frequency, cents, and tuning.

Table 58. The six-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE

NUMBER

NOTE INTERVAL RATIO

(DECIMAL)

FREQUENCY

(HERTZ)

CENTS TUNING

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

ÐÑ

C

D

E¢

F¥

A¸

B¸

C

unison

equal or just major tone

equal or Pythagorean major third, or ditone

equal or acute or large tritone, or augmented fourth

equal or Pythagorean minor sixth

equal or Pythagorean minor seventh

octave

1.000000

1.122462

1.259921

1.414214

1.587401

1.781797

2.000000

261.626

293.665

329.628

369.994

415.305

466.164

523.251

0.000

200.000

400.000

600.000

800.000

1000.000

1200.000

+00

+00

+00

+00

+00

+00

+00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to

represent sléndro and pélog scales.

188 Daniélou, Tableau Comparatif des Intervalles Musicaux 55.

118 The Equally-Tempered Archetype

Ex. 23. 6-tone equal temperament

The following tuning matrix contains the data required to represent the temperament chromatically within

the program memory of the Akai S3000XL.

Table 59. 6-tone equal temperament tuning matrix no. 1 (Program 06)

NOTE C – – D – – E¢ – – F¥ – – A¸ – – B¸ – –

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00

Seven-Tone Equal Temperament Seven-tone equal temperament is generated by the factor 7 2 , and with its seven equal tones 171.429

cents in size, which represent the 7-et grave or small tone (approximately 56295/50989),189 make it an

intonation generally associated with the system of tuning in the traditional music of Siam (Thailand). The

equidistant claim is speculative amongst ethnomusicologists, with scientific analysis revealing a much more

complex system of intervallic diversity.190 Terry E. Miller and Sam-ang Sam make the following

observations:

“In the case of Khmer music, there has never been a strong claim for equidistance, and in fact Khmer tuning is

clearly non-equidistant. Certain pitch levels are considered useable, others not. In sum, then, Thai tuning is

functionally equidistant while Khmer is only apparently non-equidistant but variable.”191

189 Daniélou, Tableau Comparatif des Intervalles Musicaux 48.

190 Picken, “The Music of Far Eastern Asia: 2. Other Countries,” Ancient and Oriental Music 163. 191 Terry E. Miller, and Sam-ang Sam, “The Classical Musics of Cambodia and Thailand: A Study of Distinctions,”

Ethnomusicology 39.2 (Spring-Summer, 1995): 237-38.

The Equally-Tempered Archetype 119

The temperament introduces some unique simultaneous sonorities to the work, which include the dyads

F¢ and AÍ, or the 7-et acute or large fourth ( 37 ]2[ , approximately 689/512, or 514.286 cents); C and

G², and D³ and A², or the 7-et grave or small fifth ( 47 ]2[ , approximately 1024/689, or 685.714 cents);

and E³ and C, or the 7-et grave or small major sixth ( 57 ]2[ , approximately 105/64, or 857.143 cents).192

At 685.714 cents, the 7-et grave or small fifth ( 47 ]2[ ) is at the lower limits of Blackwood’s criteria for a

“perfect fifth within the range of recognizability,” and represents the just perfect fifth (3/2) with a falsity of

ß16.241 cents. The 7-et grave or small major sixth ( 57 ]2[ ) and 7-et acute or large fourth ( 37 ]2[ )

represent deviations of just intervals (5/3 and 4/3) by ß27.216 and +16.241 cents on each count.193

Ex. 24. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 18

The following table depicts the tonal resources of seven-tone equal temperament, indicating degree,

notation, interval, ratio, frequency, cents, and tuning.

Table 60. The seven-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE

NUMBER

NOTE INTERVAL RATIO

(DECIMAL)

FREQUENCY

(HERTZ)

CENTS TUNING

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐÑ

C

D³

E³

F¢

G²

A²

AÍ / Bí

C

unison

7-et grave or small tone

7-et neutral third

7-et acute or large fourth

7-et grave or small fifth

7-et grave or small major sixth

7-et twenty-one quarter-tones

octave

1.000000

1.104090

1.219014

1.345900

1.485994

1.640671

1.811447

2.000000

261.626

288.858

318.925

352.122

388.774

429.241

473.921

523.251

0.000

171.429

342.857

514.286

685.714

857.143

1028.571

1200.000

+00

ß29

ß57

+14

ß14

ß43

+29

+00

192 Daniélou, Tableau Comparatif des Intervalles Musicaux 89-127.

193 Blackwood, The Structure of Recognizable Diatonic Tunings 197.

120 The Equally-Tempered Archetype

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to

represent sléndro and pélog scales.

Ex. 25. 7-tone equal temperament

The following tuning matrix contains the data required to represent the temperament chromatically within

the program memory of the Akai S3000XL.

Table 61. 7-tone equal temperament tuning matrix no. 1 (Program 07)

NOTE C – – D³ E³ – – F¢ – – G² – – A² AÍ – –

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +00 ß29 +43 +00 +14 +00 ß14 +00 ß43 +29 +00

Eight-Tone Equal Temperament Eight-tone equal temperament ( 8 2 ) presents a series of eight equal tones 150.000 cents in size

(approximately 1024/939), and therefore the juxtaposition of two sets of four-tone equally-tempered

intervals, or four sets of two-tone equally-tempered intervals interlocked an equal three-quarter-tone

( 8 2 ) apart. The exercise produces the equally-tempered three-quarter-tone octatonic scale, which

features eight equal three-quarter-tones, and hence, the addition of the equal three-quarter-tone ( 8 2 ),

nine quarter-tones ( 38 ]2[ , approximately 83/64, or 450.000 cents), fifteen quarter-tones ( 58 ]2[ ,

approximately 128/83, or 750.000 cents), and twenty-one quarter-tones ( 78 ]2[ , approximately

939/512, or 1050.000 cents) to the vocabulary of equal intervals.194 The simultaneous sonorities of E¸

and F¥, CÍ and Fí, Fí and G£, and C and E¸ highlight the utilization of the equal minor third ( 4 2 ) in the

194 Daniélou, Tableau Comparatif des Intervalles Musicaux 42-113.

The Equally-Tempered Archetype 121

work; while C and Fí, E¸ and G£, and F¥ and B³, the nine equal quarter-tones ( 38 ]2[ ), which is one of

these very distinct equal quarter-tone intervals.

Ex. 26. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 22-24

The following table depicts the tonal resources of eight-tone equal temperament, indicating degree,

notation, interval, ratio, frequency, cents, and tuning.

Table 62. The eight-tone equally-tempered division of the octave

Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE

NUMBER

NOTE INTERVAL RATIO

(DECIMAL)

FREQUENCY

(HERTZ)

CENTS TUNING

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÑ

C

CÍ / Dí

E¸

EË / Fí

F¥

G£

A¢

B³

C

unison

equal three-quarter-tone

equal or Pythagorean minor third, or trihemitone

nine equal quarter-tones

equal or acute or large tritone, or augmented fourth

fifteen equal quarter-tones

equal or Pythagorean major sixth

twenty-one equal quarter-tones

octave

1.000000

1.090508

1.189207

1.296840

1.414214

1.542211

1.681793

1.834008

2.000000

261.626

285.305

311.127

339.286

369.994

403.482

440.000

479.823

523.251

0.000

150.000

300.000

450.000

600.000

750.000

900.000

1050.000

1200.000

+00

+50

+00

ß50

+00

+50

+00

ß50

+00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to

represent sléndro and pélog scales. Ex. 27. 8-tone equal temperament

122 The Equally-Tempered Archetype

The following tuning matrix contains the data required to represent the temperament chromatically within

the program memory of the Akai S3000XL.

Table 63. 8-tone equal temperament tuning matrix no. 1 (Program 08)

NOTE C CÍ – – E¸ – – Fí F¥ G£ – – A¢ – – B³

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +50 +00 +00 +00 ß50 +00 +50 +00 +00 +00 ß50

Nine-Tone Equal Temperament Nine-tone equal temperament ( 9 2 ), with its twofold tripartite division of the octave, presents a

subdivision of the equal major third ( 3 2 ) into three equal tones 133.333 cents in size, which represent

the 9-et great limma (approximately 553/512). The twofold tripartite division of the octave is in reference

to the juxtaposition of three sets of three-tone equally-tempered intervals interlocked an equal 9-et great

limma ( 9 2 ) apart, which delivers not only the first triad in the work, but also the first complete exposition

of the pélog scale with the pitches C, DÝ, Eì, F£, G³, A¸, and B². The triad consists of the pitches C, Eì,

and G³, and presents a minor third 266.667 cents in size ( 29 ]2[ , 9-et five quarter tones, or

approximately 7/6) and a perfect fifth 666.667 cents in size ( 59 ]2[ , 9-et subfifth, or approximately

147/100).195 Eì, although equal to 9-et five quarter tones ( 29 ]2[ ), functions harmonically as a minor third,

and therefore in association with the other pitches forms a harmonic structure that may be regarded as

being essentially a nine-tone equally-tempered minor triad. In comparison with the just minor triad (equal

to the proportions 10:12:15), the nine-tone equally-tempered minor triad features two acutely dissonant

intervals, which form the complex ratio 1.000000:1.166529:1.469734. The fundamental C4 at standard

pitch (A=440Hz) may be utilized to illustrate that the third harmonic of C4 does not correspond with the

second harmonic of G³4 (769.040Hz), or the 9-et subfifth ( 59 ]2[ ). A just perfect fifth with a 3/2

proportion would have presented identical frequencies, and hence a ‘zero beating’ condition between

195 Daniélou, Tableau Comparatif des Intervalles Musicaux 38-102.

The Equally-Tempered Archetype 123

the two intervals. In striking contrast, nine-tone equal temperament presents 15.836 beats, with 31.673

beats between the sixth harmonic of C4 and the fourth harmonic of G³4 (1538.080Hz). Eì4

(305.194Hz), or 9-et five quarter tones ( 29 ]2[ ), presents 43.784 beats between the sixth harmonic of

C4 and the fifth harmonic of Eì4 (1525.969Hz). The level of dissonance is clearly within the sonic realm

of ‘roughness’, and therefore outside of the 20-25Hz threshold that Doty defines as a distinctly audible

rate of beating.196

Table 64. The beating characteristics of the nine-tone equally-tempered minor triad

C4

(PARTIAL)

FREQUENCY

(HERTZ)

Eì4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

G³4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

– – – –

5

6

– – – –

7

– – – –

8

261.626

– – – –

– – – –

523.251

– – – –

784.877

– – – –

1046.502

– – – –

– – – –

1308.128

1569.753

– – – –

1831.379

– – – –

2093.005

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

– – – –

5

6

– – – –

– – – –

– – – –

– – – –

305.194

– – – –

– – – –

610.388

– – – –

915.581

– – – –

– – – –

1220.775

– – – –

1525.969

1831.163

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

43.784

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

– – – –

4

– – – –

– – – –

5

– – – –

– – – –

– – – –

384.520

– – – –

– – – –

769.040

– – – –

– – – –

1153.560

– – – –

– – – –

1538.080

– – – –

– – – –

1922.601

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

15.836

– – – –

– – – –

– – – –

– – – –

– – – –

31.673

– – – –

– – – –

– – – –

– – – –

The pélog scale is outlined with the pitches C, DÝ, Eì, F£, G³, A¸, and B², and equal to 0.000, 133.333,

266.667, 533.333, 666.667, 800.000, and 1066.667 cents. The pélog tuning of the gamelan gedhé

analyzed by Perlman presents the following series of unequal intervals: 116.000, 165.000, 269.000,

119.000, 100.000, 192.000, and 239.000 cents. The mean of this example (and the division of the

octave into seven equal tones) is represented by the figure of 171.429 cents (with a standard deviation

196 “Beats can be perceived clearly when the difference is less than 20-25Hz, but as the difference increases

beyond this point the beats blend together, giving rise to a general sensation of roughness. This roughness gradually

decreases as the difference increases, persisting until the difference exceeds the critical band, which, for most of the

audio range falls between a whole-tone and a minor third,” explains Doty. For a further discussion, see Doty, The Just

Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 20-22.

124 The Equally-Tempered Archetype

value of 65.158415), and therefore with some certainty, it may be stated that with the falsities of

ß17.333, +14.333, +16.667, +2.333, ß31.000, and ß106.667 cents on each count, the intervals of

the gamelan gedhé only remotely resemble those of nine-tone equal temperament. The inclusion of A£

in the pélog system – although far from being representative of any legitimate form of a minor seventh –

replaces the last value of contention (B²) with a falsity of +27.667 cents. In consideration of the

pentatonic outline, or principal tones of the pélog scale (the pitches C, DÝ, Eì, G³, and A¸) – just like the

previous sléndro example – the mean value of 240.00 cents (with a standard deviation of 157.310203)

may be ascertained, but with an even greater degree of inequality.197

Simultaneous sonorities include the triad G³, C, and G³; and the inversion F£, C, and F£; which

may be represented by the 9-et superfourth ( 49 ]2[ , or approximately 200/147) and 9-et subfifth

( 59 ]2[ ).198 The two intervals represent an extremely dissonant just perfect fourth (4/3) and just perfect

fifth (3/2) with a falsity +35.288 and ß35.288 on each count.

Ex. 28. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 27

Other simultaneous sonorities include the dyads Eì and G³, B² and Eì, and G³ and B²; all equal to the

interval of an equal major third ( 3 2 ); with a passage spelling F£ and G³, G³ and A£, and a melodic

descent to A¸, marking the intervals of the 9-et great limma ( 9 2 ), 9-et five quarter tones ( 29 ]2[ ), and 9-

et great limma ( 9 2 ) respectively.

Ex. 29. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 28-29

197 Perlman, Unplayed Melodies: Javanese Gamelan and the Genesis of Music Theory 41.

198 Daniélou, Tableau Comparatif des Intervalles Musicaux 130-131.

The Equally-Tempered Archetype 125

The following table depicts the tonal resources of nine-tone equal temperament, indicating degree,

notation, interval, ratio, frequency, cents, and tuning.

Table 65. The nine-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE

NUMBER

NOTE INTERVAL RATIO

(DECIMAL)

FREQUENCY

(HERTZ)

CENTS TUNING

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÐÑ

C

DÝ

DË / Eì

E¢

F£

G³

A¸

AË / Bì

B²

C

unison

9-et great limma, or large half-tone

9-et five quarter-tones

equal or Pythagorean major third, or ditone

9-et superfourth

9-et subfifth

equal or Pythagorean minor sixth

9-et nineteen quarter-tones

9-et grave or small major seventh

octave

1.000000

1.080060

1.166529

1.259921

1.360790

1.469734

1.587401

1.714488

1.851749

2.000000

261.626

282.571

305.194

329.628

356.017

384.520

415.305

448.554

484.465

523.251

0.000

133.333

266.667

400.000

533.333

666.667

800.000

933.333

1066.667

1200.000

+00

+33

ß33

+00

+33

ß33

+00

+33

ß33

+00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to

represent sléndro and pélog scales.

Ex. 30. 9-tone equal temperament

The following tuning matrix contains the data required to represent the temperament chromatically within

the program memory of the Akai S3000XL.

126 The Equally-Tempered Archetype

Table 66. 9-tone equal temperament tuning matrix no. 1 (Program 09)

NOTE C DÝ – – Eì E¢ F£ – – G³ A¸ AË – – B²

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +33 +00 ß33 +00 +33 +00 ß33 +00 +33 +00 ß33

Ten-Tone Equal Temperament Ten-tone equal temperament ( 10 2 ) presents a subdivision of the 5-et supermajor second ( 5 2 ) into

two equal tones 120.000 cents in size, which represent the 10-et just diatonic semitone (approximately

15/14). The temperament may be alternatively expressed as the juxtaposition of two sets of five-tone

equally-tempered intervals, or five sets of two-tone equally-tempered intervals interlocked a 10-et just

diatonic semitone ( 10 2 ) apart. The intonation scheme therefore produces the identical pitches for

sléndro as five-tone equal temperament, although additional pitches generate possible auxiliary pitches of

an extended pentatonic form. Sethares makes the following observations: “The 10-tet tuning has no fifth,

no third, no major seconds, and no dominant sevenths. The only interval common to both 10-tet and

12-tet (other than the octave) is the 600-cent interval normally called the tritone, augmented fourth, or

diminished fifth.”199

A variety of simultaneous sonorities are explored in the work within the scope of this temperament,

which include the dyads E² and G¢, and F² and A³, or the 10-et grave or small major third ( 310 ]2[ ,

approximately 16/13, or 360.000 cents), and C and AÚ, or the 5-et augmented sixth ( 45 ]2[ ,

approximately 47/27, or 960.000 cents).200 The two intervals represent the just major third (5/4) and the

acute or large minor seventh (9/5) with a falsity of ß26.314 and ß57.596 cents on each count.

Ex. 31. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 32

199 Sethares, Tuning, Timbre, Spectrum, Scale 291.

200 Daniélou, Tableau Comparatif des Intervalles Musicaux 34-92.

The Equally-Tempered Archetype 127

Other simultaneous sonorities include the dyads D£ and A³, or the equal tritone ( 2 2 ), and C and G¢, or

the 5-et acute or large fifth ( 35 ]2[ ).

Ex. 32. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 34

The following table depicts the tonal resources of ten-tone equal temperament, indicating degree,

notation, interval, ratio, frequency, cents, and tuning.

Table 67. The ten-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE

NUMBER

NOTE INTERVAL RATIO

(DECIMAL)

FREQUENCY

(HERTZ)

CENTS TUNING

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÐÑ

C

D¸

D£

E²

F²

F¥

G¢

A³

AÚ

B

C

unison

10-et just diatonic semitone, or major half-tone

5-et supermajor second

10-et grave or small major third

5-et grave or small fourth

equal or acute or large tritone, or augmented fourth

5-et acute or large fifth

10-et neutral sixth

5-et augmented sixth

10-et just diatonic major seventh

octave

1.000000

1.071773

1.148698

1.231144

1.319508

1.414214

1.515717

1.624505

1.741101

1.866066

2.000000

261.626

280.403

300.529

322.099

345.217

369.994

396.550

425.012

455.517

488.211

523.251

0.000

120.000

240.000

360.000

480.000

600.000

720.000

840.000

960.000

1080.000

1200.000

+00

+20

+40

ß40

ß20

+00

+20

ß60

ß40

ß20

+00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to

represent sléndro and pélog scales.

Ex. 33. 10-tone equal temperament

128 The Equally-Tempered Archetype

The following tuning matrix contains the data required to represent the temperament chromatically within

the program memory of the Akai S3000XL.

Table 68. 10-tone equal temperament tuning matrix no. 1 (Program 10)

NOTE C D¸ D£ – – E² F² F¥ G¢ A³ – – AÚ B

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +20 +40 +00 ß40 ß20 +00 +20 +40 +00 ß40 ß20

Eleven-Tone Equal Temperament Eleven-tone equal temperament is generated by the factor 112 , and presents eleven equal intervals

109.091 cents in size, which represent the 11-et just diatonic semitone (approximately 82/77). An

implied triad makes its appearance in the composition in bar 38. This triad (essentially an eleven-tone

equally-tempered major triad) consists of the pitches C, EË, and G³, and presents a major third 436.364

cents in size ( 411 ]2[ , 11-et nine quarter-tones, or approximately 659/512) and a perfect fifth 654.545

cents in size ( 611 ]2[ , 11-et subfifth, or approximately 54/37); hence an extremely dissonant harmonic

construct.201 EË, although equal to 11-et nine quarter-tones ( 411 ]2[ ), functions harmonically as a major

third. In comparison with the just major triad (equal to the proportions 4:5:6), the eleven-tone equally-

tempered major triad forms the complex ratio 1.000000:1.286665:1.459480. The temperament

produces the just major third (5/4) and just perfect fifth (3/2) with a falsity of +50.050 and ß47.410

cents, which is approximately an equal quarter-tone ( 24 2 ) deviation on each count. G³4 (381.837Hz),

or the 11-et subfifth ( 611 ]2[ ), presents 21.202 beats between the third harmonic of C4 and the second

harmonic of G³4 (763.675Hz), and 42.404 beats between the sixth harmonic of C4 and the fourth

harmonic of G³4 (1527.349Hz); while EË4 (336.624Hz), or 11-et nine quarter-tones ( 411 ]2[ ), 38.370

beats between the fifth harmonic of C4 and the fourth harmonic of EË4 (1346.498Hz).

201 Daniélou, Tableau Comparatif des Intervalles Musicaux 30-135.

The Equally-Tempered Archetype 129

Table 69. The beating characteristics of the eleven-tone equally-tempered major triad

C4

(PARTIAL)

FREQUENCY

(HERTZ)

EË4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

G³4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

5

6

– – – –

7

– – – –

8

261.626

– – – –

– – – –

523.251

– – – –

784.877

– – – –

1046.502

– – – –

1308.128

1569.753

– – – –

1831.379

– – – –

2093.005

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

– – – –

5

– – – –

6

– – – –

– – – –

336.624

– – – –

– – – –

673.249

– – – –

1009.873

– – – –

– – – –

1346.498

– – – –

1683.122

– – – –

2019.747

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

38.370

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

4

– – – –

– – – –

5

– – – –

– – – –

– – – –

381.837

– – – –

– – – –

763.675

– – – –

– – – –

1145.512

– – – –

1527.349

– – – –

– – – –

1909.187

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

21.202

– – – –

– – – –

– – – –

– – – –

42.404

– – – –

– – – –

– – – –

– – – –

Simultaneous sonorities introduced in the work with this temperament include the dyads F£ and G³, or

the 11-et just diatonic semitone ( 112 ), and B¹ and C, or the 11-et acute or large tone ( 211 ]2[ ,

approximately 245/216, or 218.182 cents).202

Ex. 34. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 41

Other simultaneous sonorities include the dyads D¢ and F£, EË and Aì, and G³ and B¹, or 11-et seven

quarter-tones ( 311 ]2[ , approximately 29/24, or 327.273 cents), and C and EË, and B¹ and D¢, or 11-et

nine quarter-tones ( 411 ]2[ ).203

202 Daniélou, Tableau Comparatif des Intervalles Musicaux 58.

203 Daniélou, Tableau Comparatif des Intervalles Musicaux 84.

130 The Equally-Tempered Archetype

Ex. 35. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 42

The following table depicts the tonal resources of eleven-tone equal temperament, indicating degree,

notation, interval, ratio, frequency, cents, and tuning.

Table 70. The eleven-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE

NUMBER

NOTE INTERVAL RATIO

(DECIMAL)

FREQUENCY

(HERTZ)

CENTS TUNING

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÐÑ

C

D¸

D¢

DÍ / Eí

EË / Fí

F£

G³

GË / Aì

A²

B¹

B

C

unison

11-et just diatonic semitone, or major half-tone

11-et acute or large tone

11-et seven quarter-tones

11-et nine quarter-tones

11-et superfourth

11-et subfifth

11-et fifteen quarter-tones

11-et grave or small major sixth

11-et grave or small minor seventh

11-et just diatonic major seventh

octave

1.000000

1.065041

1.134313

1.208089

1.286665

1.370351

1.459480

1.554406

1.655507

1.763183

1.877862

2.000000

261.626

278.642

296.765

316.067

336.624

358.519

381.837

406.672

433.123

461.294

491.297

523.251

0.000

109.091

218.182

327.273

436.364

545.455

654.545

763.636

872.727

981.818

1090.909

1200.000

+00

+09

+18

+27

+36

+45

ß45

ß36

ß27

ß18

ß09

+00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to

represent sléndro and pélog scales.

Ex. 36. 11-tone equal temperament

The Equally-Tempered Archetype 131

The following tuning matrix contains the data required to represent the temperament chromatically within

the program memory of the Akai S3000XL.

Table 71. 11-tone equal temperament tuning matrix no. 1 (Program 11)

NOTE C D¸ D¢ DÍ EË F£ – – G³ Aì A² B¹ B

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +09 +18 +27 +36 +45 +00 ß45 ß36 ß27 ß18 ß09

Twelve-Tone Equal Temperament Twelve-tone equal temperament ( 12 2 ), with its division of the octave into twelve equally-tempered

semitones 100.000 cents in size, marks the central structural climax of the composition. The

temperament may alternatively be expressed as the juxtaposition of two sets of six-tone equally-

tempered intervals, three sets of four-tone equally-tempered intervals, four sets of three-tone equally-

tempered intervals, or six sets of two-tone equally-tempered intervals interlocked an equal semitone

( 12 2 ) apart. Philippe De Vitry (1291-1361) has the following to say about the semitone: “The semitone,

as Bernardus said, is the sugar and spice of all music, and without it song is corrupted, altered, and

destroyed,” while theorizing the interval of the semitone as “the interval between two unisons, which in

the human voice is incapable of, and will admit of, division or the interpolation of a middle sound.”204

With regards to the rationale behind the adoption of twelve-tone equal temperament as the intonation

standard of Western music, Blackwood offers the following explanation: “There is no doubt in my mind

that of all equal tunings, twelve is the most versatile and most expressive. It’s no accident that we have

twelve instead of thirteen.”205

204 Philippe de Vitry, “Philippe de Vitry’s ‘Ars Nova’: A Translation,” Journal of Music Theory 5.2 (Winter, 1961): 10.

205 Douglas Keislar, Easley Blackwood, John Eaton, Lou Harrison, Ben Johnston, Joel Mandelbaum, and William

Schottstaedt, “Six American Composers on Nonstandard Tunings,” Perspectives of New Music 29.1 (Winter, 1991): 180.

132 The Equally-Tempered Archetype

The first simultaneous sonority of interest makes its appearance in the first beat of bar 44, which

incorporates the pitches C, G, F, and C, and may be theoretically explained as two equal perfect fourths

( 512 ]2[ ) juxtaposed an equal minor seventh apart ( 56 ]2[ ), or C(add 11/omit 3). Consequential sonorities

B¢ and F, E¢ and B¸, and F¥ and C (F¥ alternatively expressed via its enharmonic equivalent of G¸), all

equal to 600.000 cents, or the equal tritone ( 2 2 ), while E¢ and C, and C and B¸, the equal minor sixth

( 23 ]2[ ) and equal minor seventh ( 56 ]2[ ).

Ex. 37. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 44-45

Other sonorities include the dyads F and G¸, and B¢ and C, which represent the equal semitone ( 12 2 ).

In comparison with the just diatonic semitone (16/15), the equal semitone ( 12 2 ) has a falsity of ß11.731

cents, while its inversion, the equal major seventh ( 1112 ]2[ , approximately 967/512, or 1100.000

cents),206 a falsity of +11.731 cents with the just diatonic major seventh (15/8, or 884.359 cents)

Twelve-tone equal temperament is the first temperament to contain what Blackwood terms as

recognizable diatonic scales and “perfect fifths within the range of recognizability,”207 and produces the

just perfect fifth (3/2) and just perfect fourth (4/3) with a falsity of +1.955 and ß1955 cents on each

count.

Ex. 38. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 46

The following table depicts the tonal resources of twelve-tone equal temperament, indicating degree,

notation, interval, ratio, frequency, cents, and tuning.

206 Daniélou, Tableau Comparatif des Intervalles Musicaux 29.

207 Blackwood, The Structure of Recognizable Diatonic Tunings 197.

The Equally-Tempered Archetype 133

Table 72. The twelve-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE

NUMBER

NOTE INTERVAL RATIO

(DECIMAL)

FREQUENCY

(HERTZ)

CENTS TUNING

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÐÑ

C

D¹

D

E¸

E¢

F

F¥

G

A¸

A¢

B¸

B¢

C

unison

equal semitone, or Pythagorean limma

equal or just major tone

equal or Pythagorean minor third, or trihemitone

equal or Pythagorean major third, or ditone

equal, or just and Pythagorean perfect fourth

equal or acute or large tritone, or augmented fourth

equal, or just and Pythagorean perfect fifth

equal or Pythagorean minor sixth

equal or Pythagorean major sixth

equal or Pythagorean minor seventh

equal or Pythagorean major seventh

octave

1.000000

1.059463

1.122462

1.189207

1.259921

1.334840

1.414214

1.498307

1.587401

1.681793

1.781797

1.887749

2.000000

261.626

277.183

293.665

311.127

329.628

349.228

369.994

391.995

415.305

440.000

466.164

493.883

523.251

0.000

100.000

200.000

300.000

400.000

500.000

600.000

700.000

800.000

900.000

1000.000

1100.000

1200.000

+00

+00

+00

+00

+00

+00

+00

+00

+00

+00

+00

+00

+00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to

represent sléndro and pélog scales.

Ex. 39. 12-tone equal temperament

The following tuning matrix contains the data required to represent the temperament chromatically within

the program memory of the Akai S3000XL.

134 The Equally-Tempered Archetype

Table 73. 12-tone equal temperament tuning matrix no. 1 (Program 12)

NOTE C D¹ D E¸ E¢ F F¥ G A¸ A¢ B¸ B¢

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00

Thirteen-Tone Equal Temperament Thirteen-tone equal temperament is generated by the factor 13 2 , and presents thirteen equal intervals

92.308 cents in size, which represent the 13-et Pythagorean limma (approximately 77/73), and

approximate the Pythagorean limma (256/243) with a falsity of +2.083 cents. The thirteen-tone equally-

tempered division of the octave now for the first time presents more than one possibility for the

approximation of a particular tone. In this case, that tone being the fifth, with the possible representation

as G£, 13-et thirteen quarter-tones ( 713 ]2[ , approximately 61/42, or 646.154 cents), or the alternative of

Gí, the 13-et superfifth ( 813 ]2[ , approximately 72/47, or 738.462 cents).208 Throughout the work, the

criteria for accessing whether this second tone is incorporated into the scalar scheme as a primary or

secondary ‘auxiliary’ tone will be dependent on the accuracy of the approximation. In thirteen-tone

equal temperament Gí ( 713 ]2[ ), hence accorded as an auxiliary tone, due to the fact that G£ ( 813 ]2[ ) is

36.507 cents higher than the just perfect fifth (3/2), while Gí ( 713 ]2[ ), 55.801 cents lower, or with a

falsity exceeding that of the primary fifth G£ ( 813 ]2[ ) by 19.294 cents. In the case of more than two

possibilities for the approximation of one tone, only the closest two will be taken into account. Melodic

development in bar 51 of the work exploits the intervallic nuances of the two microtonal inflections of

the fifth via the intonational reinterpretation of pitch recapitulation.

Ex. 40. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 51

208 Daniélou, Tableau Comparatif des Intervalles Musicaux 26-137.

The Equally-Tempered Archetype 135

American composer Easley Blackwood (1933-) composed Twelve Microtonal Etudes for Electronic

Music Media in 1979 and 1980 as part of a research project supported by the National Endowment for

the Humanities in association with Webster College in St. Louis, USA. The twelve-movement work

explored the tonal and modal characteristic of all the equal divisions of the octave between thirteen and

twenty-four, stimulating the devising of appropriate notational practices, and culminating in the compact

disc release of Microtonal Compositions by Easley Blackwood, which feature Blackwood on the

polyfusion synthesizer. It is therefore most appropriate to hereon incorporate some of the comments

expressed by the composer in relation to each temperament. According to Blackwood, the thirteen-

tone equally-tempered division of the octave is “the most alien tuning of all; so dissonant that no three-

note combination sounds like major or minor triad.”209 Two new triads are introduced to the work via this

temperament. The first triad (essentially a thirteen-tone equally-tempered major triad) consists of the

pitches C, E², and G£, and presents the 13-et grave or small major third ( 413 ]2[ , approximately 26/21, or

369.231 cents) and the 13-et superfifth ( 813 ]2[ ). The second triad consists of the pitches C, DÚ, and

Gí, and presents the 13-et augmented second ( 313 ]2[ , approximately 601/512, or 276.923 cents) and

13-et thirteen quarter-tones ( 713 ]2[ ). Enharmonically reinterpreted as C, E¹, and Gí, the latter triad

serves as an implied thirteen-tone equally-tempered minor triad.210

Ex. 41. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 55

The temperament produces the just major third (5/4) and just perfect fifth (3/2) with a falsity of ß17.083

and +36.507 cents, or approximate equal sixth-tone ( 36 2 , approximately 261/256, or 33.333 cents)

and third-tone ( 18 2 , 133/128, or 66.667 cents) deviations on each count.211 G£4 (400.802Hz), or the

13-et superfifth ( 813 ]2[ ), presents 16.726 beats between the third harmonic of C4 and the second

harmonic of G£4 (801.603Hz), and 33.453 beats between the sixth harmonic of C4 and the fourth

harmonic of G£4 (1603.206Hz); while E²4 (323.821Hz), or the 13-et grave or small major third ( 413 ]2[ ),

209 Easley Blackwood, liner notes, Microtonal Compositions by Easley Blackwood, perf. Easley Blackwood

(polyfusion synthesizer), and Jeffrey Kust (guitar), rec. 16 Sep. 1990, Cedille, 1994, CDR 90000 018, n. pag.

210 Daniélou, Tableau Comparatif des Intervalles Musicaux 74-94.

211 Daniélou, Tableau Comparatif des Intervalles Musicaux 8-18.

136 The Equally-Tempered Archetype

12.844 beats between the fifth harmonic of C4 and the fourth harmonic of E²4 (1295.283Hz). This triad

is certainly more consonant than the eleven-tone equally-tempered major triad previously presented (C,

EË, and G³), with a major third 436.364 cents in size a perfect fifth 654.545 cents in size, although when

compared with the theoretical major triad of twelve-tone equal temperament (C, E¢, and G), which

presents an equal major third ( 3 2 ) and perfect fifth ( 712 ]2[ ) 400.000 and 700.000 cents in size, it is

quite dissonant in character.

Table 74. The beating characteristics of the thirteen-tone equally-tempered major triad

C4

(PARTIAL)

FREQUENCY

(HERTZ)

E²4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

G£4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

5

6

– – – –

7

– – – –

8

261.626

– – – –

– – – –

523.251

– – – –

784.877

– – – –

1046.502

– – – –

1308.128

1569.753

– – – –

1831.379

– – – –

2093.005

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

– – – –

5

– – – –

6

– – – –

– – – –

323.821

– – – –

– – – –

647.642

– – – –

971.463

– – – –

– – – –

1295.283

– – – –

1619.104

– – – –

1942.925

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

12.844

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

4

– – – –

– – – –

5

– – – –

– – – –

– – – –

400.802

– – – –

– – – –

801.603

– – – –

– – – –

1202.405

– – – –

1603.206

– – – –

– – – –

2004.008

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

16.726

– – – –

– – – –

– – – –

– – – –

33.453

– – – –

– – – –

– – – –

– – – –

The two microtonal inflections of 3/2 are also explored as simultaneous sonorities, with the dyads C and

Gí, and Gì and C, or 13-et thirteen quarter-tones ( 713 ]2[ ), and C and G£, or the 13-et superfifth ( 813 ]2[ ).

Ex. 42. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 50

The Equally-Tempered Archetype 137

Other sonorities include the dyads G£ and B¸, or the 13-et augmented second ( 313 ]2[ ); Gí and B¸, or

the 13-et grave or small major third ( 413 ]2[ ); E² and GÍ, or the 13-et subfourth ( 513 ]2[ , approximately

47/36, or 461.538 cents); DÚ and GÍ, or 13-et eleven quarter-tones ( 613 ]2[ , approximately 705/512, or

553.846 cents); D² and A£, or the 13-et superfifth ( 813 ]2[ ), E² and C, or the 13-et seventeen quarter-

tones ( 913 ]2[ , approximately 21/13, or 830.769 cents); and DÚ and C, or the 13-et acute or large major

sixth ( 1013 ]2[ , approximately 1024/601, or 923.077 cents).212 The following table depicts the tonal

resources of thirteen-tone equal temperament, indicating degree, notation, interval, ratio, frequency,

cents, and tuning.

Table 75. The thirteen-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE

NUMBER

NOTE INTERVAL RATIO

(DECIMAL)

FREQUENCY

(HERTZ)

CENTS TUNING

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÑÓ

ÐÑ

C

D¹

D²

DÚ

E²

F³

FË / Gì

FÍ / Gí

G£

GÍ / Aí

A£

B¸

B¢

C

unison

13-et Pythagorean limma

13-et just minor tone

13-et augmented second

13-et grave or small major third

13-et subfourth

13-et eleven quarter-tones

13-et thirteen quarter-tones

13-et superfifth

13-et seventeen quarter-tones

13-et acute or large major sixth

13-et Pythagorean minor seventh

13-et Pythagorean major seventh

octave

1.000000

1.054766

1.112531

1.173460

1.237726

1.305512

1.377009

1.452423

1.531966

1.615866

1.704361

1.797702

1.896155

2.000000

261.626

275.954

291.067

307.007

323.821

341.555

360.261

379.991

400.802

422.752

445.904

470.325

496.083

523.251

0.000

92.308

184.616

276.923

369.231

461.538

553.846

646.154

738.462

830.769

923.077

1015.385

1107.692

1200.000

+00

ß08

ß15

ß23

ß31

ß38

ß46

ß54

+38

+31

+23

+15

+08

+00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to

represent sléndro and pélog scales.

Ex. 43. 13-tone equal temperament

212 Daniélou, Tableau Comparatif des Intervalles Musicaux 75-34.

138 The Equally-Tempered Archetype

The following two tuning matrixes contain the data required to represent the temperament chromatically

within the program memory of the Akai S3000XL.

Table 76. 13-tone equal temperament tuning matrix no. 1 (Program 13)

NOTE C D¸ D² DÚ E² F³ Gì G£ GÍ A£ B¸ B¢

KEY C C! D D! E F F! G G! A A! B

CENTS +00 ß08 ß15 ß23 ß31 ß38 ß46 +38 +31 +23 +15 +08

Table 77. 13-tone equal temperament tuning matrix no. 2 (Program 14)

NOTE – – – – – – – – – – – – Gí – – – – – – – – – –

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +00 +00 +00 +00 +00 +46 +00 +00 +00 +00 +00

Fourteen-Tone Equal Temperament Fourteen-tone equal temperament ( 14 2 ) presents a subdivision of the 7-et grave or small tone ( 7 2 )

into two equal tones 85.714 cents in size, which represent the 14-et Pythagorean limma (approximately

269/256). The temperament may alternatively be expressed as the juxtaposition of two sets of seven-

tone equally-tempered intervals, or seven sets of two-tone equally-tempered intervals interlocked a 14-et

Pythagorean limma ( 14 2 ) apart. Within the sléndro scalar scheme two options become available for the

approximation of the major second, being D³ or DË, or the 7-et grave or small tone ( 7 2 ) and 14-et five

quarter-tones ( 314 ]2[ , approximately 297/256, or 257.143 cents); two for the major third, being E³ or

E£, or the 7-et neutral third ( 27 ]2[ , approximately 128/105, or 342.857 cents) and 14-et acute or large

major third ( 514 ]2[ , approximately 16807/13122, or 428.571 cents); as well as two options for the

augmented sixth, being AÚ or AÍ, or the 14-et augmented sixth ( 1114 ]2[ , approximately 512/297, or

942.857 cents) and 7-et twenty-one quarter-tones ( 67 ]2[ , approximately 50989/28147, or 1028.571

The Equally-Tempered Archetype 139

cents).213 The pélog scalar scheme accommodates only the latter set of intervals. Bar 58 of the work

presents D³ and DË as an intonational reinterpretation of a melodic sequence, while bar 59 alternatively

presents C and D³, and C and DË as intonational reinterpretations of a simultaneous sonority.

Ex. 44. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 58-59

A further example of an intonational reinterpretation of a melodic sequence is bar 61, which highlights AÚ

and AÍ.

Ex. 45. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 61

In spite of E£ ( 514 ]2[ ), or the 14-et acute or large major third being technically closer to the just major

third (5/4) – E³ ( 27 ]2[ ), or the 7-et neutral third and E£ ( 514 ]2[ ) represent 5/4 with a falsity of ß43.457

and +42.258 on each count – E³ ( 27 ]2[ ) is relegated with the role of primary tone (and not the

expected secondary ‘auxiliary’ tone) for simple aesthetic considerations; hereby presenting an exception

to the rule where the accorded status of a tone is dependent on the accuracy of the approximation to

just intervals. It must be nevertheless noted that the 1.199 cents difference between the tones is

negligible, and therefore not a point of theoretical contention.

Blackwood offers the following description of fourteen-tone equal temperament: “This very

discordant tuning offers two highly contrasting modal arrangements. One is the division of an octave into

213 Daniélou, Tableau Comparatif des Intervalles Musicaux 24-108.

140 The Equally-Tempered Archetype

seven equal parts, sounding like a diatonic scale with no distinctions of major and minor. The other is a

combination of two differently tuned diminished seventh chords.”214

One new triad (essentially a fourteen-tone equally-tempered major triad) is introduced to the work

via this temperament, which consists of the pitches C, E³, and G², and presents a major third 342.857

cents in size ( 27 ]2[ , or 7-et neutral third) and a perfect fifth 685.714 cents in size ( 47 ]2[ , or 7-et grave

or small fifth). The temperament produces the just major third (5/4) and just perfect fifth (3/2) with a

falsity of ß43.457 and ß16.241 cents on each count. G²4 (388.774Hz), or 7-et grave or small fifth

( 47 ]2[ ), presents 7.329 beats between the third harmonic of C4 and the second harmonic of G²4

(777.548Hz), and 14.658 beats between the sixth harmonic of C4 and the fourth harmonic of G²4

(1555.096Hz); while E³4 (318.925Hz), or the 7-et neutral third ( 27 ]2[ ), 32.427 beats between the fifth

harmonic of C4 and the fourth harmonic of E³4 (1275.700Hz).

Table 78. The beating characteristics of the fourteen-tone equally-tempered major triad

C4

(PARTIAL)

FREQUENCY

(HERTZ)

E³4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

G²4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

5

6

– – – –

7

– – – –

8

261.626

– – – –

– – – –

523.251

– – – –

784.877

– – – –

1046.502

– – – –

1308.128

1569.753

– – – –

1831.379

– – – –

2093.005

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

– – – –

5

– – – –

6

– – – –

– – – –

318.925

– – – –

– – – –

637.850

– – – –

956.776

– – – –

– – – –

1275.700

– – – –

1594.626

– – – –

1913.551

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

32.427

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

4

– – – –

– – – –

5

– – – –

– – – –

– – – –

388.774

– – – –

– – – –

777.548

– – – –

– – – –

1166.322

– – – –

1555.096

– – – –

– – – –

1943.870

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

7.329

– – – –

– – – –

– – – –

– – – –

14.658

– – – –

– – – –

– – – –

– – – –

The concept of exploiting the intervallic nuances of two microtonal inflections is developed further in

fourteen-tone equal temperament, with its incorporation into simultaneous sonorities as a structural

compositional device. Bar 62 features a repeat of the harmonic sequence of C and GÚ, or the 14-et

augmented fifth ( 914 ]2[ , approximately 26244/16807, or 771.429 cents); AÚ and G², or the 14-et

214 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag.

The Equally-Tempered Archetype 141

augmented sixth ( 1114 ]2[ ); A² and F¢, or the 7-et grave or small major sixth ( 57 ]2[ ); and G² and E£, and

F¢ and DË, or the or 14-et augmented sixth ( 1114 ]2[ ); with the two final dyads replaced by G² and E³,

and F¢ and D³, or the 7-et grave or small major sixth ( 57 ]2[ ). The recapitulation of the harmonic

sequence emphasizes intervallic diversity available within the two alternatives for major seconds and

thirds.215

Ex. 46. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 62

The following two bars outline a fourteen-tone equally-tempered sléndro scalar descent with the pitches

C, AÍ, A², G², F¢, E³, D³, and C – equal to 1200.000, 1028.571, 857.143, 685.714, 514.286, 342.857,

and 171.429 cents respectively. The melodic line is further supported via the harmonic progression of

E³ and C, D³ and AÍ, C and A², or the 7-et grave or small major sixth ( 57 ]2[ ); and C and G², or the 7-et

grave or small fifth ( 47 ]2[ ); followed singularly by F¢, or the 7-et acute or large fourth ( 37 ]2[ ); E³, or the

7-et neutral third ( 27 ]2[ ); and D³, or the 7-et grave or small tone ( 7 2 ), to a resolution provided by the

simultaneous sonority of C and E³, or the 7-et neutral third ( 27 ]2[ ), which implies a fourteen-tone

equally-tempered major triad.

Ex. 47. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 63-64

Additional simultaneous sonorities include the pitches C and F¢, and F¢ and AÍ, or the 7-et acute or large

fourth ( 37 ]2[ ); D¹ and GÚ, or the 7-et grave or small fifth ( 47 ]2[ ); and D¹ and AÍ, or the 14-et

215 Daniélou, Tableau Comparatif des Intervalles Musicaux 109.

142 The Equally-Tempered Archetype

augmented sixth ( 1114 ]2[ ). The following table depicts the tonal resources of fourteen-tone equal

temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.

Table 79. The fourteen-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE

NUMBER

NOTE INTERVAL RATIO

(DECIMAL)

FREQUENCY

(HERTZ)

CENTS TUNING

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÑÓ

ÑÔ

ÐÑ

C

D¹

D³

DË / Eì

E³

E£

F¢

F¥

G²

GÚ

A²

AÚ

AÍ / Bí

B¢

C

unison

14-et Pythagorean limma

7-et grave or small tone

14-et five quarter-tones

7-et neutral third

14-et acute or large major third

7-et acute or large fourth

equal or acute or large tritone, or augmented fourth

7-et grave or small fifth

14-et augmented fifth

7-et grave or small major sixth

14-et augmented sixth

7-et twenty-one quarter-tones

14-et Pythagorean major seventh

octave

1.000000

1.050757

1.104090

1.160129

1.219014

1.280887

1.345900

1.414214

1.485994

1.561418

1.640671

1.723946

1.811447

1.903390

2.000000

261.626

274.905

288.858

303.520

318.925

335.113

352.122

369.994

388.774

408.507

429.241

451.028

473.921

497.976

523.251

0.000

85.714

171.429

257.143

342.857

428.571

514.286

600.000

685.714

771.429

857.143

942.857

1028.571

1114.286

1200.000

+00

ß14

ß29

+57

ß57

+29

+14

+00

ß14

ß29

ß43

ß57

+29

+14

+00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to

represent sléndro and pélog scales.

Ex. 48. 14-tone equal temperament

The Equally-Tempered Archetype 143

The following two tuning matrixes contain the data required to represent the temperament chromatically

within the program memory of the Akai S3000XL.

Table 80. 14-tone equal temperament tuning matrix no. 1 (Program 15)

NOTE C D¹ D³ E³ E£ F¢ F¥ G² GÚ A² AÍ B¢

KEY C C! D D! E F F! G G! A A! B

CENTS +00 ß14 ß29 +43 +29 +14 +00 ß14 ß29 ß43 +29 +14

Table 81. 14-tone equal temperament tuning matrix no. 2 (Program 16)

NOTE – – – – – – DË – – – – – – – – – – AÚ – – – –

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +00 +00 ß43 +00 +00 +00 +00 +00 +43 +00 +00

Fifteen-Tone Equal Temperament Fifteen-tone equal temperament ( 15 2 ), with its twofold tripartite division of the octave, presents a

subdivision of the 5-et supermajor second ( 5 2 ) into three equal tones 80.000 cents in size, which

represent the 15-et grave or small just chromatic semitone (approximately 22/21).216 The temperament

may alternatively be expressed as the juxtaposition of three sets of five-tone equally-tempered intervals,

or five sets of three-tone equally-tempered intervals interlocked a 15-et grave or small just chromatic

semitone ( 15 2 ) apart. Blackwood describes the temperament thus: “One of the most fascinating of the

equal tunings, this contains triads that are sufficiently in tune to serve to serve as the final harmony in

cadences. The major scale, however, is so strange that even the most common diatonic progressions are

disturbing unless precautions are taken.”217

One new triad (essentially a fifteen-tone equally-tempered minor triad) is introduced to the work

via this temperament, which consists of the pitches C, EÝ, and G¢, and presents a minor third 320.000

cents in size ( 415 ]2[ , approximately 77/64, or 15-et just minor third) and a perfect fifth 720.000 cents in

size ( 35 ]2[ , or 5-et acute or large fifth).218 The temperament produces the just minor third (6/5) and just

perfect fifth (3/2) with a falsity of +4.359 and +18.045 cents on each count. G¢4 (396.550Hz), or 5-et

acute or large fifth ( 35 ]2[ ), presents 8.224 beats between the third harmonic of C4 and the second

216 Daniélou, Tableau Comparatif des Intervalles Musicaux 22.

217 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag.

218 Daniélou, Tableau Comparatif des Intervalles Musicaux 82.

144 The Equally-Tempered Archetype

harmonic of G¢4 (793.100Hz), and 16.447 beats between the sixth harmonic of C4 and the fourth

harmonic of G¢4 (1586.201Hz); while EÝ4 (314.742Hz), or the 15-et just minor third ( 415 ]2[ ), 3.957

beats between the sixth harmonic of C4 and the fifth harmonic of EÝ4 (1573.711Hz).

Ex. 49. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 66

Table 82. The beating characteristics of the fifteen-tone equally-tempered minor triad

C4

(PARTIAL)

FREQUENCY

(HERTZ)

EÝ4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

G¢4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

– – – –

5

6

– – – –

7

– – – –

8

261.626

– – – –

– – – –

523.251

– – – –

784.877

– – – –

1046.502

– – – –

– – – –

1308.128

1569.753

– – – –

1831.379

– – – –

2093.005

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

– – – –

5

6

– – – –

– – – –

– – – –

– – – –

314.742

– – – –

– – – –

629.484

– – – –

944.226

– – – –

– – – –

1258.968

– – – –

1573.711

1888.453

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

3.957

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

– – – –

4

– – – –

– – – –

5

– – – –

– – – –

– – – –

396.550

– – – –

– – – –

793.100

– – – –

– – – –

1189.651

– – – –

– – – –

1586.201

– – – –

– – – –

1982.751

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

8.224

– – – –

– – – –

– – – –

– – – –

– – – –

16.447

– – – –

– – – –

– – – –

– – – –

The closest approximation of the major scale available in fifteen-tone equal temperament in both sléndro

and pélog forms is considered as presenting the series: 0.000, 240.000, 400.000, 480.000, 720.000,

880.000, 1120.000, and 1200.000 cents, and therefore representing the just major tone (9/8), just major

The Equally-Tempered Archetype 145

third (5/4), just perfect fourth (4/3), just perfect fifth (3/2), just major sixth (5/3), and just major seventh

(15/8) with falsity of +36.090, +13.686, ß18.045, +18.045, ß4.359, and +31.731 cents on each

count. Major, minor, and diminished triads produced via this temperament presents major thirds 400.000

cents in size, minor thirds with a range between 240.000 and 320.000 cents, perfect fifths with a range

between 640.000 and 720.000 cents, and a diminished fifth 560.000 cents in size. The most

problematic triad being D minor, or the pitches D£, F², and A, with a minor third and perfect fifth

represented by the 5-et supermajor second ( 5 2 ) and 15-et thirteen quarter-tones ( 815 ]2[ , 741/512, or

640.000 cents),219 and a deviation from just intervals of ß75.641 and ß61.955 cents. In harmonic terms,

the two intervals of the D minor triad essentially approximate a supermajor second (59049/51300, or

243.545 cents) and the division of the disjunction between GÝ and G³, or thirteen quarter-tones (90/62,

or 645.188 cents), and therefore contributing nothing with any possibility of being orally interpreted as

any form of a major triad.

Ex. 50. The fifteen-tone equally-tempered major scale

The closest approximation of the minor scale on the other hand presents the series: 0.000, 240.000,

320.000, 480.000, 720.000, 800.000, 1120.000, and 1200.000 cents, and therefore representing the

just major tone (9/8), just minor third (6/5), just perfect fourth (4/3), just perfect fifth (3/2), just minor sixth

(8/5), and acute or large minor seventh (9/5) with a falsity of +36.090, +4.359, ß18.045, +18.045,

ß13.686, and +2.404 cents on each count.

Ex.51. The fifteen-tone equally-tempered minor scale

The sléndro scalar scheme provides two options for the approximation of the major second, being D³

or D£, or the 15-et grave or small tone ( 215 ]2[ , approximately 34/31, or 160.000 cents) and 5-et

supermajor second ( 5 2 ); while pélog, two options for the tritone, being FË or FÍ, or 15-et eleven

quarter-tones ( 715 ]2[ , 1024/741, or 560.000 cents) and 15-et thirteen quarter-tones ( 815 ]2[ ), as well as

two options for the major seventh, being B³ or B£, or the 15-et neutral seventh ( 1315 ]2[ , approximately

31/17, or 1040.000 cents) and 15-et just diatonic major seventh ( 1415 ]2[ , approximately 489/256, or

219 Daniélou, Tableau Comparatif des Intervalles Musicaux 137.

146 The Equally-Tempered Archetype

1120.000 cents).220 Bar 68 is representative of intonational reinterpretations of a simultaneous sonority

based on the two major seconds, which represent the just major tone (9/8) with a falsity of ß43.013

and +39.635 cents on each count. The latter interval is further represented by the simultaneous

sonorities of D£ and F², and G¢ and AÚ in bar 66. The mean deviation of the two intervals from the just

major tone (9/8) is significantly close to the interval of the great diesis (128/125), or “the defect of three

major thirds from an octave.”221

Ex. 52. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 68

Bars 69 and 70 depicts the two microtonal inflections of the major seventh, or the 15-et neutral seventh

( 1315 ]2[ ) and 15-et just diatonic major ( 1415 ]2[ ).

Ex. 53. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 69-70

Bar 72 on the other hand, depicts the two microtonal inflections of the tritone, 15-et eleven quarter-

tones ( 715 ]2[ ) and 15-et thirteen quarter-tones ( 815 ]2[ ); while the following bar outlines a fifteen-tone

equally-tempered pélog scalar descent (omitting CÚ) with the pitches G¢, FÍ, FË, F², and EÝ – equal to

720.000, 640.000, 560.000, 480.000, and 320.000 cents respectively, and therefore a scalar

progression with equal tones 80.000 cents in size, or the 15-et grave or small just chromatic semitone

220 Daniélou, Tableau Comparatif des Intervalles Musicaux 23-136.

221 The juxtaposition of three just major thirds (5/4), equal to 1158.941 cents is a figure that diminishes the octave

(2/1) by 41.059 cents, and the interval termed as the ‘great diesis’ with a frequency ratio of 128/125. For a further

discussion, see Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 453.

The Equally-Tempered Archetype 147

( 15 2 ). Framed as simultaneous sonorities with repeated C's, the resolution is provided by the

resounding of the fundamental in the downbeat of sixteen-tone equal temperament.

Ex. 54. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 72-73

The following table depicts the tonal resources of fifteen-tone equal temperament, indicating degree,

notation, interval, ratio, frequency, cents, and tuning.

Table 83. The fifteen-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE

NUMBER

NOTE INTERVAL RATIO

(DECIMAL)

FREQUENCY

(HERTZ)

CENTS TUNING

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÑÓ

ÑÔ

ÑÕ

ÐÑ

C

CÚ

D³

D£

EÝ

E¢

F²

FË / Gì

FÍ / Gí

G¢

A¸

A

AÚ

B³

B£

C

unison

15-et grave or small just chromatic semitone

15-et grave or small tone

5-et supermajor second

15-et just minor third

equal or Pythagorean major third, or ditone

5-et grave or small fourth

15-et eleven quarter-tones

15-et thirteen quarter-tones

5-et acute or large fifth

equal or Pythagorean minor sixth

15-et just major sixth

5-et augmented sixth

15-et neutral seventh

15-et just diatonic major seventh

octave

1.000000

1.047294

1.096825

1.148698

1.203025

1.259921

1.319508

1.381913

1.447269

1.515717

1.587401

1.662476

1.741101

1.823445

1.909683

2.000000

261.626

273.999

286.957

300.529

314.742

329.628

345.217

361.544

378.643

396.550

415.305

434.946

455.517

477.060

499.622

523.251

0.000

80.000

160.000

240.000

320.000

400.000

480.000

560.000

640.000

720.000

800.000

880.000

960.000

1040.000

1120.000

1200.000

+00

ß20

ß40

+40

+20

+00

ß20

+60

+40

+20

+00

ß20

ß40

ß60

+20

+00

148 The Equally-Tempered Archetype

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to

represent sléndro and pélog scales.

Ex. 55. 15-tone equal temperament

The following two tuning matrixes contain the data required to represent the temperament chromatically

within the program memory of the Akai S3000XL.

Table 84. 15-tone equal temperament tuning matrix no. 1 (Program 17)

NOTE C CÚ D£ EÝ E¢ F² FË G¢ A¸ A A! B£

KEY C C! D D! E F F! G G! A A! B

CENTS +00 ß20 +40 +20 +00 ß20 ß40 +20 +00 ß20 ß40 +20

Table 85. 15-tone equal temperament tuning matrix no. 2 (Program 18)

NOTE – – – – D³ – – – – – – FÍ – – – – – – B³ – –

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +00 ß40 +00 +00 +00 +40 +00 +00 +00 +40 +00

The Equally-Tempered Archetype 149

Sixteen-Tone Equal Temperament Sixteen-tone equal temperament ( 16 2 ) presents a series of sixteen equal tones 75.000 cents in size,

which represent the 16-et grave or small just chromatic semitone (approximately 47/45),222 and is what

may be interpreted as being a scale made up of sixteen equal three-quarter-tone steps. The

temperament may alternatively be expressed as the juxtaposition of two sets of eight-tone equally-

tempered intervals, four sets of four-tone equally-tempered intervals, or eight sets of two-tone equally-

tempered intervals interlocked a 16-et grave or small just chromatic semitone ( 16 2 ) apart. Blackwood

makes the following observations about the temperament: “Triads in 16-note tuning, although

recognizable, are too discordant to serve as the final harmony in cadences. But keys can still be

established by successions of altered subdominant and dominant harmonies.”223

The work outlines a sixteen-tone equally-tempered minor triad, within an added-note chord

incorporating CÚ, and therefore essentially what could be interpreted as being a form of a Cmin(add"9)

tetrad if enharmonically reinterpreted. The triad consists of the pitches C, E¸, and G², and presents a

minor third 300.000 cents in size ( 4 2 , or the equal minor third) and a perfect fifth 675.000 cents in size

( 916 ]2[ , 16-et grave or small fifth, or approximately 189/128).224 The comparison with the just minor third

(6/5) and just perfect fifth (3/2) reveal a falsity of +4.359 and +18.045 cents on each count. G²4

(386.375Hz), or 16-et grave or small fifth ( 916 ]2[ ), presents 12.126 beats between the third harmonic of

C4 and the second harmonic of G²4 (772.751Hz), and 24.251 beats between the sixth harmonic of C4

and the fourth harmonic of G¢4 (1545.502Hz); while E¸4, or the equal minor third ( 4 2 ), 14.118 beats

between the sixth harmonic of C4 and the fifth harmonic of E¸4.

Ex. 56. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 78

222 Daniélou, Tableau Comparatif des Intervalles Musicaux 20.

223 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag.

224 Daniélou, Tableau Comparatif des Intervalles Musicaux 129.

150 The Equally-Tempered Archetype

Table 86. The beating characteristics of the sixteen-tone equally-tempered minor triad

C4 (PARTIAL)

FREQUENCY

(HERTZ)

E¸4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

G²4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

– – – –

5

6

7

– – – –

– – – –

8

261.626

– – – –

– – – –

523.251

– – – –

784.877

– – – –

1046.502

– – – –

– – – –

1308.128

1569.753

1831.379

– – – –

– – – –

2093.005

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

– – – –

5

– – – –

6

– – – –

– – – –

– – – –

311.127

– – – –

– – – –

622.254

– – – –

933.381

– – – –

– – – –

1244.508

– – – –

1555.635

– – – –

1866.762

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

14.118

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

– – – –

4

– – – –

– – – –

5

– – – –

– – – –

– – – –

386.375

– – – –

– – – –

772.751

– – – –

– – – –

1159.126

– – – –

– – – –

1545.502

– – – –

– – – –

1931.877

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

12.126

– – – –

– – – –

– – – –

– – – –

– – – –

24.251

– – – –

– – – –

– – – –

– – – –

The pélog scalar scheme now provides two options for the approximation of the augmented unison,

being CÚ or CÍ, or the 16-et grave or small just chromatic semitone ( 16 2 ) and equal three-quarter-tone

( 8 2 ); two for the fourth, being Fí or F¢, or nine equal quarter-tones ( 38 ]2[ ) and the 16-et acute or

large fourth 716 ]2[ , approximately 256/189, or 525.000 cents); two for the fifth, being G² or G£, or the

16-et grave or small fifth ( 916 ]2[ ) and fifteen equal quarter-tones ( 58 ]2[ ); as well as two options for the

major seventh, being B³ or B£, or twenty-one equal quarter-tones ( 78 ]2[ ) and the 16-et acute or large

major seventh ( 1516 ]2[ , approximately 90/47, or 1125.000 cents). Sléndro accommodates the fourth

and fifth. The microtonal nuances of these pitches are exploited both melodically and harmonically

throughout the section of the work demarcated as ‘16-tone equal temperament’. Simultaneous

sonorities include the dyads F¢ and G², or the equal three-quarter-tone ( 8 2 ); F¢ and G£, or 16-et acute

or large tone ( 316 ]2[ , approximately 41/36, or 225.000 cents); CÚ and F¢, or the equal minor third ( 4 2 ),

CÍ and F¢, or the 16-et grave or small major third ( 316 ]2[ , approximately 77/62, or 375.000 cents); D¢ and G², and CÚ and F¥, or nine equal quarter-tones ( 38 ]2[ ); D¢ and G£, and E¸ and B¹, or the 16-et

acute or large fourth ( 716 ]2[ ); CÚ and GÍ, or the equal tritone ( 2 2 ); C and G², and F¢ and C, or the 16-

et grave or small fifth ( 916 ]2[ ); C and G£, or fifteen equal quarter-tones ( 58 ]2[ ); E¸ and B£, C and GÍ,

B¹ and F¥, and G² and E¸, or 16-et seventeen quarter-tones ( 1116 ]2[ , approximately 124/77, or

825.000 cents); as well as C and B¹, or the 16-et grave or small minor seventh ( 1316 ]2[ , approximately

The Equally-Tempered Archetype 151

72/41, or 975.000 cents).225 The following table depicts the tonal resources of sixteen-tone equal

temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.

Table 87. The sixteen-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE

NUMBER

NOTE INTERVAL RATIO

(DECIMAL)

FREQUENCY

(HERTZ)

CENTS TUNING

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÑÓ

ÑÔ

ÑÕ

ÑÖ

ÐÑ

C

CÚ

CÍ / Dí

D¢

E¸

E²

EË / Fí

F¢

F¥

G²

G£

GÍ / Aí

A¢

B¹

B³

B£

C

unison

16-et grave or small just chromatic semitone

equal three-quarter-tone

16-et acute or large tone

equal or Pythagorean minor third, or trihemitone

16-et grave or small major third

nine equal quarter-tones

16-et acute or large fourth

equal or acute or large tritone, or augmented fourth

16-et grave or small fifth

fifteen equal quarter-tones, or superfifth

16-et seventeen quarter-tones

equal or Pythagorean major sixth

16-et grave or small minor seventh

twenty-one equal quarter-tones, or neutral seventh

16-et acute or large major seventh

octave

1.000000

1.044274

1.090508

1.138789

1.189207

1.241858

1.296840

1.354256

1.414214

1.476826

1.542211

1.610490

1.681793

1.756252

1.834008

1.915207

2.000000

261.626

273.209

285.305

297.936

311.127

324.902

339.286

354.308

369.994

386.375

403.482

421.345

440.000

459.480

479.823

501.067

523.251

0.000

75.000

150.000

225.000

300.000

375.000

450.000

525.000

600.000

675.000

750.000

825.000

900.000

975.000

1050.000

1125.000

1200.000

+00

ß25

+50

+25

+00

ß25

ß50

+25

+00

ß25

+50

+25

+00

ß25

ß50

+25

+00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to

represent sléndro and pélog scales.

Ex. 57. 16-tone equal temperament

225 Daniélou, Tableau Comparatif des Intervalles Musicaux 21-128.

152 The Equally-Tempered Archetype

The following two tuning matrixes contain the data required to represent the temperament chromatically

within the program memory of the Akai S3000XL.

Table 88. 16-tone equal temperament tuning matrix no. 1 (Program 19)

NOTE C CÚ D¢ E¸ E² F¢ F¥ G² GÍ A¢ B¹ B£

KEY C C! D D! E F F! G G! A A! B

CENTS +00 ß25 +25 +00 ß25 +25 +00 ß25 +25 +00 ß25 +25

Table 89. 16-tone equal temperament tuning matrix no. 2 (Program 20)

NOTE – – CÍ – – – – – – Fí – – G£ – – – – – – B³

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +50 +00 +00 +00 ß50 +00 +50 +00 +00 +00 ß50

Seventeen-Tone Equal Temperament Seventeen-tone equal temperament is generated by the factor 17 2 , and with its seventeen equal tones

70.588 cents in size, which represent the 17-et grave or small just chromatic semitone, is able to

approximate the important interval of just intonation termed as the grave or small just chromatic semitone

(25/24) with extreme accuracy – a value expressing a falsity of ß0.084 cents.226 This is the ratio in just

intonation “applied to any diatonic note to produce the sharpened form of that note.”227 Blackwood

describes the temperament thus: “17-note triads are very discordant due to the large major third, so the

fundamental consonant harmony of the tuning is a minor triad with an added minor seventh. The scale is

very good due to the relatively small minor second.”228 Blackwood is of course referring to the major

third 423.529 cents in size (E£), which may be referred to as the 17-et acute or large major third

( 617 ]2[ , or approximately 327/256), and deviates from the just major third (5/4) by +37.216 cents. The

present scheme relegates this third to the position of auxiliary tone, due to the better approximation of

226 Daniélou, Tableau Comparatif des Intervalles Musicaux 20.

227 John Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters,” Perspectives of New Music

29.2 (Summer, 1991): 109.

228 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag.

The Equally-Tempered Archetype 153

the just major third (5/4) provided by E³ ( 517 ]2[ , or approximately 38/31), the 17-et neutral third, at

352.941 cents.229 The alternative theorization produces a dissonant triad nevertheless, yet seventeen-

tone equal temperament is the second temperament to contain what Blackwood terms as recognizable

diatonic scales and “perfect fifths within the range of recognizability”.230

The seventeen-tone equally-tempered major triad consists of the pitches C, E³, and G, and

presents a major third 352.941 cents in size ( 517 ]2[ , or 17-et neutral third) and a perfect fifth 705.882

cents in size ( 1017 ]2[ , or 17-et just perfect fifth). The temperament produces the just major third (5/4)

and just perfect fifth (3/2) with a falsity of ß33.373 and +3.927 cents on each count. G4 (393.330Hz),

or 17-et just perfect fifth ( 1017 ]2[ ), presents 1.783 beats between the third harmonic of C4 and the

second harmonic of G4 (786.659Hz), and 3.565 beats between the sixth harmonic of C4 and the fourth

harmonic of G4 (1573.318Hz); while E³4 (307.972Hz), or the 17-et neutral third ( 517 ]2[ ), 76.241 beats

between the fifth harmonic of C4 and the fourth harmonic of E³4 (1231.189Hz).

Ex. 58. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 87

229 Daniélou, Tableau Comparatif des Intervalles Musicaux 90-106.

230 Blackwood, The Structure of Recognizable Diatonic Tunings 197.

154 The Equally-Tempered Archetype

Table 90. The beating characteristics of the seventeen-tone equally-tempered major triad

C4

(PARTIAL)

FREQUENCY

(HERTZ)

E³4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

G4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

5

– – – –

6

7

– – – –

8

261.626

– – – –

– – – –

523.251

– – – –

784.877

– – – –

1046.502

– – – –

1308.128

– – – –

1569.753

1831.379

– – – –

2093.005

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

5

– – – –

– – – –

6

– – – –

– – – –

307.972

– – – –

– – – –

615.943

– – – –

923.915

– – – –

– – – –

1231.189

1539.858

– – – –

– – – –

1847.830

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

76.241

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

– – – –

4

– – – –

5

– – – –

– – – –

– – – –

393.330

– – – –

– – – –

786.659

– – – –

– – – –

1179.989

– – – –

– – – –

1573.318

– – – –

1966.648

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1.783

– – – –

– – – –

– – – –

– – – –

– – – –

3.565

– – – –

– – – –

– – – –

Blackwood’s consonant minor triad is located in this scheme within the pitches C, DÚ, and G, which when

enharmonically reinterpreted as C, EÝ, and G produces a minor third 282.353 cents in size, and therefore

a just minor third (6/5) with a falsity of ß33.288 cents. The exercise illustrates that the principal minor

triad is no more consonant than the principal major triad in seventeen-tone equal temperament.

Seventeen-tone equal temperament provides two options for the approximation of the

augmented unison, being CÚ or CÍ, or the 17-et grave or small just chromatic semitone ( 17 2 ) and 17-et

three-quarter-tone ( 217 ]2[ , approximately 243/224, or 141.176 cents); two for the major third, being E³

or E£, or the 17-et neutral third ( 517 ]2[ ) and 17-et acute or large major third ( 617 ]2[ ); two for the

tritone, being FÚ or GÝ, or the 17-et grave or small augmented fourth ( 817 ]2[ , approximately 709/512, or

564.706 cents) and 17-et acute or large diminished fifth ( 917 ]2[ , approximately 739/512, or 635.294

cents); two for the major sixth, being A³ or A£, or the 17-et neutral sixth ( 1217 ]2[ , approximately

835/512, or 847.059 cents, and 17-et acute or large major sixth ( 1317 ]2[ , approximately 435/256, or

917.647 cents); as well as two options for the major seventh, being B² or B£, or the 17-et grave or small

major seventh ( 1517 ]2[ , approximately 448/243, or 1058.824 cents) and 17-et grave or small major

seventh ( 1617 ]2[ , approximately 48/25, or 1129.412 cents).

Simultaneous sonorities include the dyads A£ and C, or the 17-et augmented second ( 417 ]2[ ,

approximately 512/435, or 282.353 cents); G and C, and B¸ and DÚ, or the 17-et just perfect fourth

The Equally-Tempered Archetype 155

( 717 ]2[ , approximately 681/512, or 494.118 cents); A³ and C, and C and E³, or the 17-et neutral third

( 517 ]2[ ); C and E£, or the 17-et acute or large major third ( 617 ]2[ ); C and FÚ, F and B², FÚ and B£, GÝ

and C, G and CÚ, and GÚ and CÍ, or the 17-et grave or small augmented fourth ( 817 ]2[ ); C and GÝ, CÚ

and G, and CÍ and GÚ, or the 17-et acute or large diminished fifth ( 917 ]2[ ); C and G, and DÚ and B¸, or

17-et just perfect fifth ( 1017 ]2[ ); DÚ and B£, and C and A³, or the 17-et neutral sixth ( 1217 ]2[ ); C and

A£, or the 17-et acute or large major sixth ( 1317 ]2[ ); CÚ and B², and C and B¸, or the 17-et Pythagorean

minor seventh ( 1417 ]2[ , approximately 3584/2025, or 988.235); as well as C and B², or the 17-et grave

or small major seventh ( 1517 ]2[ ).231 The following table depicts the tonal resources of seventeen-tone

equal temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.

Table 91. The seventeen-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE

NUMBER

NOTE INTERVAL RATIO

(DECIMAL)

FREQUENCY

(HERTZ)

CENTS TUNING

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÑÓ

ÑÔ

ÑÕ

ÑÖ

Ñ×

ÐÑ

C

CÚ

CÍ / Dí

D

DÚ

E³

E£

F

FÚ

GÝ

G

GÚ

A³

A£

B¸

B²

B£

C

unison

17-et grave or small just chromatic semitone

17-et three-quarter-tone

17-et just major tone

17-et augmented second

17-et neutral third

17-et acute or large major third

17-et just and Pythagorean perfect fourth

17-et grave or small augmented fourth

17-et acute or large diminished fifth

17-et just and Pythagorean perfect fifth

17-et augmented fifth

17-et neutral sixth

17-et acute or large major sixth

17-et Pythagorean minor seventh

17-et grave or small major seventh

17-et acute or large major seventh

octave

1.000000

1.041616

1.084964

1.130116

1.177147

1.226135

1.277162

1.330312

1.385674

1.443341

1.503407

1.565972

1.631142

1.699024

1.769730

1.843379

1.920093

2.000000

261.626

272.513

283.854

295.667

307.972

320.788

334.138

348.044

362.528

377.615

393.330

409.698

426.748

444.508

463.007

482.275

502.346

523.251

0.000

70.588

141.176

211.765

282.353

352.941

423.529

494.118

564.706

635.294

705.882

776.471

847.059

917.647

988.235

1058.824

1129.412

1200.000

+00

ß29

+41

+12

ß18

ß47

+24

ß06

ß35

+35

+06

ß24

ß53

+18

ß12

ß41

+29

+00

231 Daniélou, Tableau Comparatif des Intervalles Musicaux 21-139.

156 The Equally-Tempered Archetype

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to

represent sléndro and pélog scales.

Ex. 59. 17-tone equal temperament

The following two tuning matrixes contain the data required to represent the temperament chromatically

within the program memory of the Akai S3000XL.

Table 92. 17-tone equal temperament tuning matrix no. 1 (Program 21)

NOTE C CÍ D D! E³ F F! G G! A£ B¸ B²

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +41 +12 ß18 ß47 ß06 ß35 +06 ß24 +18 ß12 ß41

Table 93. 17-tone equal temperament tuning matrix no. 2 (Program 22)

NOTE – – CÚ – – – – E£ – – GÝ – – A³ – – – – B£

KEY C C! D D! E F F! G G! A A! B

CENTS +00 ß29 +00 +00 +24 +00 +35 +00 +47 +00 +00 +29

Eighteen-Tone Equal Temperament Eighteen-tone equal temperament ( 18 2 ) presents a subdivision of the equal major third ( 3 2 ) into six

equal tones 66.667 cents in size, or equal third-tones, which represent the 18-et grave or small just

chromatic semitone (approximately 133/128).232 The temperament may alternatively be expressed as the

juxtaposition of two sets of nine-tone equally-tempered intervals, three sets of six-tone equally-tempered

232 Daniélou, Tableau Comparatif des Intervalles Musicaux 18.

The Equally-Tempered Archetype 157

intervals, six sets of three-tone equally-tempered intervals, or nine sets of two-tone equally-tempered

intervals interlocked an 18-et grave or small just chromatic semitone ( 18 2 ) apart. Blackwood makes the

following observations about the temperament: “The perfect fifths are so out of tune that even seventh

chords are disturbingly discordant.”233 Major, minor, and diminished triads produced via the major scale

derived from this temperament presents major thirds with a range between 333.333 and 400.000 cents;

minor thirds between 266.667 and 333.333 cents; perfect fifths between 666.667 and 733.333 cents;

and a diminished fifth 600.000 cents in size. In comparison with the just perfect fifth (3/2), fifths within

the specified range present a deviation of between ß31.378 and +31.378 cents, which is roughly an

equal sixth-tone ( 36 2 ), and approaching two syntonic commas (6561/6400, or 43.013) – clearly

outside of the boundaries set by Blackwood in his theory of “perfect fifths within the range of

recognizability” and its formula: 74 a<v< 5

3 a, or 685.714<v<720.000.234

All triads prove to be problematic, and due to the fact of either producing the just major third

(5/4) and just perfect fifth (3/2) with a falsity of +13.683 and +31.378 cents, as in C major (C, E¢, and

G£) and F major (F², A², and C); or ß52.980 and ß35.288 cents, as in G major (G£, B², and D). Minor

triads produce the just minor third (6/5) and just perfect fifth (3/2) with a falsity of ß48.975 and

ß35.288 cents, as in D minor (D, F², and A²); +17.692 and ß35.288 cents, as in E minor (E¢, G£, and

B²); and +17.692 and +31.378 cents, as in A minor (A², C, and E¢); while the B diminished triad (B², D,

and F²) produces the just minor third (6/5) and just tritone (45/32) with a falsity of +17.692 and +9.776

cents.

Ex. 60. The eighteen-tone equally-tempered major scale

The work outlines an implied eighteen-tone equally-tempered minor triad, within an arpeggiated added-

note chord incorporating B¸, and therefore essentially a Cmin7 tetrad. The triad consists of the pitches C,

Eí and G£, and presents a minor third 333.333 cents in size ( 518 ]2[ , 18-et seven quarter-tones, or

approximately 40/33) and a perfect fifth 733.333 cents in size ( 1118 ]2[ , 18-et superfifth, or

approximately 189/128).235 Eí, although equal to 18-et seven quarter-tones ( 518 ]2[ ), functions

harmonically as a minor third. The comparison with the just minor third (6/5) and just perfect fifth (3/2)

reveal a falsity of +17.692 and +31.378 cents on each count. G£4 (399.616Hz), or the 18-et superfifth

( 1118 ]2[ ), presents 14.355 beats between the third harmonic of C4 and the second harmonic of G£4

233 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag.

234 Blackwood, The Structure of Recognizable Diatonic Tunings 197.

235 Daniélou, Tableau Comparatif des Intervalles Musicaux 18-129.

158 The Equally-Tempered Archetype

(799.232Hz), and 28.711 beats between the sixth harmonic of C4 and the fourth harmonic of G£4

(1598.464Hz); while Eí4 (317.175Hz), or 18-et seven quarter-tones ( 518 ]2[ ), 16.124 beats between

the sixth harmonic of C4 and the fifth harmonic of Eí4 (1585.877Hz). Bar 88 also serves to highlight the

highly dissonant nature of the two fifths available in eighteen-tone equal temperament.

Ex. 61. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 88

Table 94. The beating characteristics of the eighteen-tone equally-tempered minor triad

C4

(PARTIAL)

FREQUENCY

(HERTZ)

Eí4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

G£4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

– – – –

5

6

7

– – – –

– – – –

8

261.626

– – – –

– – – –

523.251

– – – –

784.877

– – – –

1046.502

– – – –

– – – –

1308.128

1569.753

1831.379

– – – –

– – – –

2093.005

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

– – – –

5

– – – –

6

– – – –

– – – –

– – – –

317.175

– – – –

– – – –

634.351

– – – –

951.526

– – – –

– – – –

1268.702

– – – –

1585.877

– – – –

1903.053

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

16.124

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

– – – –

4

– – – –

– – – –

5

– – – –

– – – –

– – – –

399.616

– – – –

– – – –

799.232

– – – –

– – – –

1198.848

– – – –

– – – –

1598.464

– – – –

– – – –

1998.080

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

14.355

– – – –

– – – –

– – – –

– – – –

– – – –

28.711

– – – –

– – – –

– – – –

– – – –

Eighteen-tone equal temperament provides two options for the approximation of the augmented

unison/minor second, being CÚ or DÝ, or the 18-et grave or small just chromatic semitone ( 18 2 ) and 9-et

The Equally-Tempered Archetype 159

great limma ( 9 2 ); two for the minor third, being Eì or Eí, or 9-et five quarter tones ( 29 ]2[ ) and 18-et

seven quarter-tones ( 518 ]2[ ); two for the fourth, being F² or F£, or the 18-et grave or small fourth

( 718 ]2[ , approximately 512/391, or 466.667 cents) and 9-et superfourth ( 49 ]2[ ); two for the fifth, being

G³ or G£, or the 9-et subfifth ( 59 ]2[ ) and 18-et superfifth ( 1118 ]2[ ); two for the major sixth, being A² or

AË, or the 18-et grave or small major sixth ( 1318 ]2[ , approximately 33/20, or 866.667 cents) and 9-et

nineteen quarter-tones ( 79 ]2[ , approximately 12/7, or 933.333 cents); as well as two options for the

major seventh, being B² or B£, or the 9-et grave or small major seventh ( 89 ]2[ , approximately 1024/553,

or 1066.667 cents) and 18-et acute or large major seventh ( 1718 ]2[ , approximately 256/133, or

1133.333 cents).236 Bars 89 and 90 illustrate the melodic utilization of the two alternative fifths, fourths

and minor thirds, each deviating approximately an equal sixth-tone ( 36 2 ) in either direction of its

associated justly intoned interval.

Ex. 62. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 89-90

The utilization of simultaneous sonorities in this temperament are somewhat limited due to the focus on

melodic development, although includes the dyads C and Eì, or the 9-et five quarter tones ( 29 ]2[ ), C

and A², or the 18-et grave or small major sixth ( 1318 ]2[ ), and C and AË, or 9-et nineteen quarter-tones

( 1418 ]2[ ). The following table depicts the tonal resources of eighteen-tone equal temperament,

indicating degree, notation, interval, ratio, frequency, cents, and tuning.

236 Daniélou, Tableau Comparatif des Intervalles Musicaux 31-116.

160 The Equally-Tempered Archetype

Table 95. The eighteen-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE

NUMBER

NOTE INTERVAL RATIO

(DECIMAL)

FREQUENCY

(HERTZ)

CENTS TUNING

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÑÓ

ÑÔ

ÑÕ

ÑÖ

Ñ×

ÑØ

ÐÑ

C

CÚ

DÝ

D

DË / Eì

DÍ / Eí

E¢

F²

F£

F¥

G³

G£

A¸

A²

AË / Bì

B¸

B²

B£

C

unison

18-et grave or small just chromatic semitone

9-et great limma, or large half-tone

equal or just major tone

9-et five quarter-tones

18-et seven quarter-tones

equal or Pythagorean major third, or ditone

18-et grave or small fourth

9-et superfourth

equal or acute or large tritone, or augmented fourth

9-et subfifth

18-et superfifth

equal or Pythagorean minor sixth

18-et grave or small major sixth

9-et nineteen quarter-tones

equal or Pythagorean minor seventh

9-et grave or small major seventh

18-et acute or large major seventh

octave

1.000000

1.039259

1.080060

1.122462

1.166529

1.212326

1.259921

1.309385

1.360790

1.414214

1.469734

1.527435

1.587401

1.649721

1.714488

1.781797

1.851749

1.924448

2.000000

261.626

271.897

282.571

293.665

305.194

317.175

329.628

342.568

356.017

369.994

384.520

399.616

415.305

431.609

448.554

466.164

484.465

503.485

523.251

0.000

66.667

133.333

200.000

266.667

333.333

400.000

466.667

533.333

600.000

666.667

733.333

800.000

866.667

933.333

1000.000

1066.667

1133.333

1200.000

+00

ß33

+33

+00

+67

+33

+00

ß33

+33

+00

ß33

+33

+00

ß33

+33

+00

ß33

+33

+00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to

represent sléndro and pélog scales.

Ex. 63. 18-tone equal temperament

The Equally-Tempered Archetype 161

The following two tuning matrixes contain the data required to represent the temperament chromatically

within the program memory of the Akai S3000XL.

Table 96. 18-tone equal temperament tuning matrix no. 1 (Program 23)

NOTE C DÝ D DÍ E¢ F² F¥ G³ A¸ A² B¸ B²

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +33 +00 +33 +00 ß33 +00 ß33 +00 ß33 +00 ß33

Table 97. 18-tone equal temperament tuning matrix no. 2 (Program 24)

NOTE – – CÚ – – DË – – F£ – – G£ – – AË – – B£

KEY C C! D D! E F F! G G! A A! B

CENTS +00 ß33 +00 ß33 +00 +33 +00 +33 +00 +33 +00 +33

Nineteen-Tone Equal Temperament Nineteen-tone equal temperament is generated by the factor 19 2 , and presents nineteen equal tones

63.158 cents in size, which represent the 19-et just diatonic semitone (approximately 531/512).237 It is a

temperament advocated by American musicologist, organist, and conductor Joseph Yasser (1893-

1981), and theorized in his book of 1932 entitled Theory of Evolving Tonality. A. R. McClure offers the

following discussion:

“Yasser’s speculations on the evolution of music led him to conclude that the diatonic scale of seven tones

plus five accidentals would, in time, give place to a supra-diatonic scale of twelve tones plus seven

accidentals. Then, on certain premises of his own and from assumptions of physical theorists, he

developed, by most ingenious logic, a fanciful scheme with fresh notation and terminology, designed to

serve the needs of composers in the future – the detailed plan of a hypothetical new order.”238

The temperament was theoretically perceived by Joseph Yasser as the historical expansion of a diatonic

system into a chromatic one (with the addition of five auxiliary tones), and a chromatic one into a ‘supra-

diatonic scale’ (with the further addition of seven auxiliary tones). The nineteen-tone scale is

accommodated by Yasser with a notation system incorporating a ten-line staff and two additional

symbols for accidentals. “On the theoretical side there are many points of interest, for the nineteen-tone

237 Daniélou, Tableau Comparatif des Intervalles Musicaux 18.

238 A. R. McClure, “Studies in Keyboard Temperaments,” The Galpin Society Journal 1 (Mar., 1948): 32-34

162 The Equally-Tempered Archetype

system allowed one to actually distinguish augmented intervals and their inversions, which on the normal

pianoforte have no separate existence,” points out McClure.239 Joseph Yasser envisaged systems of

temperament beyond the ‘supra-diatonic scale’, with the theorization of the ‘ultra-diatonics’ and systems

with tonal structures such as 19+12=31, and 31+19=50. The complete evolutionary chain is

expressed as the following combinations of scale degrees: 2+3=5, 5+2=7, 7+5=12, 12+7=19,

19+12=31, and 31+19=50.240

With regards to the aesthetic qualities of nineteen-tone equal temperament, Blackwood makes the

following observations: “Triads are smooth, but the scale sounds slightly out of tune because the leading

tone seems low with respects to the tonic. Diatonic behaviour is virtually identical to that of 12-note

tuning, but chromatic behaviour is very different. For example, a perfect fourth is divisible into two equal

parts, while an augmented sixth and a diminished seventh sound identical.”241 The pitch allocated as the

leading-tone that Blackwood refers to is B² at 1073.684 cents ( 1719 ]2[ , 19-et grave or small major

seventh, or approximately 119/64),242 which represents the just diatonic major seventh (15/8) with a

falsity of ß14.585 cents. Blackwood also makes note that “nineteen-note equal tuning contains diatonic

scales in which a major second spans three chromatic degrees, while a minor second spans two.”243

The nineteen-tone division of the octave (along with five, seven, twelve, thirty-one, forty-one, fifty-

three, three-hundred and six, and three-hundred and forty-seven) are represented in the list of nine

seminal propositions by theorists with regards to the cyclic division of the octave, or the Pythagorean

cycle of fifths. Although, the fact that the nineteenth 3/2 produces a tone 137.145 cents (3ñù/2óð, or

cyclic great limma) above the eleventh 2/1 in the nineteen cycle, reveals somewhat of an inferiority when

compared to the twelve-tone cycle, where the twelfth 3/2 produces a tone 23.460 cents (3ñò/2ñù, or

Pythagorean comma) above the seventh 2/1.

Just minor thirds and major sixths (6/5s and 5/3s) are represented well in this temperament, with a

falsity of +0.148 and ß0.148 cents on each count, while just major thirds and minor sixths (5/4s and

8/5s), represented adequately, with a falsity +7.366 and ß7.366 cents on each count. This is a

significant improvement to the falsities of 15.641 cents for 6/5 and 5/3, and 13.686 cents for 5/4 and 8/5

expounded by twelve-tone equal temperament.244 Nineteen-tone equal temperament is the third

239 Read, 20th-Century Microtonal Notation 98.

240 Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 431-32.

241 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag.

242 Daniélou, Tableau Comparatif des Intervalles Musicaux 37.

243 Easley Blackwood, “Modes and Chord Progressions in Equal Tunings,” Perspectives of New Music 29.2

(Summer, 1991): 168. 244 Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 400-32.

The Equally-Tempered Archetype 163

temperament to contain what Blackwood terms as recognizable diatonic scales and “perfect fifths within

the range of recognisability.”245

The nineteen-tone equally-tempered major scale generates consistent major triads, which produce

the just major third (5/4) and just perfect fifth (3/2) with a falsity of +7.366 and ß7.218 cents, as in C

major (C, E, and G), F major (F, A, and C), and G major (G, B², and D²). Equally consistent minor triads are

available, which produce the just minor third (6/5) and just perfect fifth with a falsity of +0.148 and

ß7.218 cents, as in D minor (D, F, and A), E minor (E, G, and B²), and A minor (A, C, and E); while the B

diminished triad (B², D², and F), produces the just minor third (6/5) and just tritone (45/32) with a falsity

of +0.148 and +41.355 cents. “In sum, all diatonic progressions of triads and seventh chords have the

same behaviour and produce the same musical effect in twelve-note and nineteen-note tuning,”

comments Blackwood, “save for slight differences only, the most noticeable being the peculiar tuning of

the nineteen-note major scale.” In striking contrast, “many nineteen-note chromatic progressions will

bring about alien melodic intervals.”246

Ex. 64. The nineteen-tone equally-tempered major scale

The work outlines a nineteen-tone equally-tempered minor triad in first inversion, which includes the

pitches C, EÝ, and G, and presents a minor third 315.789 cents in size ( 519 ]2[ , 19-et just minor third, or

approximately 6/5) and a perfect fifth 694.737 cents in size ( 1119 ]2[ , 19-et just perfect fifth, or

approximately 115/77).247 The comparison with the just minor third (6/5) and perfect fifth (3/2) reveal a

falsity of +0.148 and ß7.218 cents on each count. G4 (390.806Hz), or the 19-et just perfect fifth

( 1119 ]2[ ), presents 3.266 beats between the third harmonic of C4 and the second harmonic of G4

(781.611Hz), and 6.531 beats between the sixth harmonic of C4 and the fourth harmonic of G4 (1563.222Hz); while EÝ4 (313.978Hz), or the 19-et just minor third ( 519 ]2[ ), 0.134 beats between the

sixth harmonic of C4 and the fifth harmonic of EÝ4 (1569.888Hz).

245 Blackwood, The Structure of Recognizable Diatonic Tunings 197.

246 Blackwood, “Modes and Chord Progressions in Equal Tunings,” Perspectives of New Music 169-72.

247 Daniélou, Tableau Comparatif des Intervalles Musicaux 82-125.

164 The Equally-Tempered Archetype

Ex. 65. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 97

Table 98. The beating characteristics of the nineteen-tone equally-tempered minor triad

C4

(PARTIAL)

FREQUENCY

(HERTZ)

EÝ4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

G4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

– – – –

5

6

7

– – – –

– – – –

8

261.626

– – – –

– – – –

523.251

– – – –

784.877

– – – –

1046.502

– – – –

– – – –

1308.128

1569.753

1831.379

– – – –

– – – –

2093.005

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

– – – –

5

– – – –

6

– – – –

– – – –

– – – –

313.978

– – – –

– – – –

627.955

– – – –

941.933

– – – –

– – – –

1255.910

– – – –

1569.888

– – – –

1883.865

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

0.134

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

– – – –

4

– – – –

– – – –

5

– – – –

– – – –

– – – –

390.806

– – – –

– – – –

781.611

– – – –

– – – –

1172.417

– – – –

– – – –

1563.222

– – – –

– – – –

1954.028

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

3.266

– – – –

– – – –

– – – –

– – – –

– – – –

6.531

– – – –

– – – –

– – – –

– – – –

Nineteen-tone equal temperament provides two options for the approximation of the minor second,

being D¸ or DÝ, or the 19-et just diatonic semitone ( 19 2 ) and 19-et great limma 219 ]2[ , approximately

128/119, or 126.316 cents); two for the major second, being D² or DË, or the 19-et just minor tone

( 319 ]2[ , approximately 512/459, or 189.474 cents) and 19-et five quarter-tones ( 419 ]2[ , approximately

81/70, or 252.632 cents); two for the major third, being E or EË, or the 19-et just major third ( 619 ]2[ ,

approximately 61/49, or 378.947 cents) and 19-et nine quarter-tones ( 719 ]2[ , approximately 661/512,

or 442.105 cents); two for the tritone, being FÚ or GÝ, or the 19-et grave or small augmented fourth

( 919 ]2[ , approximately 711/512, or 568.421 cents) and 19-et acute or large diminished fifth ( 1019 ]2[ ,

The Equally-Tempered Archetype 165

approximately 1024/711, or 631.579 cents); two for the minor sixth, being A¸ or Aí, or the 19-et

Pythagorean minor sixth ( 1219 ]2[ , approximately 793/512, or 757.895 cents) and 19-et seventeen

quarter-tones ( 1319 ]2[ , approximately 98/61, or 821.053 cents); two for the major sixth, being A or AË,

or the 19-et just major sixth ( 1419 ]2[ , approximately 5/3, or 884.211 cents) and 19-et nineteen quarter-

tones ( 1519 ]2[ , approximately 140/81, or 947.368 cents); as well as two options for the major seventh,

being B² or BË, or the 19-et grave or small major seventh ( 1719 ]2[ ) and 19-et twenty-three quarter-tones

( 1819 ]2[ , approximately 27/14, or 1136.842 cents).248

Simultaneous sonorities include the dyads A and C, or the 19-et just minor third ( 519 ]2[ ); DÚ and

G, EÚ and A, Aí and D¸, or 19-et nine quarter-tones ( 619 ]2[ ); C and F, D² and G, EÝ and Aí, E and A, F

and BÝ, G and C, or the 19-et just perfect fourth ( 819 ]2[ , approximately 154/115, or 505.263 cents); Aí

and D², BÝ and E, AÚ and EÝ, C and FÚ, DÝ and G, or the 19-et grave or small augmented fourth ( 919 ]2[ );

A¸ and D², A and EÝ, C and GÝ, or the 19-et acute or large diminished fifth ( 1019 ]2[ ); C and G, or the 19-

et just perfect fifth ( 1119 ]2[ ); and C and Aí, or 19-et seventeen quarter-tones ( 1319 ]2[ ). Bar 94 and 95

highlights intonational reinterpretations of simultaneous sonorities based on the two options for the major

second (D² and G, and DÚ and G); two for the major third (E and A, and EÚ and A); two for the minor

sixth (A¸ and D², and Aí and D²); two for the major sixth (A and EÝ, and AÚ and EÝ); and the two

options for the tritone (C and FÚ, and C and GÝ).249

Ex. 66. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 94-95

The following table depicts the tonal resources of nineteen-tone equal temperament, indicating degree,

notation, interval, ratio, frequency, cents, and tuning.

248 Daniélou, Tableau Comparatif des Intervalles Musicaux 17-139.

249 The juxtaposition of three just major thirds (5/4), equal to 1158.941 cents is a figure that diminishes the octave

(2/1) by 41.059 cents, and the interval termed as the ‘great diesis’ with a frequency ratio of 128/125. For a further

discussion, see Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 453.

166 The Equally-Tempered Archetype

Table 99. The nineteen-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE

NUMBER

NOTE INTERVAL RATIO

(DECIMAL)

FREQUENCY

(HERTZ)

CENTS TUNING

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÑÓ

ÑÔ

ÑÕ

ÑÖ

Ñ×

ÑØ

ÑÙ

ÐÑ

C

D¸

DÝ

D²

DË / Eì

EÝ

E

EË / Fí

F

FÚ

GÝ

G

A¸

GÍ / Aí

A

AË / Bì

BÝ

B²

BË / Cí

C

unison

19-et just diatonic semitone, or major half-tone

19-et great limma, or large half-tone

19-et just minor tone

19-et five quarter-tones

19-et just minor third

19-et just major third

19-et nine quarter-tones

19-et just and Pythagorean perfect fourth

19-et grave or small augmented fourth

19-et acute or large diminished fifth

19-et just and Pythagorean perfect fifth

19-et Pythagorean minor sixth

19-et seventeen quarter-tones

19-et just major sixth

19-et nineteen quarter-tones

19-et acute or large minor seventh

19-et grave or small major seventh

19-et twenty-three quarter-tones

octave

1.000000

1.037155

1.075691

1.115658

1.157110

1.200103

1.244693

1.290939

1.338904

1.388651

1.440247

1.493759

1.549260

1.606822

1.666524

1.728444

1.792664

1.859271

1.928352

2.000000

261.626

271.346

281.428

291.885

302.730

313.978

325.643

337.743

350.292

363.307

376.805

390.806

405.326

420.386

436.005

452.205

469.007

486.433

504.506

523.251

0.000

63.158

126.316

189.474

252.632

315.789

378.947

442.105

505.263

568.421

631.579

694.737

757.895

821.053

884.211

947.368

1010.526

1073.684

1136.842

1200.000

+00

ß37

+26

ß11

+53

+16

ß21

+42

+05

ß32

+32

ß05

ß42

ß79

ß16

+47

+11

ß26

+37

+00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to

represent sléndro and pélog scales.

Ex. 67. 19-tone equal temperament

The Equally-Tempered Archetype 167

The following two tuning matrixes contain the data required to represent the temperament chromatically

within the program memory of the Akai S3000XL.

Table 100. 19-tone equal temperament tuning matrix no. 1 (Program 25)

NOTE C DÝ D² EÝ E F F! G Aí A BÝ B²

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +26 ß11 +16 ß21 +05 ß32 ß05 +21 ß16 +11 ß26

Table 101. 19-tone equal temperament tuning matrix no. 2 (Program 26)

NOTE – – D¸ – – DË EË – – GÝ – – A¸ AË – – BË

KEY C C! D D! E F F! G G! A A! B

CENTS +00 ß37 +00 ß47 +42 +00 +32 +00 ß42 +47 +00 +37

Twenty-Tone Equal Temperament Twenty-tone equal temperament ( 20 2 ) presents a subdivision of the 5-et supermajor second ( 5 2 ) into

four equal tones 60.000 cents in size, which represent the 20-et Greek enharmonic quarter-tone

(approximately 265/256).250 The temperament may alternatively be expressed as the juxtaposition of two

sets of ten-tone equally-tempered intervals, four sets of five-tone equally-tempered intervals, five sets of

four-tone equally-tempered intervals, or ten sets of two-tone equally-tempered intervals interlocked a

20-et Greek enharmonic quarter-tone ( 20 2 ) apart. Blackwood makes the following statement with

regards to the temperament: “Triads are very bad, and the most consonant harmony this tuning offers is a

minor triad with an added major sixth, along with its inversions.”251 The minor triad with an added major

sixth that Blackwood refers to includes the pitches C, E¸, G¢, and A¢, and is made up of the equal major

third ( 3 2 ), the 5-et acute or large fifth ( 35 ]2[ ), and the equal major sixth ( 34 ]2[ ). All these pitches are

also available in four-tone, eight-tone, twelve-tone, sixteen-tone, and twenty-four-tone equal

temperaments, but for G¢ ( 35 ]2[ , or 5-et acute or large fifth), which is only available in five-tone, ten-

tone, fifteen-tone, and twenty-tone equal temperaments.

250 Daniélou, Tableau Comparatif des Intervalles Musicaux 16.

251 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag.

168 The Equally-Tempered Archetype

Twenty-tone equal temperament provides two options for the approximation of the minor second,

being Dì or D¸, or the 20-et Greek enharmonic quarter-tone ( 20 2 ) and 10-et just diatonic semitone

( 10 2 ); two for the major second, being D² or D£, or the 20-et just minor tone ( 320 ]2[ , approximately

81/73, or 180.000 cents) and 5-et supermajor second ( 5 2 ); two for the major third, being E² or E£, or

the 10-et grave or small major third ( 310 ]2[ ) and 20-et acute or large major third ( 720 ]2[ , approximately

25088/19683, or 420.000 cents); two for the fifth, being G³ or G¢, or the 20-et subfifth ( 1120 ]2[ ,

approximately 41/28, or 660.000 cents) and 5-et acute or large fifth ( 35 ]2[ ); two for the major sixth,

being A³ or A¢, or the 10-et neutral sixth ( 710 ]2[ , or approximately 13/8) and equal major sixth ( 34 ]2[ );

two for the augmented sixth/minor seventh, being AÚ or BÝ, or the 5-et augmented sixth ( 45 ]2[ ) and 20-

et acute or large minor seventh ( 1720 ]2[ , approximately 146/81, or 1020.000 cents); as well as two

options for the major seventh, being B or BË, or the 10-et just diatonic major seventh ( 910 ]2[ ,

approximately 28/15, or 1080.000 cents) and 20-et twenty-three quarter-tones ( 1920 ]2[ , approximately

989/512, or 1140.000 cents).252 Simultaneous sonorities include the dyads C and E¸, or the equal minor

third ( 4 2 ); E¸ and G³, or the 10-et grave or small major third ( 310 ]2[ ); E¸ and G¢, and C and E£, or the

20-et acute or large major third ( 720 ]2[ ); as well as C and A³, or the 10-et neutral sixth ( 710 ]2[ ). Bar

100 illustrates the utilization of intonational reinterpretations of simultaneous sonorities based on the two

options for the fifth, G³ and G¢, or the 20-et subfifth ( 1120 ]2[ ) and 5-et acute or large fifth ( 35 ]2[ ). In

this case, the juxtaposition with E¸, or the equal minor third ( 4 2 ) generates the 20-et acute or large

major third ( 720 ]2[ ) and 10-et grave or small major third ( 310 ]2[ ), which in each case present the just

major third (5/4) with a falsity of +33.686 and ß26.314 cents.

Ex. 68. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 100

The following table depicts the tonal resources of twenty-tone equal temperament, indicating degree,

notation, interval, ratio, frequency, cents, and tuning.

252 Daniélou, Tableau Comparatif des Intervalles Musicaux 17-133.

The Equally-Tempered Archetype 169

Table 102. The twenty-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE

NUMBER

NOTE INTERVAL RATIO

(DECIMAL)

FREQUENCY

(HERTZ)

CENTS TUNING

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÑÓ

ÑÔ

ÑÕ

ÑÖ

Ñ×

ÑØ

ÑÙ

ÒÐ

ÐÑ

C

CË / Dì

D¸

D²

D£

E¸

E²

E£

F²

F£

F¥

G³

G¢

GÚ

A³

A¢

AÚ

BÝ

B

BË / Cí

C

unison

20-et Greek enharmonic or septimal quarter-tone

10-et just diatonic semitone, or major half-tone

20-et just minor tone

5-et supermajor second

equal or Pythagorean minor third, or trihemitone

10-et grave or small major third

20-et acute or large major third

5-et grave or small fourth

20-et superfourth

equal or acute or large tritone, or augmented fourth

20-et subfifth

5-et acute or large fifth

20-et augmented fifth

10-et neutral sixth

equal or Pythagorean major sixth

5-et augmented sixth

20-et acute or large minor seventh

10-et just diatonic major seventh

20-et twenty-three quarter-tones

octave

1.000000

1.035265

1.071773

1.109569

1.148698

1.189207

1.231144

1.274561

1.319508

1.366040

1.414214

1.464086

1.515717

1.569168

1.624505

1.681793

1.741101

1.802501

1.866066

1.931873

2.000000

261.626

270.852

280.403

290.292

300.529

311.127

322.099

333.458

345.217

357.391

369.994

383.042

396.550

410.535

425.012

440.000

455.517

471.580

488.211

505.427

523.251

0.000

60.000

120.000

180.000

240.000

300.000

360.000

420.000

480.000

540.000

600.000

660.000

720.000

780.000

840.000

900.000

960.000

1020.000

1080.000

1140.000

1200.000

+00

ß40

+20

ß20

+40

+00

ß40

+20

ß20

+40

+00

ß40

+20

ß20

ß60

+00

ß40

+20

ß20

+40

+00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to

represent sléndro and pélog scales.

Ex. 69. 20-tone equal temperament

170 The Equally-Tempered Archetype

The following two tuning matrixes contain the data required to represent the temperament chromatically

within the program memory of the Akai S3000XL.

Table 103. 20-tone equal temperament tuning matrix no. 1 (Program 27)

NOTE C D¸ D² E¸ E² F² F¥ G¢ G! A£ BÝ B

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +20 ß20 +00 ß40 ß20 +00 +20 ß20 +00 +20 ß20

Table 104. 20-tone equal temperament tuning matrix no. 2 (Program 28)

NOTE – – Dì D£ – – E£ F£ – – G³ A³ – – A! BË

KEY C C! D D! E F F! G G! A A! B

CENTS +00 ß40 +40 +00 +20 +40 +00 ß40 +40 +00 ß40 +40

Twenty-One-Tone Equal Temperament Twenty-one-tone equal temperament ( 212 ), with its twofold tripartite division of the octave, presents a

subdivision of the 7-et grave or small tone ( 7 2 ) into three equal tones 57.143 cents in size, which

represent the 21-et Greek enharmonic quarter-tone (approximately 1323/1280).253 The temperament

may alternatively be expressed as the juxtaposition of three sets of seven-tone equally-tempered intervals

interlocked an equal 21-et Greek enharmonic quarter-tone ( 212 ) apart. According to Blackwood,

“Major and minor triads and keys are relatively consonant here. Scales sound somewhat out of tune,

however, due to the impossibility of dividing a major third into two equal parts.”254 The twenty-one-tone

equally-tempered major scale generates inconsistent major triads, which produce the just major third

(5/4) and just perfect fifth (3/2) with a falsity of +13.686 cents and ß16.241 cents, as in C major (C, E¢,

and G²); ß43.457 and ß16.241 cents, as in F major (F¢, A², and C); and +13.686 and +40.902 cents,

as in G major (G², B, and D¢). Equally inconsistent minor triads are available, deviating from the just minor

third (6/5) and just perfect fifth (3/2) by ß29.927 and ß73.384 cents, as in D minor (D, F¢, and A²);

ß29.927 and ß16.241 cents, as in E minor (E¢, G², and B); and +27.216 and +40.902 cents, as in A

253 Daniélou, Tableau Comparatif des Intervalles Musicaux 16.

254 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag.

The Equally-Tempered Archetype 171

minor (A², C, and E¢); while the B diminished triad (B, D¢, and F¢), deviating from the just minor third (6/5)

and just tritone (45/32) by +27.216 and +38.348 cents. The D minor triad produces a just perfect fifth

(3/2) with a falsity of ß73.384 cents, which is an extremely dissonant interval 628.571 cents in size

( 1121 ]2[ , 21-et acute or large diminished fifth, or approximately 23/16), and therefore certainly no form of

recognizable perfect fifth.255

Ex. 70. The twenty-one-tone equally-tempered major scale

Twenty-one-tone equal temperament provides two options for the approximation of the minor second,

being Dì or D¸, or the 21-et Greek enharmonic quarter-tone ( 212 ) and 21-et just diatonic semitone

( 221 ]2[ , approximately 47/44, or 114.286 cents); two for the major second, being D³ or D¢, or the 7-et

grave or small tone ( 7 2 ) and 21-et acute or large tone ( 421 ]2[ , approximately 81/71, or 228.571

cents); two for the major third, being E³ or E¢, or the 7-et neutral third ( 27 ]2[ ) and equal major third

( 3 2 ); two for the fourth, being F³ or F¢, or the 21-et subfourth ( 821 ]2[ , approximately 125/96, or

457.143 cents) and 7-et acute or large fourth ( 37 ]2[ ); two for the tritone, being FÚ or GÝ, or the 21-et

grave or small augmented fourth ( 1021 ]2[ , approximately 32/23, or 571.429 cents) and 21-et acute or

large diminished fifth ( 1121 ]2[ ); two for the fifth, being G² or G£, or the 7-et grave or small fifth ( 47 ]2[ )

and 21-et superfifth ( 1321 ]2[ , approximately 43/28, or 742.857 cents); two for the major sixth, being A²

or A¢, or the 7-et grave or small major sixth ( 57 ]2[ ) and 21-et Pythagorean major sixth ( 1621 ]2[ ,

approximately 27783/16384, or 914.286 cents); two for the minor seventh, being B¹ or Bí, or the 21-et

grave or small minor seventh ( 1721 ]2[ , approximately 142/81, or 971.429 cents) and 7-et twenty-one

quarter-tones ( 67 ]2[ ); as well as two options for the major seventh, being B or BË, or the 21-et just

diatonic major seventh ( 1921 ]2[ , approximately 88/47, or 1085.714 cents) and 21-et twenty-three

quarter-tones ( 2021 ]2[ , approximately 2560/1323, or 1142.857 cents).256 Bar 107 highlights a

descending melodic passage, which incorporates all the primary, secondary, and auxiliary pitches

available within the pélog scheme – the series C, (BË), B, Bí, (B¹), A¸, (G£), G², (GÝ), FÚ, F¢, (F³), E¸, D¸,

(Dì), and C.

255 Daniélou, Tableau Comparatif des Intervalles Musicaux 141.

256 Daniélou, Tableau Comparatif des Intervalles Musicaux 17-140.

172 The Equally-Tempered Archetype

Ex. 71. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 107

Simultaneous sonorities utilized in this temperament are limited to C and G², or the 7-et grave or small fifth

( 47 ]2[ ), and F¢ and Bí, and F³ and B¹, or the 7-et acute or large fourth ( 37 ]2[ ). The following table

depicts the tonal resources of twenty-one-tone equal temperament, indicating degree, notation, interval,

ratio, frequency, cents, and tuning.

Table 105. The twenty-one-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE

NUMBER

NOTE INTERVAL RATIO

(DECIMAL)

FREQUENCY

(HERTZ)

CENTS TUNING

ÐÑ ÐÒ ÐÓ ÐÔ ÐÕ ÐÖ Ð× ÐØ ÐÙ ÑÐ ÑÑ ÑÒ ÑÓ ÑÔ ÑÕ ÑÖ Ñ× ÑØ ÑÙ ÒÐ ÒÑ

ÐÑ

C CË / Dì

D¸ D³ D¢ EÝ E³ E¢ F³ F¢ FÚ GÝ G² G£ A¸ A² A¢ B¹

AÍ / Bí B

BË / Cí

C

unison

21-et Greek enharmonic or septimal quarter-tone

21-et just diatonic semitone, or major half-tone

7-et grave or small tone

21-et acute or large tone

21-et just minor third

7-et neutral third

equal or Pythagorean major third, or ditone

21-et subfourth

7-et acute or large fourth

21-et grave or small augmented fourth

21-et acute or large diminished fifth

7-et grave or small fifth

21-et superfifth

equal or Pythagorean minor sixth

7-et grave or small major sixth

21-et Pythagorean major sixth

21-et grave or small minor seventh

7-et twenty-one quarter-tones

21-et just diatonic major seventh

21-et twenty-three quarter-tones

octave

1.000000

1.033558

1.068242

1.104090

1.141140

1.179434

1.219014

1.259921

1.302201

1.345900

1.391066

1.437747

1.485994

1.535861

1.587401

1.640671

1.695728

1.752633

1.811447

1.872235

1.935064

2.000000

261.626

270.405

279.479

288.858

298.551

308.570

318.925

329.628

340.689

352.122

363.938

376.151

388.774

401.820

415.305

429.241

443.646

458.534

473.921

489.825

506.262

523.251

0.000

57.143

114.286

171.429

228.571

285.714

342.857

400.000

457.143

514.286

571.429

628.571

685.714

742.857

800.000

857.143

914.286

971.429

1028.571

1085.714

1142.857

1200.000

+00

ß43

+14

ß29

+29

ß14

+43

+00

ß43

+14

ß29

+29

ß14

+43

+00

ß43

+14

ß29

ß71

ß14

+43

+00

The Equally-Tempered Archetype 173

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to

represent sléndro and pélog scales.

Ex. 72. 21-tone equal temperament

The following two tuning matrixes contain the data required to represent the temperament chromatically

within the program memory of the Akai S3000XL.

Table 106. 21-tone equal temperament tuning matrix no. 1 (Program 29)

NOTE C D¸ D¢ E¸ E¢ F¢ F! G² A¸ A² Bí B

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +14 +29 ß14 +00 +14 ß29 ß14 +00 ß43 +29 ß14

Table 107. 21-tone equal temperament tuning matrix no. 2 (Program 30)

NOTE – – Dì D³ E³ – – F³ GÝ G£ – – A¢ B¹ BË

KEY C C! D D! E F F! G G! A A! B

CENTS +00 ß43 ß29 +43 +00 ß43 +29 +43 +00 +14 ß29 +43

Twenty-Two-Tone Equal Temperament Twenty-two-tone equal temperament ( 22 2 ) presents a subdivision of the 11-et just diatonic semitone

( 112 ) into two equal tones 54.545 cents in size, which represent the 22-et Greek enharmonic quarter-

tone (approximately 4096/3969).257 The temperament may alternatively be expressed as the

juxtaposition of two sets of eleven-tone equally-tempered intervals, or eleven sets of two-tone equally-

257 Daniélou, Tableau Comparatif des Intervalles Musicaux 14.

174 The Equally-Tempered Archetype

tempered intervals interlocked a 22-et Greek enharmonic quarter-tone ( 22 2 ) apart. Blackwood makes

the following observations: “This tuning contains triads that are very smooth – in some respects, even a

bit smoother than those of 12-note tuning. The smoothness of the triads is offset, however, by the out-

of-tune scale which sharply restricts the tuning’s diatonic vocabulary.”258 Twenty-two-tone equal

temperament is the fourth temperament to contain what Blackwood terms as recognizable diatonic

scales and “perfect fifths within the range of recognisability.”259 The twenty-two-tone equally-tempered

major scale generates consistent major triads, which produce the just major third (5/4) and just perfect

fifth (3/2) with a falsity of ß4.496 cents and +7.136 cents, as in C major (C, E, and G), F major (F, A², and

C), and G major (G, B, and D¢). A very dissonant minor triad is available, which presents the just minor

third (6/5) and perfect fifth (3/2) with a falsity of by ß42.914 and ß47.410 cents, as in D minor (D, F, and A²); while another two, display more consistency, as well as consonance, with a deviation of

+11.631 and +7.136 cents, as in E minor (E, G, and B) and A minor (A², C, and E). The B diminished

triad (B, D¢, and F) presents the just minor third (6/5) and just tritone (45/32) with a falsity of +11.631

and ß9.776 cents. The out-of-tune scale Blackwood is referring to is made up of tones and semitones

deviating from just major tones (9/8s), just minor tones (10/9s), and just diatonic semitones (16/15s) by

+14.272, ß18.767, and ß2.640 cents. Twelve-tone equal temperament yields ß3.910, +17.596, and

ß11.731 cents on the same basis.

Ex. 73. The twenty-two-tone equally-tempered major scale

The work outlines the triad C, DÚ, and G, which when enharmonically reinterpreted as C, E¹, and G may be stated as being a twenty-two-tone equally-tempered minor triad with the minor third and perfect fifth

272.727 and 709.091 cents in size respectively (the interval of the augmented second converted into a

subminor third). The comparison with the just minor third (6/5) and just perfect fifth (3/2) reveal a falsity

of ß42.914 and +7.136 cents on each count. G4 (394.059Hz), or the 22-et just perfect fifth ( 1322 ]2[ ,

or approximately 122/81), presents 3.242 beats between the third harmonic of C4 and the second

harmonic of G4 (788.119Hz), and 6.484 beats between the sixth harmonic of C4 and the fourth

harmonic of G4 (1576.237Hz); while E¹4 (306.264Hz), or 22-et subminor third ( 522 ]2[ , or

approximately 2560/2187), 38.433 beats between the sixth harmonic of C4 and the fifth harmonic of

E¹4 (1531.320Hz).260

258 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag.

259 Blackwood, The Structure of Recognizable Diatonic Tunings 197.

260 Daniélou, Tableau Comparatif des Intervalles Musicaux 72-123.

The Equally-Tempered Archetype 175

Ex. 74. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 109

Table 108. The beating characteristics of the nineteen-tone equally-tempered minor triad

C4

(PARTIAL)

FREQUENCY

(HERTZ)

E¹4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

G4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

– – – –

5

6

7

– – – –

– – – –

8

261.626

– – – –

– – – –

523.251

– – – –

784.877

– – – –

1046.502

– – – –

– – – –

1308.128

1569.753

1831.379

– – – –

– – – –

2093.005

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

– – – –

5

– – – –

6

– – – –

– – – –

– – – –

306.264

– – – –

– – – –

612.528

– – – –

918.792

– – – –

– – – –

1225.056

– – – –

1531.320

– – – –

1837.585

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

38.433

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

– – – –

4

– – – –

– – – –

5

– – – –

– – – –

– – – –

394.059

– – – –

– – – –

788.119

– – – –

– – – –

1182.178

– – – –

– – – –

1576.237

– – – –

– – – –

1970.296

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

3.242

– – – –

– – – –

– – – –

– – – –

– – – –

6.484

– – – –

– – – –

– – – –

– – – –

Twenty-two-tone equal temperament provides two options for the approximation of the major second,

being D³ or D¢, or the 22-et grave or small tone ( 322 ]2[ , approximately 256/233, or 163.636 cents) and

11-et acute or large tone ( 211 ]2[ ); two for the augmented second, being DÚ or DÍ, or the 22-et

augmented second ( 522 ]2[ ) and 11-et seven quarter-tones ( 311 ]2[ ); two for the major third, being E or

EË, or the 22-et just major third ( 722 ]2[ , approximately 96/77, or 381.818 cents) and 11-et nine quarter-

tones ( 411 ]2[ ); two for the fourth, being F or F£, or the 22-et just perfect fourth ( 922 ]2[ , approximately

81/61, or 490.909 cents) and 11-et superfourth ( 511 ]2[ , approximately 37/27, or 545.455 cents); two

for the fifth, being G³ or G, or the 11-et subfifth ( 611 ]2[ ) and 22-et just perfect fifth ( 1322 ]2[ ); two for

the minor sixth, being Aì or AÝ, or 11-et fifteen quarter-tones ( 711 ]2[ , approximately 199/128, or

763.636 cents) and the 22-et just minor sixth ( 1522 ]2[ , approximately 77/48, or 818.182 cents); two for

176 The Equally-Tempered Archetype

the major sixth, being A² or A£, or the 11-et grave or small major sixth ( 811 ]2[ , approximately 48/29, or

872.727 cents) and 22-et acute or large major sixth ( 1722 ]2[ , approximately 2187/1280, or 927.273

cents); two for the minor seventh, being B¹ or Bí, or the 11-et grave or small minor seventh ( 911 ]2[ ,

approximately 432/245, or 981.818 cents) and 22-et twenty-one quarter-tones ( 1922 ]2[ , approximately

233/128, or 1036.364 cents); as well as two options for the major seventh, being B or BË, or the 11-et

just diatonic major seventh ( 1011 ]2[ , approximately 77/41, or 1090.909 cents) and 22-et twenty-three

quarter-tones 2122 ]2[ , approximately 3969/2048, or 1145.455 cents).261 A microtonal cluster in bar 108

made up of the pitches D¸, Eí, F', F¥, G, AÝ, Bí, and B' highlights four significant twenty-two-tone

equally-tempered intervals, measuring 54.545, 109.091, 163.636, and 218.182 cents.

Ex. 75. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 108

The only simultaneous sonority utilized in this temperament is Bí and F, or the 11-et subfifth ( 611 ]2[ ).

The following table depicts the tonal resources of twenty-two-tone equal temperament, indicating

degree, notation, interval, ratio, frequency, cents, and tuning.

261 Daniélou, Tableau Comparatif des Intervalles Musicaux 31-134.

The Equally-Tempered Archetype 177

Table 109. The twenty-two-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE

NUMBER

NOTE INTERVAL RATIO

(DECIMAL)

FREQUENCY

(HERTZ)

CENTS TUNING

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÑÓ

ÑÔ

ÑÕ

ÑÖ

Ñ×

ÑØ

ÑÙ

ÒÐ

ÒÑ

ÒÒ

ÐÑ

C

CË / Dì

D¸

D³

D¢

DÚ

DÍ / Eí

E

EË / Fí

F

F£

F¥

G³

G

GË / Aì

AÝ

A²

A£

B¹

AÍ / Bí

B

BË / Cí

C

unison

22-et Greek enharmonic or septimal quarter-tone

11-et just diatonic semitone, or major half-tone

22-et grave or small tone

11-et acute or large tone

22-et augmented second

11-et seven quarter-tones

22-et just major third

11-et nine quarter-tones

22-et just and Pythagorean perfect fourth

11-et superfourth

equal or acute or large tritone, or augmented fourth

11-et subfifth

22-et just and Pythagorean perfect fifth

11-et fifteen quarter-tones

22-et just minor sixth

11-et grave or small major sixth

22-et acute or large major sixth

11-et grave or small minor seventh

22-et twenty-one quarter-tones

11-et just diatonic major seventh

22-et twenty-three quarter-tones

octave

1.000000

1.032008

1.065041

1.099131

1.134313

1.170620

1.208089

1.246758

1.286665

1.327849

1.370351

1.414214

1.459480

1.506196

1.554406

1.604160

1.655507

1.708496

1.763183

1.819619

1.877862

1.937969

2.000000

261.626

270.000

278.642

287.561

296.765

306.264

316.067

326.184

336.624

347.399

358.519

369.994

381.837

394.059

406.672

419.689

433.123

446.986

461.294

476.059

491.297

507.022

523.251

0.000

54.545

109.091

163.636

218.182

272.727

327.273

381.818

436.364

490.909

545.455

600.000

654.545

709.091

763.636

818.182

872.727

927.273

981.818

1036.364

1090.909

1145.455

1200.000

+00

+55

+09

ß36

+18

ß27

+27

ß18

+36

ß09

+45

+00

ß45

+09

ß36

+18

ß27

+27

ß18

ß64

ß09

+45

+00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to

represent sléndro and pélog scales.

Ex. 76. 22-tone equal temperament

178 The Equally-Tempered Archetype

The following two tuning matrixes contain the data required to represent the temperament chromatically

within the program memory of the Akai S3000XL.

Table 110. 22-tone equal temperament tuning matrix no. 1 (Program 31)

NOTE C D¸ D¢ DÍ E F F¥ G AÝ A² Bí B

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +09 +18 +27 ß18 ß09 +00 +09 +18 ß27 +36 ß09

Table 111. 22-tone equal temperament tuning matrix no. 2 (Program 32)

NOTE – – CË D³ D! EË F£ – – G³ Aì A£ B¹ BË

KEY C C! D D! E F F! G G! A A! B

CENTS +00 ß45 ß36 ß27 +36 +45 +00 ß45 ß36 +27 ß18 +45

Twenty-Three-Tone Equal Temperament Twenty-three-tone equal temperament is generated by the 23 2 , and presents twenty-three equal tones

52.174 cents in size, which represent the 23-et Greek enharmonic quarter-tone (approximately 34/33),

and approximate the equal quarter-tone ( 24 2 ) with a falsity of +2.174 cents.262 According to

Blackwood, twenty-three-tone equal temperament presents the following characteristics: “A particular

challenge, 23-note tuning contains no diatonic configurations and no chromatic structures in common

with any of the other tunings of this study. However, it does contain an intriguing arrangement of the two

distinct pentatonic modes of Java and Bali, known as sléndro and pélog – modes that cannot be

realistically approximated in 12-note tuning.”263 The twenty-three-tone equally-tempered major scale

generates inconsistent major triads, which produce the just major third (5/4) and just perfect fifth (3/2)

262 Daniélou, Tableau Comparatif des Intervalles Musicaux 14.

263 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag.

The Equally-Tempered Archetype 179

with a falsity of ß21.096 cents and ß23.694 cents, as in C major (C, E², and G²) and F major (F¢, A, and

C); and with +31.078 cents and +28.480 cents, as in G major (G, B, and D). Somewhat more

consonant minor triads are available, which produce the just minor third (6/5) and just perfect fifth (3/2)

with a falsity of ß2.598 and ß23.694 cents, as in D minor (D, F¢, and A) and A minor (A, C, and E²); and

with ß2.598 and +28.480 cents, as in E minor (E, G², and B); while the B diminished triad (B, D, and F¢)

produces the just minor third (6/5) and just tritone (45/32) with a falsity of ß2.598 and +35.863 cents.

Ex. 77. The twenty-three-tone equally-tempered major scale

The sléndro scale is represented in twenty-three-tone equal temperament with the pitches C, D, F¢, G²,

and B¸, and equal to 0.000, 208.696, 521.739, 678.261, and 991.304 cents; while the pélog scale, with

the pitches C, D¸, EÝ, F¢, G², A¸, and B, and equal to 0.000, 104.348, 313.043, 521.739, 678.261,

782.609, and 1095.652 cents. A comparison with the tuning of the gamelan gedhé analyzed by Perlman

(presenting sléndro measurements equal to 0.000, 238.000, 475.000, 727.000, and 950.000 cents; and

pélog measurements equal to 0.000, 116.000, 281.000, 550.000, 669.000, 769.000, and 861.000

cents) reveals a falsity of between 9.261 and 32.043 cents on the first count (not accounting for the

+134.652 cent falsity of B), and a falsity between 29.304 and 48.739 cents on the second count.264 A

further comparison of the two scales with five-tone and seven-tone equal temperaments produces an

even greater level of error.

The work outlines a twenty-three-tone equally-tempered minor triad, which includes the pitches C,

EÝ, and G², and produces a minor third 313.043 cents in size ( 623 ]2[ , 23-et just minor third, or

approximately 1024/855) and a perfect fifth 678.261 cents in size ( 1323 ]2[ , 23-et grave or small fifth, or

approximately 262144/177147).265 The comparison with the just minor third (6/5) and perfect fifth (3/2)

reveal a falsity of ß2.598 and ß23.694 cents on each count. G²4 (387.104Hz), or the 23-et grave or

small fifth ( 1323 ]2[ ), presents 10.669 beats between the third harmonic of C4 and the second harmonic

of G²4 (774.208Hz), and 21.338 beats between the sixth harmonic of C4 and the fourth harmonic of G²4

(1548.416Hz); while EÝ4 (313.480Hz), or the 23-et just minor third ( 623 ]2[ ), 2.354 beats between the

sixth harmonic of C4 and the fifth harmonic of EÝ4 (1567.400Hz).

264 Perlman, Unplayed Melodies: Javanese Gamelan and the Genesis of Music Theory 41.

265 Daniélou, Tableau Comparatif des Intervalles Musicaux 82-129.

180 The Equally-Tempered Archetype

Ex. 78. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 112

Table 112. The beating characteristics of the nineteen-tone equally-tempered minor triad

C4

(PARTIAL)

FREQUENCY

(HERTZ)

EÝ4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

G²4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

– – – –

5

6

7

– – – –

– – – –

8

261.626

– – – –

– – – –

523.251

– – – –

784.877

– – – –

1046.502

– – – –

– – – –

1308.128

1569.753

1831.379

– – – –

– – – –

2093.005

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

– – – –

5

– – – –

6

– – – –

– – – –

– – – –

313.480

– – – –

– – – –

626.960

– – – –

940.440

– – – –

– – – –

1253.920

– – – –

1567.400

– – – –

1880.880

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

2.354

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

– – – –

4

– – – –

– – – –

5

– – – –

– – – –

– – – –

387.104

– – – –

– – – –

774.208

– – – –

– – – –

1161.312

– – – –

– – – –

1548.416

– – – –

– – – –

1935.520

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

10.669

– – – –

– – – –

– – – –

– – – –

– – – –

21.338

– – – –

– – – –

– – – –

– – – –

Twenty-three-tone equal temperament provides two options for the approximation of the major second,

being D³ or D, or the 23-et grave or small tone ( 323 ]2[ , approximately 81/74, or 156.522 cents) and 23-

et just major tone ( 423 ]2[ , approximately 44/39, or 208.696 cents); two for the augmented

second/minor third, being DË or EÝ, or 23-et five quarter-tones ( 523 ]2[ , approximately 93/80, or 260.870

cents) and the 23-et just minor third ( 623 ]2[ ); two for the major third, being E² or E¢, or the 23-et grave

or small major third ( 723 ]2[ , approximately 2401/1944, or 365.217 cents) and 23-et Pythagorean major

third ( 823 ]2[ , approximately 14/11, or 417.391 cents); two for the fourth, being F² or F¢, or the 23-et

grave or small fourth ( 923 ]2[ , approximately 101/77, or 469.565 cents) and 23-et acute or large fourth

( 1023 ]2[ , approximately 173/128, or 521.739 cents); two for the tritone, being FÚ or GÝ, or the 23-et

grave or small augmented fourth ( 1123 ]2[ , approximately 1024/735, or 573.913 cents) and 23-et acute

The Equally-Tempered Archetype 181

or large diminished fifth ( 1223 ]2[ , approximately 735/512, or 626.087 cents); two for the fifth, being G²

or G¢, or the 23-et grave or small fifth ( 1323 ]2[ ) and 23-et acute or large fifth ( 1423 ]2[ , approximately

154/101, or 730.435 cents); two for the major sixth, being A³ or A, or the 23-et neutral sixth ( 1623 ]2[ ,

approximately 3888/2401, or 834.783 cents) and 23-et just major sixth ( 1723 ]2[ , approximately

855/512, or 886.957 cents); as well as two options for the major seventh, being B³ or B, or the 23-et

neutral seventh ( 2023 ]2[ , approximately 148/81, or 1043.478 cents) and 23-et just diatonic major

seventh ( 2123 ]2[ , approximately 145/77, or 1095.652 cents). The two approximations for the

augmented second/minor third and fifth is also highlighted in bar 112 with the twenty-three-tone equally-

tempered minor triad variant of C, DË, and G¢, which presents a triad with a just minor third (6/5) and just

perfect fifth (3/2) with a falsity of ß2.598 and ß23.694 cents.266

Simultaneous sonorities include the dyads F¢ and A¸, or 23-et five quarter-tones ( 523 ]2[ ); G¢ and

B³, and B¸ and D¸, or the 23-et just minor third ( 623 ]2[ ); GÝ and B¸, or the 23-et grave or small major

third ( 723 ]2[ ); A¸ and C, and B³ and DË, or the 23-et Pythagorean major third ( 823 ]2[ ); C and F², and DË

and G¢, or the 23-et grave or small fourth ( 923 ]2[ ); G² and C, and D¸ and GÝ, or the or 23-et acute or

large fourth ( 1023 ]2[ ). The following table depicts the tonal resources of twenty-three-tone equal

temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.

266 Daniélou, Tableau Comparatif des Intervalles Musicaux 29-141.

182 The Equally-Tempered Archetype

Table 113. The twenty-three-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE

NUMBER

NOTE INTERVAL RATIO

(DECIMAL)

FREQUENCY

(HERTZ)

CENTS TUNING

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÑÓ

ÑÔ

ÑÕ

ÑÖ

Ñ×

ÑØ

ÑÙ

ÒÐ

ÒÑ

ÒÒ

ÒÓ

ÐÑ

C

CË / Dì

D¸

D³

D

DË / Eì

EÝ

E²

E¢

F²

F¢

FÚ

GÝ

G²

G¢

A¸

A³

A

AË / Bì

B¸

B³

B

BË / Cí

C

unison

23-et Greek enharmonic or septimal quarter-tone

23-et just diatonic semitone, or major half-tone

23-et grave or small tone

23-et just major tone

23-et five quarter-tones

23-et just minor third

23-et grave or small major third

23-et Pythagorean major third, or ditone

23-et grave or small fourth

23-et acute or large fourth

23-et grave or small augmented fourth

23-et acute or large diminished fifth

23-et grave or small fifth

23-et acute or large fifth

23-et Pythagorean minor sixth

23-et neutral sixth

23-et just major sixth

23-et nineteen quarter-tones

23-et Pythagorean minor seventh

23-et neutral seventh

23-et just diatonic major seventh

23-et twenty-three quarter-tones

octave

1.000000

1.030596

1.062127

1.094624

1.128114

1.162629

1.198201

1.234860

1.272642

1.311579

1.351707

1.393063

1.435685

1.479610

1.524880

1.571534

1.619616

1.669169

1.720239

1.772870

1.827112

1.883014

1.940626

2.000000

261.626

269.630

277.880

286.382

295.143

304.174

313.480

323.071

332.956

343.143

353.641

364.461

375.612

387.104

398.948

411.154

423.733

436.697

450.058

463.828

478.019

492.645

507.717

523.251

0.000

52.174

104.348

156.522

208.696

260.870

313.043

365.217

417.391

469.565

521.739

573.913

626.087

678.261

730.435

782.609

834.783

886.957

939.130

991.304

1043.478

1095.652

1147.826

1200.000

+00

+52

+04

ß43

+09

ß39

+13

ß35

+17

ß30

+22

ß26

+26

ß22

+30

ß17

ß65

ß13

+39

ß09

ß56

ß04

ß52

+00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to

represent sléndro and pélog scales.

Ex. 79. 23-tone equal temperament

The Equally-Tempered Archetype 183

The following two tuning matrixes contain the data required to represent the temperament chromatically

within the program memory of the Akai S3000XL.

Table 114. 23-tone equal temperament tuning matrix no. 1 (Program 33)

NOTE C D¸ D EÝ E² F¢ F! G² A¸ A B¸ B

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +04 +09 +13 ß35 +22 ß26 ß22 ß17 ß13 ß09 ß04

Table 115. 23-tone equal temperament tuning matrix no. 2 (Program 34)

NOTE – – CË D³ DÍ E¢ F² GÝ G¢ A³ AË B³ Cí

KEY C C! D D! E F F! G G! A A! B

CENTS +00 ß48 ß43 ß39 +17 ß30 +26 +30 +35 +39 +44 +48

Twenty-Four-Tone Equal Temperament Twenty-four-tone equal temperament ( 24 2 ), with its division of the octave into twenty-four equally-

tempered quarter-tones 50.000 cents in size, marks the concluding tuning modulation of the work. The

temperament may alternatively be expressed as the juxtaposition of two sets of twelve-tone equally-

tempered intervals, three sets of eight-tone equally-tempered intervals, four sets of six-tone equally-

tempered intervals, six sets of four-tone equally-tempered intervals, eight sets of three-tone equally-

tempered intervals, or twelve sets of two-tone equally-tempered intervals interlocked an equal quarter-

tone ( 24 2 ) apart. Blackwood makes the following observations: “This familiar ‘quarter-tone’ tuning is

actually one of the most difficult to deal with in a practical situation. Without exception, the notes ‘in the

cracks’ make extreme discords with the other notes, and there are only a very few satisfactory harmonies

184 The Equally-Tempered Archetype

that combine the two.”267 Twenty-four-tone equal temperament is the fifth temperament to contain what

Blackwood terms as recognizable diatonic scales and “perfect fifths within the range of recognisability.”268

Twenty-four-tone equal temperament provides two options for the approximation of the minor

second, being Dì or D¹ , or the equal quarter-tone ( 24 2 ) and equal semitone ( 12 2 ); two for the major

second, being D or D£, or the equal major tone ( 6 2 ) and five equal quarter-tones ( 524 ]2[ ,

approximately 52/45, or 250.000 cents); two for the major third, being E³ or E, or seven equal quarter-

tones ( 724 ]2[ , approximately 60/49, or 350.000 cents) and the equal major third ( 3 2 ); two for the

fourth, being Fí or F, or nine equal quarter-tones ( 38 ]2[ ) and the equal perfect fourth ( 512 ]2[ ); two for

the tritone, being FË or F¥, or eleven equal quarter-tones ( 1124 ]2[ , approximately 703/512, or 550.000

cents) and the equal tritone ( 2 2 ); two for the fifth, being G or G£, or the equal perfect fifth ( 712 ]2[ )

and fifteen equal quarter-tones ( 58 ]2[ ); two for the minor sixth, being A² or A¢, or seventeen equal

quarter-tones ( 1724 ]2[ , approximately 49/30, or 850.00 cents) and the equal major sixth ( 34 ]2[ ); two

for the augmented sixth/minor seventh, being AÚ or B¸, or nineteen equal quarter-tones ( 1924 ]2[ ,

approximately 45/26, or 950.000 cents) and the equal minor seventh ( 56 ]2[ ); as well as two options for

the major seventh, being B³ or B¢, or twenty-one equal quarter-tones ( 78 ]2[ ) and the equal major

seventh ( 1112 ]2[ ).269 A five-figure rhythmically perpetuating descending melodic passage in bar 115

highlights all the primary, secondary, and auxiliary pitches available within the pélog scheme.

Ex. 80. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 115

Simultaneous sonorities in this temperament are limited to the dyads G and C, or the equal perfect fourth

( 512 ]2[ ), and its inversion of C and G, or the equal perfect fifth ( 712 ]2[ ). American composer Charles

Edward Ives’s (1874-1954) experiments with twenty-four-tone equal temperament conducted during

the 1925-62 period revealed that the division of the equal perfect fifth ( 712 ]2[ ) into seven equal quarter-

tones, or neutral thirds ( 724 ]2[ ) to produce a twenty-four-tone equally-tempered neutral triad, resulted

in a “chord that is heard as an ‘out of tune’ major or minor triad, not as an entity in itself,” but by the

addition of a fourth tone (“in a quarter-tone relationship to the root or fifth”) the simultaneous sonority

267 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag.

268 Blackwood, The Structure of Recognizable Diatonic Tunings 197.

269 Daniélou, Tableau Comparatif des Intervalles Musicaux 66-134.

The Equally-Tempered Archetype 185

“seemed to establish an identity of its own.”270 The following table depicts the tonal resources of twenty-

four-tone equal temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.

Table 116. The twenty-four-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz

DEGREE

NUMBER

NOTE INTERVAL RATIO

(DECIMAL)

FREQUENCY

(HERTZ)

CENTS TUNING

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÑÓ

ÑÔ

ÑÕ

ÑÖ

Ñ×

ÑØ

ÑÙ

ÒÐ

ÒÑ

ÒÒ

ÒÓ

ÒÔ

ÐÑ

C

CË / Dì

D¹

CÍ / Dí

D

D£

E¸

E³

E¢

EË / Fí

F

FË / Gì

F¥

FÍ / Gí

G

G£

A¸

A²

A¢

AÚ

B¸

B³

B¢

BË / Cí

C

unison

equal, Greek enharmonic or septimal quarter-tone

equal semitone, or Pythagorean limma

equal three-quarter-tone

equal or just major tone

five equal quarter-tones, or supermajor second

equal or Pythagorean minor third, or trihemitone

seven equal quarter-tones, or neutral third

equal or Pythagorean major third, or ditone

nine equal quarter-tones

equal, or just and Pythagorean perfect fourth

eleven equal quarter-tones

equal or acute or large tritone, or augmented fourth

thirteen equal quarter-tones

equal, or just and Pythagorean perfect fifth

fifteen equal quarter-tones, or superfifth

equal or Pythagorean minor sixth

seventeen equal quarter-tones

equal or Pythagorean major sixth

nineteen equal quarter-tones, or augmented sixth

equal or Pythagorean minor seventh

twenty-one equal quarter-tones, or neutral seventh

equal or Pythagorean major seventh

twenty-three equal quarter-tones

octave

1.000000

1.029302

1.059463

1.090508

1.122462

1.155353

1.189207

1.224054

1.259921

1.296840

1.334840

1.373954

1.414214

1.455653

1.498307

1.542211

1.587401

1.633915

1.681793

1.731073

1.781797

1.834008

1.887749

1.943064

2.000000

261.626

269.292

277.183

285.305

293.665

302.270

311.127

320.244

329.628

339.286

349.228

359.461

369.994

380.836

391.995

403.482

415.305

427.474

440.000

452.893

466.164

479.823

493.883

508.355

523.251

0.000

50.000

100.000

150.000

200.000

250.000

300.000

350.000

400.000

450.000

500.000

550.000

600.000

650.000

700.000

750.000

800.000

850.000

900.000

950.000

1000.000

1050.000

1100.000

1150.000

1200.000

+00

+50

+00

ß50

+00

+50

+00

ß50

+00

ß50

+00

ß50

+00

ß50

+00

+50

+00

ß50

+00

ß50

+00

ß50

+00

ß50

+00

The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to

represent sléndro and pélog scales.

270 Wolf, “Alternative Tunings, Alternative Tonalities,” Contemporary Music Review 5.

186 The Equally-Tempered Archetype

Ex. 81. 24-tone equal temperament

The following two tuning matrixes contain the data required to represent the temperament chromatically

within the program memory of the Akai S3000XL.

Table 117. 24-tone equal temperament tuning matrix no. 1 (Program 35)

NOTE C D¹ D E¸ E¢ F F¥ G A¸ A¢ B¸ B¢

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00

Table 118. 24-tone equal temperament tuning matrix no. 2 (Program 36)

NOTE CË Dí D£ E³ Fí FÍ Gí G£ A² A! B³ Cí

KEY C C! D D! E F F! G G! A A! B

CENTS +50 +50 +50 +50 +50 +50 +50 +50 +50 +50 +50 +50

The Equally-Tempered Archetype 187

Blackwood’s Dictum Blackwood offers the following summary of some the characteristics of equal tuning principles, and ‘the

equally-tempered archetype’:

“Mostly, I’ve explored the equal tunings from twelve to twenty-four notes per octave. The choice of which

tuning to use depends largely on the desired style. Certain tunings are more versatile or prettier than

others. Twelve, seventeen, nineteen, twenty-two, and twenty-four contain recognizable diatonic scales. If

random dissonance is what you want, then the worst of all tunings is twelve-note equal, because it

contains a greater concentration of relatively consonant intervals and harmonies in a smaller number of

notes. The most effective temperament for random dissonance is eleven notes. There aren’t two notes in

that tuning that make any kind of a consonance. Certain others tend toward modal arrangements that

coexist in twelve-note equal. For example, if the number of notes is divisible by four, you always have

families of octatonic scales. If the number of notes is divisible by six, there are always families of whole-

tone scales that can arise as altered chords, as they normally do in twelve-note tuning. If the number of

notes is divisible by three, there is a symmetric mode that alternates minor thirds with minor seconds,

creating a chromatic world all its own.”271

271 Keislar, Blackwood, Eaton, Harrison, Johnston, Mandelbaum, and Schottstaedt, “Six American Composers on

Nonstandard Tunings,” Perspectives of New Music 177.

188 The Equally-Tempered Archetype

3. The Harmonic Consideration: La Homa Kanto for Harmonically Tuned Synthesizer Quartet

Just Intonation

“Extended just intonation is a tuning system based on the ‘pure’ intervals of the overtone series: intervals

generated therefore from whole number ratios. This is in contrast to temperaments, both equal and

unequal, which use compromised intervals, often based on irrational numbers,” explains John Fonville.

American composer and theorist Benjamin Burwell Johnston (1926-)272 personally elaborates on the

matter within a statement contained in the performance notes of the score to his String Quartet No. 9,

headed On the Performance Practice of Extended Just Intonation:

“Just intonation is simply the easiest way to tune musical intervals by ear. It results in greatly heightened

purity and clarity of sound for two reasons: it eliminates acoustic beats to the maximum possible, and

second, it exploits resonances by utilizing harmonically simple combinations of pitches. The term

extended refers to the use of higher overtones than the first six partials.”273

In order to further understand the basic concept of just intonation, one must refer to the natural acoustic

phenomenon known as the ‘harmonic series’. The overtones of a specific pitch are generally referred to

as the ‘harmonic series’, and the musical scale derived from this series is constructed around ‘pure’ or

‘just’ intervals. This system of just intonation is strikingly dissimilar to the twelve-tone equally-tempered

division of the octave, which is based on the division of the octave into twelve equal intervals; the

frequency ratio of each semitone therefore mathematically representing the twelfth root of two.274 Leta E.

Miller and Fredric Lieberman describe just intonation thus: “Pure intervals arise when the frequencies of

the individual tones reflect the precise mathematical proportions that occur in the series: 3/2 for the fifth,

4/3 for the fourth, etcetera.” Intervals manifested naturally within the harmonic series are particularly

favourable in just intonation, and certainly ones with “superparticular vibration ratios,” where the

272 “Johnston’s credentials are impressive. He studied with Harry Partch, Darius Milhaud, John Cage, and Burrill

Phillips, and although some of these teachers influenced Johnston’s early music, he has remained an individual and

followed his own creative path. Yet he has written in many different styles: jazz resulting from his Naval service,

neoclassicism, and serialism (which he has adapted to serve a functional purpose in just intonation). Much of his work has

intuitively predated important trends in contemporary music. He used combinatoriality in the late 1950’s, quotations in

the 1960’s, and returned to tonality in the early 1970’s. He even tried indeterminacy and electronic music but has been

less successful in these areas.” For a further discussion, see Heidi Von Gunden, The Music of Ben Johnston (Metuchen,

N.J.:The Scarecrow Press, 1986) vii.

273 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters,” 106-07.

274 Olson, Music, Physics and Engineering 46-47.

“numerator exceeds the denominator by one” such as the just perfect fifth (3/2), just perfect fourth (4/3),

just major third (5/4) and just minor third (6/5).275

The Harmonic and Subharmonic Series

The concept of the vibrating string and its connection to music – via the sounding of an open string to

produce the fundamental (or the 1/1 ratio), with the string stopped at the midpoint producing the

octave (or the 2/1 ratio), and at the two-third point, the perfect fifth (or the 3/2 ratio) – is first presented

in the West by Pythagoras of Samos (570-504 B.C.), although it is not until Galileo Galilei (1564-1642) and

Marin Mersenne (1588-1648) that the connection between the actual time or period of the cycle or

vibration, and string length, tension and density is made. John Wallis (1616-1703) and Joseph Sauveur

(1653-1716) are acknowledged for contributing further to the study with their discovery that

proportional vibrations (in the ratios of 1/1, 1/2, 1/3, 1/4, and so on) are also simultaneously produced –

this fact elucidating the notion of the vibrating string as an amalgam of the fundamental and octave

proposed by Aristotle (382-322 B.C.), and consequently stimulating scientific study that in time reveals

more and more upper partials. “Thus the note produced in the ear by a vibrating string was shown to be

composed of a series of pure tones (partials), sounding simultaneously, and corresponding to

component or partial vibrations of the main vibration whose periods (or string lengths) formed an

harmonic series,” explains Lloyd and Boyle.276 The following table illustrates the harmonic characteristics of

the first eight partials of the harmonic series.277

Table 119. The beating characteristics of the first eight partials of the harmonic series

PARTIAL

NUMBER

NOTE INTERVAL RATIO

(FRACTION)

RATIO

(DECIMAL)

FREQUENCY

(HERTZ)

CENTS

1

2

3

4

5

6

7

8

C4 C5 G5 C6 E6 G6 B;6 C7

fundamental

octave

twelfth

fifteenth, or double octave

tierce (octave tenth)

octave twelfth

harmonic seventh

triple octave

1/1

2/1

6/2

4/1

20/4

12/2

28/4

8/1

1.000000

2.000000

3.000000

4.000000

5.000000

6.000000

7.000000

8.000000

261.626

523.251

784.877

1046.502

1308.128

1569.753

1831.379

2093.005

0.000

1200.000

1901.955

2400.000

2786.314

3101.955

3368.826

3600.000

275 Leta E. Miller, and Fredric Lieberman, Lou Harrison: Composing a World (New York: Oxford U. Press, 1998) 107.

276 Lloyd, and Boyle, Intervals, Scales and Temperaments: An Introduction to the Study of Musical Intonation 2-4.

277 R. H. M. Bosanquet, An Elementary Treatise on Musical Intervals and Temperament, ed. Rudolf Rasch (Utrecht,

The Netherlands: Diapason Press, 1987) 16.

190 The Harmonic Consideration

When just intonation practitioners speak of eliminating “acoustic beats to the maximum possible” they are

of course referring to the beats that occur when unmatched pure tones sounding simultaneously move in

and out-of-phase, which generate shifts in amplitude, and hence difference tones that are manifested as

beats per second.278 The chart for the beating characteristics of the mistuned and properly tuned unison

illustrates the fact that the partials of a properly tuned unison correspond precisely with those generated

by the fundamental, and result in an amalgam “absolutely smooth and free from any disturbance.” In

striking contrast, the consequence of a mistuned unison (1.000Hz sharp) is a continuous series of

mistuned pairs of partials.279

Table 120. The beating characteristics of the mistuned and properly tuned unison

FUNDAMENTAL MISTUNED UNISON PROPERLY TUNED UNISON

C4

(PARTIAL)

FREQUENCY

(HERTZ)

C4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

C4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

2

3

4

5

6

7

8

261.626

523.251

784.877

1046.502

1308.128

1569.753

1831.379

2093.005

1

2

3

4

5

6

7

8

262.626

525.251

787.877

1050.502

1313.128

1575.753

1838.379

1050.502

1.000

2.000

3.000

4.000

5.000

6.000

7.000

8.000

1

2

3

4

5

6

7

8

261.626

523.251

784.877

1046.502

1308.128

1569.753

1831.379

2093.005

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

The case for the octave is identical but for the fact that every second partial of a properly tuned octave

corresponds precisely with those generated by the fundamental, while a mistuned octave (1.000Hz

sharp) exponentially beats at rate of an additional 1.000Hz every second partial.280

278 Roderick D. Gordon, The World of Musical Sound (Dubuque, IA: Kendall/Hunt, 1979) 41.

279 “Beats, which have now to be considered, are essentially distinguished from combinational tones as follows: In

combinational tones the composition of vibrations in the elastic vibrating bodies which are either within or without the

ear, undergoes certain disturbances, although the ear resolves the motion which is finally conducted to it, into a series of

simple tones, according to the usual law. In beats, on the contrary, the objective motions of the elastic bodies follow the

simple law; but the composition of the sensations is disturbed. As long as several simple tones of a sufficiently different

pitch enter the ear together, the sensation due to each remains undisturbed in the ear, probably because entirely different

bundles of nerve fibres are affected. But tones of the same, or of nearly the same pitch, which therefore affect the same

nerve fibres, do not produce a sensation which is the sum of the two they would have separately excited, but new and

peculiar phenomena arises which we term ‘interference’, when caused by two perfectly equal simple tones, and ‘beats’

when due to nearly equal simple tones.” For a further discussion, see Helmholtz, On the Sensations of Tone: As a

Physiological Basis for the Theory of Music 159-60.

280 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 20.

The Harmonic Consideration 191

Table 121. The beating characteristics of the mistuned and properly tuned octave

FUNDAMENTAL MISTUNED OCTAVE PROPERLY TUNED OCTAVE

C4

(PARTIAL)

FREQUENCY

(HERTZ)

C5

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

C5

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

2

3

4

5

6

7

8

261.626

523.251

784.877

1046.502

1308.128

1569.753

1831.379

2093.005

– – – –

1

– – – –

2

– – – –

3

– – – –

4

– – – –

524.251

– – – –

1048.502

– – – –

1572.753

– – – –

2097.005

– – – –

1.000

– – – –

2.000

– – – –

3.000

– – – –

4.000

– – –

1

– – – –

2

– – – –

3

– – – –

4

– – – –

523.251

– – – –

1046.502

– – – –

1569.753

– – – –

2093.005

– – – –

0.000

– – – –

0.000

– – – –

0.000

– – – –

0.000

According to Helmholtz, ‘dissonance’ is explained as a beating condition between the corresponding

partials of a simultaneous sonority, and “there are certain determinate ratios between pitch numbers, for

which this rule suffers an exception, and either no beats at all are formed, or at least only such as have so

little intensity that they produce no unpleasant disturbance of the united sound. These exceptional cases

are called consonances.” James Tenney explains: “Helmholtz equates the dissonance of a simultaneous

aggregate with ‘roughness’ of the sensation caused by beats between adjacent partials (and to a lesser

extent, between ‘combinational tones’) in the combined spectrum of the tones forming the aggregate.”281

Helmholtz categorizes consonances as “the most perfect consonances,” or “absolute,” which include

the octave (2/1), twelfth (6/2), and double octave (4/1). Next come “perfect consonances” such as the

fifth (3/2) and fourth (4/3), which are followed by “medial consonances” such as major sixth (5/3) and

major third (5/4); with the minor third (6/5) and minor sixth (8/5) categorized as “imperfect

consonances.” The “essence of dissonance” is ultimately summarized by Helmholtz as follows:

“It is apparent to the simplest natural observation that the essence of dissonance consists merely in very

rapid beats. The nerves of hearing feel these rapid beats as ‘rough’ and unpleasant, because every

intermittent excitement of any nervous apparatus affects us more powerfully than one that lasts unaltered.

The individual pulses of tone in a dissonant combination give us certainly the same impression of separate

pulses as slow beats, although we are unable to recognize them separately and count them; hence they

form a ‘tangled’ mass of tone, which cannot be analyzed into its constituents. The cause of the

unpleasantness of dissonance we attribute to this ‘roughness’ and ‘entanglement’. The meaning of this

distinction may be thus briefly stated: ‘Consonance is a continuous, dissonance an intermittent sensation of

tone.’ Two consonant tones flow quietly side by side in an undisturbed stream; dissonant tones cut one

another up into separate pulses of tone. This description of the distinction at which we have arrived

281 James Tenney, A History of ‘Consonance’ and ‘Dissonance’ (New York: Excelsior Music Publishing Co., 1988)

87-88.

192 The Harmonic Consideration

agrees precisely with Euclid’s old definition, ‘Consonance is the blending of a higher with a lower tone.

Dissonance is incapacity to mix, when two tones cannot blend, but appear rough to the ear.’”282

Tenney offers the following observations with regards to the general acceptance of Helmholtz’s beat

theory by music theorists: “The fact that the consonance or dissonance predicted by the beat theory for

a given dyad would vary with the absolute frequencies of its tones, rather than simply the intervals

between them, has been pointed out by many other writers – and generally used as an argument against

the validity of Helmholtz’s theory.”283 It is interesting to note that Bosanquet makes a distinction “between

‘beating dissonances’ and ‘unsatisfied combinations’,” classifying the harmonic seventh (7/4), along with

the just perfect fourth (4/3) dyads in the latter group;284 while Johnston, “between different kinds of

dissonance: the dissonance of complex ratios and the dissonance of higher prime numbers,” utilizing the

numerical similarity of the grave or small just chromatic semitone (25/24) and tridecimal third tone (26/25,

or 67.900 cents) as examples of the relationship between the dissonant curve and prime limit.285

An important factor in just intonation rationale is the further notion of the complement or mirror

image of the harmonics series, which is referred to as the ‘subharmonic series’, and represents the

foundation of Partch’s theory of ‘otonalities’ (pitches derived from the ascending series) and ‘utonalities’

(pitches derived from the descending series). “Unlike the harmonic series, the subharmonic series is not

represented in the partials of any known sounding bodies,” notes Doty. “Theorists in earlier centuries

anxiously sought sounds in nature with subharmonic partials, but none were ever discovered.”286 Henry

Cowell presents the following discussion on the musical significance of research conducted on the

subharmonic series:

“A very interesting approach to the theoretical explanation of minor is a consideration of the theory of

undertones. Until recently undertones were a theory only. Their existence was contested by scientists on

the ground that a string or vibrating body could not vibrate at a length greater than its complete length,

which gives the fundamental tone. Hence, it was contended, no deeper tones in such a series would be

possible of formation on the string. Now, however, Professor Nicolas A. Garbusov, of the Moscow State

Institute for Musicology, has built an instrument on which at least the first nine undertones are easily heard

without the aid of resonators. The principle is not that the original sounding body produces the

undertones, but that it is difficult to avoid them in resonation.”287

282 Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 194-226.

283 Tenney, A History of ‘Consonance’ and ‘Dissonance’ 90. 284 R. H. M. Bosanquet, “Temperament; Or, the Division of the Octave (Part II),” Proceedings of the Musical

Association, 2nd Sess. (1874-75): 127.

285 Keislar, Blackwood, Eaton, Harrison, Johnston, Mandelbaum, and Schottstaedt, “Six American Composers on

Nonstandard Tunings,” Perspectives of New Music 202.

286 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 28-29.

287 Henry Cowell, New Musical Resources (Cambridge: Cambridge U. Press, 1996) 21-22.

The Harmonic Consideration 193

Ex. 82. The harmonics and subharmonics of the fundamental C – first partial, through the 16th partial

The following comparative table of intonation depicts the twelve basic intervals of just intonation,

Pythagorean intonation, meantone temperament, and equal temperament, indicating interval, ratio, and

cents for each system of tuning.

Table 122. Comparative table of intonation

COMPARATIVE

TABLE

JUST

INTONATION

PYTHAGOREAN

INTONATION

MEANTONE

TEMPERAMENT

EQUAL

TEMPERAMENT

INTERVAL RATIO CENTS RATIO CENTS RATIO CENTS RATIO CENTS

unison

diatonic semitone

major tone

minor third

major third

perfect fourth

tritone

perfect fifth

minor sixth

major sixth

minor seventh

major seventh

octave

1/1

16/15

9/8

6/5

5/4

4/3

45/32

3/2

8/5

5/3

9/5

15/8

2/1

0.000

111.731

203.910

315.641

386.314

498.045

590.224

701.955

813.686

884.359

1017.596

1088.269

1200.000

1/1

2187/2048

9/8

32/27

81/64

4/3

729/512

3/2

128/81

27/16

16/9

243/128

2/1

0.000

113.685

203.910

294.135

407.820

498.045

611.730

701.955

792.180

905.865

996.090

1109.775

1200.000

1.000000

1.044907

1.118034

1.196279

1.250000

1.337481

1.397542

1.495349

1.600000

1.671851

1.788854

1.869186

2.000000

0.000

76.049

193.157

310.265

386.314

503.422

579.471

696.578

813.686

889.735

1006.843

1082.892

1200.000

1.000000

1.059463

1.122462

1.189207

1.259921

1.334840

1.414214

1.498307

1.587401

1.681793

1.781797

1.887749

2.000000

0.000

100.000

200.000

300.000

400.000

500.000

600.000

700.000

800.000

900.000

1000.000

1100.000

1200.000

194 The Harmonic Consideration

Tabl

e 12

3. T

he h

arm

onic

ser

ies

– fir

st p

artia

l, th

roug

h th

e 12

8th

par

tial

Rela

tive

Pitc

h: A

4=44

0Hz

/ C4 (

mid

dle

C)=

261.

6255

654H

z DE

GRE

E

NUM

BER

NO

TE

INTE

RVA

L RA

TIO

(FRA

CTIO

N)

RATI

O

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

TUN

ING

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÑÓ

ÑÔ

ÑÕ

ÑÖ

Ñ×

ÑØ

ÑÙ

ÒÐ

ÒÑ

C C.

C,

CP

CX(

D+ D D!

EW

E©

EO E E(

F+(

uniso

n (1

st h

arm

onic

)

trid

ecim

al c

omm

a (6

5th

harm

onic

)

und

ecim

al c

omm

a (3

3rd

and

66t

h ha

rmon

ic)

67th

har

mon

ic

sep

tend

ecim

al c

hrom

atic

sem

itone

(17t

h, 3

4th

and

68t

h ha

rmon

ic)

trivi

gesim

al c

hrom

atic

sem

itone

(69

th h

arm

onic

)

sep

timal

neu

tral s

econ

d (3

5th

and

70th

har

mon

ic)

71st

har

mon

ic

just

maj

or to

ne (

9th,

18t

h, 3

6th

and

72nd

har

mon

ic)

73rd

har

mon

ic

37t

h an

d 74

th h

arm

onic

augm

ente

d se

cond

(75t

h ha

rmon

ic)

nona

dec

imal

sub

min

or, o

r ove

rtone

min

or th

ird (1

9th,

38t

h an

d 76

th h

arm

onic

)

und

ecim

al n

eutra

l thi

rd (

77th

har

mon

ic)

trid

ecim

al g

rave

or s

mal

l neu

tral t

hird

(39

th a

nd 7

8th

harm

onic

)

79th

har

mon

ic

just

maj

or th

ird (

5th,

10t

h, 2

0th,

40t

h an

d 80

th h

arm

onic

)

Pyth

agor

ean

maj

or th

ird, o

r dito

ne (8

1st h

arm

onic

)

41st

and

82n

d ha

rmon

ic

83rd

har

mon

ic

sep

timal

sub

four

th (

21st

, 42n

d an

d 84

th h

arm

onic

)

1/1

65/6

4

33/3

2 (6

6/64

)

67/6

4

17/1

6 (3

4/32

, 68/

64)

69/6

4

35/3

2 (7

0/64

)

71/6

4

9/8

(18/

16, 3

6/32

, 72/

64)

73/6

4

37/3

2 (7

4/64

)

75/6

4

19/1

6 (3

8/32

, 76/

64)

77/6

4

39/3

2 (7

8/64

)

79/6

4

5/4

(10/

8, 2

0/16

, 40/

32, 8

0/64

)

81/6

4

41/3

2 (8

2/64

)

83/6

4

21/1

6 (4

2/32

, 84/

64)

1.00

0000

1.01

5625

1.03

1250

1.04

6875

1.06

2500

1.07

8125

1.09

3750

1.10

9375

1.12

5000

1.14

0625

1.15

6250

1.17

1875

1.18

7500

1.20

3125

1.21

8750

1.23

4375

1.25

0000

1.26

5625

1.28

1250

1.29

6875

1.31

2500

261.

626

265.

713

269.

801

273.

889

277.

977

282.

065

286.

153

290.

241

294.

329

298.

417

302.

505

306.

592

310.

680

314.

768

318.

856

322.

944

327.

032

331.

120

335.

208

339.

296

343.

384

0.00

0

26.8

41

53.2

73

79.0

70

104.

955

130.

229

155.

140

179.

697

203.

910

227.

789

251.

344

274.

582

297.

513

320.

144

342.

483

364.

537

386.

314

407.

820

429.

062

450.

047

470.

781

+00

+27

ß47

ß21

+05

+30

ß45

ß20

+04

+28

ß49

ß25

ß02

+20

+42

ß35

ß14

+08

+29

ß50

ß29

The Harmonic Consideration 195

DEG

REE

NUM

BER

NO

TE

INTE

RVA

L RA

TIO

(FRA

CTIO

N)

RATI

O

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

TUN

ING

ÒÒ

ÒÓ

ÒÔ

ÒÕ

ÒÖ

Ò×

ÒØ

ÒÙ

ÓÐ

ÓÑ

ÓÒ

ÓÓ

ÓÔ

ÓÕ

ÓÖ

Ó×

ÓØ

ÓÙ

ÔÐ

ÔÑ

ÔÒ

ÔÓ

ÔÔ

EP

F6(

F, F!

(

GÞ

FX(

F`(

G2

G

AŒ(

G,

GÚ GP

AO

A+(

sep

tend

ecim

al s

uper

augm

ente

d th

ird (

85th

har

mon

ic)

43rd

and

86t

h ha

rmon

ic

nona

vige

simal

sup

erfo

urth

(87

th h

arm

onic

)

und

ecim

al s

uper

four

th (

11th

, 22n

d, 4

4th

and

88t

h ha

rmon

ic)

89th

har

mon

ic

just

trito

ne, o

r aug

men

ted

four

th (

45th

and

90t

h ha

rmon

ic)

trid

ecim

al d

imin

ished

fifth

(91

st h

arm

onic

)

trivi

gesim

al s

uper

augm

ente

d fo

urth

(23r

d, 4

6th

and

92n

d h

arm

onic

)

untri

gesim

al s

uper

augm

ente

d fo

urth

(93

rd h

arm

onic

)

47th

and

94t

h ha

rmon

ic

nona

dec

imal

sub

fifth

(95

th h

arm

onic

)

just

and

Pyt

hago

rean

per

fect

fifth

(3r

d, 6

th, 1

2th,

24t

h, 4

8th

and

96t

h ha

rmon

ic)

97th

har

mon

ic

sep

timal

dim

inish

ed s

ixth

(49

th a

nd 9

8th

harm

onic

)

und

ecim

al s

uper

fifth

(99t

h ha

rmon

ic)

augm

ente

d fi

fth (

25th

, 50t

h an

d 1

00th

har

mon

ic)

101s

t har

mon

ic

sept

ende

cim

al s

uper

augm

ente

d fif

th (5

1st a

nd 1

02nd

har

mon

ic)

103r

d ha

rmon

ic

trid

ecim

al g

rave

or s

mal

l neu

tral,

or o

verto

ne s

ixth

(13

th, 2

6th,

52n

d a

nd 1

04th

har

mon

ic)

sept

imal

neu

tral s

ixth

(10

5th

harm

onic

)

53rd

and

106

th h

arm

onic

107t

h ha

rmon

ic

85/6

4

43/3

2 (8

6/64

)

87/6

4

11/8

(22/

16, 4

4/32

, 88/

64)

89/6

4

45/3

2 (9

0/64

)

91/6

4

23/1

6 (4

6/32

, 92/

64)

93/6

4

47/3

2 (9

4/64

)

95/6

4

3/2

(6/4

, 12/

8, 2

4/16

, 48/

32, 9

6/64

)

97/6

4

49/3

2 (9

8/64

)

99/6

4

25/1

6 (5

0/32

, 100

/64)

101/

64

51/3

2 (1

02/6

4)

103/

64

13/8

(26/

16, 5

2/32

, 104

/64)

105/

64

53/3

2 (1

06/6

4)

107/

64

1.32

8125

1.34

3750

1.35

9375

1.37

5000

1.39

0625

1.40

6250

1.42

1875

1.43

7500

1.45

3125

1.46

8750

1.48

4375

1.50

0000

1.51

5625

1.53

1250

1.54

6875

1.56

2500

1.57

8125

1.59

3750

1.60

9375

1.62

5000

1.64

0625

1.65

6250

1.67

1875

347.

471

351.

559

355.

647

359.

735

363.

823

367.

911

371.

999

376.

087

380.

175

384.

263

388.

350

392.

438

396.

526

400.

614

404.

702

408.

790

412.

878

416.

966

421.

054

425.

142

429.

229

433.

317

437.

405

491.

269

511.

518

531.

532

551.

318

570.

880

590.

224

609.

354

628.

274

646.

991

665.

507

683.

827

701.

955

719.

895

737.

652

755.

228

772.

627

789.

854

806.

910

823.

801

840.

528

857.

095

873.

505

889.

760

ß09

+12

+32

ß49

ß29

ß10

+09

+28

+47

ß34

ß16

+02

+20

+38

ß45

ß27

ß10

+07

+24

+41

ß43

ß26

ß10

196 The Harmonic Consideration

DEG

REE

NUM

BER

NO

TE

INTE

RVA

L RA

TIO

(FRA

CTIO

N)

RATI

O

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

TUN

ING

ÔÕ

ÔÖ

Ô×

ÔØ

ÔÙ

ÕÐ

ÕÑ

ÕÒ

ÕÓ

ÕÔ

ÕÕ

ÕÖ

Õ×

ÕØ

ÕÙ

ÖÐ

ÖÑ

ÖÒ

ÖÓ

ÖÔ ÐÑ

A( A, B;

BW

AX(

B_

BO Bß

B BÏ)

B8

B!

C+(

C

Pyth

agor

ean

maj

or s

ixth

(27t

h, 5

4th

and

108t

h ha

rmon

ic)

109t

h ha

rmon

ic

und

ecim

al s

uper

maj

or s

ixth

(55t

h an

d 11

0th

harm

onic

)

111t

h ha

rmon

ic

sep

timal

sub

min

or s

even

th (

7th,

14t

h, 2

8th,

56t

h an

d 11

2th

harm

onic

)

113t

h ha

rmon

ic

nona

dec

imal

sub

min

or s

even

th (

57th

and

114

th h

arm

onic

)

trivi

gesim

al s

uper

augm

ente

d six

th (

115t

h ha

rmon

ic)

nona

vige

simal

gra

ve o

r sm

all n

eutra

l sev

enth

(29

th, 5

8th

and

116t

h ha

rmon

ic)

trid

ecim

al n

eutra

l sev

enth

(117

th h

arm

onic

)

59th

and

118

th h

arm

onic

sept

ende

cim

al n

eutra

l sev

enth

(11

9th

harm

onic

)

just

dia

toni

c m

ajor

sev

enth

(15

th, 3

0th,

60t

h an

d 12

0th

harm

onic

)

grav

e or

sm

all n

eutra

l sev

enth

(12

1st h

arm

onic

)

61st

and

122

nd h

arm

onic

123r

d ha

rmon

ic

untri

gesim

al s

uper

maj

or s

even

th (

31st

, 62n

d an

d 12

4th

harm

onic

)

augm

ente

d s

even

th (1

25th

har

mon

ic)

sep

timal

sub

dim

inish

ed o

ctav

e (6

3rd

and

126

th h

arm

onic

)

127t

h ha

rmon

ic

octa

ve (

2nd

har

mon

ic)

(4th

, 8th

, 16t

h, 3

2nd,

64t

h an

d 12

8th

harm

onic

)

27/1

6 (5

4/32

, 108

/64)

109/

64

55/3

2 (1

10/6

4)

111/

64

7/4

(14/

8, 2

8/16

, 56/

32, 1

12/6

4)

113/

64

57/3

2 (1

14/6

4)

115/

64

29/1

6 (5

8/32

, 116

/64)

117/

64

59/3

2 (1

18/6

4)

119/

64

15/8

(30/

16, 6

0/32

, 120

/64)

121/

64

61/3

2 (1

22/6

4)

123/

64

31/1

6 (6

2/32

, 124

/64)

125/

64

63/3

2 (1

26/6

4)

127/

64

2/1

(4/2

, 8/4

, 16/

8, 3

2/16

, 64/

32, 1

28/6

4)

1.68

7500

1.70

3125

1.71

8750

1.73

4375

1.75

0000

1.76

5625

1.78

1250

1.79

6875

1.81

2500

1.82

8125

1.84

3750

1.85

9375

1.87

5000

1.89

0625

1.90

6250

1.92

1875

1.93

7500

1.95

3125

1.96

8750

1.98

4375

2.00

0000

441.

493

445.

581

449.

669

453.

757

457.

845

461.

933

466.

021

470.

108

474.

196

478.

284

482.

372

486.

460

490.

548

494.

636

498.

724

502.

812

506.

900

510.

987

515.

075

519.

163

523.

251

905.

865

921.

821

937.

632

953.

299

968.

826

984.

215

999.

468

1014

.588

1029

.577

1044

.438

1059

.172

1073

.781

1088

.269

1102

.636

1116

.885

1131

.017

1145

.036

1158

.941

1172

.736

1186

.422

1200

.000

+06

+22

+38

ß47

ß31

ß16

ß01

+15

+30

+44

ß41

ß26

ß12

+03

+17

+31

+45

ß41

ß27

ß14

+00

The Harmonic Consideration 197

It is important to note that odd-numbered partials in the harmonic and subharmonic series represent

unique entities, while even-numbered partials, merely their respective octave duplications, and therefore

in harmonic analysis the former needs only be considered. The term designated to these unique entities

of the harmonic and subharmonic series is ‘identities’.288

The Monochord

A monochord is a simple instrument that consists of a metal string extended over two bridges, and is a

useful tool for illustrating the nature of ratios (a bamboo pipe, or air column deliver similar results). If for

example, striking the open string produced 100 cycles when set in vibration (cycles referring to the

number of vibrations per second which a tone makes), striking half of that length (the string stopped via a

third bridge placed in the centre) would produce 200 cycles, or a 200 to 100 ratio (2/1); sounding the

perfect octave of the original pitch. The logic then follows that striking a third of the full length would

produce 300 cycles, or a 300 to 200 ratio (3/2) of the half length, and hence now sounding the perfect

fifth.289 Cecil Adkins provides the following historical perspective:

“The monochord in its early form, and in the form utilized throughout the Middle Ages was a table or plank

(AC) upon which were erected two fixed bridges (EB and FD). The string was stretched across the

bridges (EF) and securely fastened at the ends (AC). A movable bridge (K) was then placed underneath

the string, dividing it into two sections (EK and KF). The marks indicating the placement of the movable

bridge were then inscribed on the table underneath the string, between the two end bridges (B and D).

The resonating box, generally considered an integral part of the instrument, is not mentioned in the treatises

of the Middle Ages, but is depicted in miniatures after the twelfth century. It was probably a late medieval

addition directed at increasing the portability of the instrument as much as enhancing its tone.”

Fig. 6. Medieval figure of the monochord

The traditional function of the monochord is the facilitation of aural representations of intervals or scales,

established via mathematical calculation, and generally articulated via the principles of proportions, string

lengths, or cents. The term ‘monochord division’ in this context refers to what Adkins prefers to call the

“manual division,” and therefore implicates the actual physical application of the mathematical formula

288 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 30.

289 Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 79-81.

198 The Harmonic Consideration

required to produce a specific interval. The Pythagorean principle of “monochord division by

proportions” takes into account the “arithmetic mean and the harmonic mean” in a system of four

principal intervals directly derived from the “smallest whole integers” that is capable of expressing the

intervallic “relationships of these two means.” In this case, the principle is represented by the numbers 6,

8, 9, and 12, and by the resulting ratios of 12/6 (the diapason, or octave), 9/6 and 12/8 (the diapente, or

fifth), 8/6 and 12/9 (the diatessaron, or fourth), and 9/8 (the tone, or major second). In different terms,

these four intervals may be further theorized as the “multiplex and superparticular proportions” of 2/1

(dupla), 3/2 (sesquialtera), 4/3 (sesquitertia), and 9/8 (sesquioctava).290 The employment of string lengths

and cents as a measuring device in the Pythagorean chromatic context represent 2/1 (octave) by the

figure 314928, or 1200.000 cents; 3/2 (just perfect fifth) by 419904, or 701.955 cents; 4/3 (just perfect

fourth) by 472392, or 498.045 cents; and 9/8 (just major tone) by 559872, or 203.910 cents. The

interval of the unison is represented by the string length of 629856, or 0.000 cents.291

Table 124. The Pythagorean monochord

DEGREE

NUMBER

NOTE LENGTHS RATIO

(DECIMAL)

CENTS FALSITY

(CENTS)

ÐÑ ÐÒ ÐÓ ÐÔ ÐÕ ÐÖ Ð× ÐØ ÐÙ ÑÐ ÑÑ ÑÒ

ÐÑ

C4

C!4

D4

D!4

E4

F4

F!4

G4

G!4

A4

A!4

B4

C5

629856

589824

559872

531441

497664

472392

442368

419904

393216

373248

354294

321776

314928

1.000000

1.067871

1.125000

1.185185

1.265625

1.333333

1.423828

1.500000

1.601807

1.687500

1.777778

1.898438

2.000000

0.000

113.685

203.910

294.135

407.820

498.045

611.730

701.955

815.640

905.865

996.090

1109.775

1200.000

+0.000

+1.954

+0.000

ß21.506

+21.506

+0.000

+21.506

+0.000

+1.954

+21.506

ß21.506

+21.506

+0.000

290 Cecil Adkins, “The Technique of the Monochord,” Acta Musicologica 39 (Jan.-Jun., 1967): 34-37.

291 Barbour, Tuning and Temperament: A Historical Survey 90.

The Harmonic Consideration 199

Combinational Tones

The discovery of combinational tones in 1745 may be attributed to German organist Georg Andreas

Sorge (1703-1778), and was followed by the later classification of Italian violinist Giuseppe Tartini (1692-

1770). According to Helmholtz, “These tones are heard whenever two musical tones of different

pitches are sounded together, loudly and continuously.” There are two classes of combinational tones,

with the first being differential tones (whereby the resultant tone equals f2 minus f1), and the second,

summation tones (whereby the resultant tone equals f2 plus f1), which represent Helmholtz’s own

contribution to the theory. Primary and upper partial tones are both capable of producing the two

classes of combinational tones. Differential tones are more prominent when the interval between the

two generating tones is inferior to an octave, as this condition produces a resultant tone below the

primary tones. The theory may be further expanded to differentiate between first and second-order

differential tones, with the latter being phenomena of an infinite nature, and the result of the secondary

differential tones produced by the combination of the primary differential tones and the actual generating

tones. Summation tones are less prominent than combinational tones; and in view of the fact that the

resultant frequency is equal to the sum of the two primary tones, the pitch of these tones will always

exceed their generators. It should be further noted that upper partial tones are also capable of

producing combinational tones.292 The following example illustrates the differential tones produced by

the octave (2/1), just perfect fifth (3/2), just perfect fourth (4/3), just major third (5/4), just minor sixth

(8/5), just minor third (6/5), and just major sixth (5/3).

Ex. 83. Differential tones produced by 2/1, 3/2, 4/3, 5/4, 8/5, 6/5, and 5/3

Summation tones for the same generating tones on the other hand present the following intervals: just

perfect fifth (3/2), just major third (5/4), septimal subminor third (7/6), just major tone (9/8), tridecimal

subdiminished fourth (13/10), undecimal acute or large neutral second (11/10), and just perfect fourth

(4/3).

292 Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 152-56.

200 The Harmonic Consideration

Ex. 84. Summation tones produced by 2/1, 3/2, 4/3, 5/4, 8/5, 6/5, and 5/3

Periodicity Pitch

“When two or more pure tones separated by intervals greater than the critical band are sounded

simultaneously, the auditory mechanism is sensitive to the period or frequency of the resulting composite

waveform. This gives rise to a phenomenon variously known as the periodicity pitch, virtual pitch,

subjective pitch, residue tone, or the missing fundamental,” explains Doty. The simultaneous sounding of

two dissimilar pitched musical tones may be mathematically explained as f2= nm Ïf1, with m and n

representative of the relevant integers (numerator and denominator) that denote the frequency ratio of

the interval. The formula for calculating the periodicity pitch of a dyad is �0= n1 �1, with n, the

denominator, utilized as the dividing factor, and applied to the fundamental, or lower tone. The process

may be illustrated via the calculations of the just perfect fifth (3/2) and just perfect fourth (4/3) in relation

to the unison (1/1), which presents 21 Ï261.626Hz (or C4)=130.8136Hz (or C3); and 3

1 Ï261.626Hz

(or C4)=87.209Hz (or F2). Doty alludes to the facts that the “smaller the value of n, the shorter the

period and the higher the frequency of the resulting pattern,” and that “musical consonance is associated

with high periodicity pitch.” Dyads generated via simple integer-ratio intervals with integers inferior to

eight or nine produce “unambiguous periodicity pitch,” while superior integers generally produce an

“ambiguous” approximation of a simpler ratio.293

Ex. 85. Periodicity pitches produced by 2/1, 3/2, 4/3, 5/3, 5/4, 7/4, 6/5, 7/5, 8/5, and 7/6

293 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 17-18.

The Harmonic Consideration 201

Sethares has the following to say about the “periodicity theory of pitch perception,” and its theoretical

significance in relation to JND, or Just Noticeable Difference:

“An alternative hypothesis (to JND, or Just Noticeable Difference), called ‘periodicity’ theory of pitch

perception suggests that information is extracted directly from the time behaviour of the sound. For

instance, the time interval over which a signal repeats may be used to determine its frequency. In fact,

there is now (and has been for the past hundred years or so) considerable controversy between

advocates of the place and periodicity theories, and it is probably safe to say that there is not enough

evidence to decide between them. Indeed, J. R. Pierce (Periodicity and Pitch Perception, 1889-93

[1991]) suggests that both mechanisms may operate in tandem, and a growing body of recent

neurophysiological research (such as P. Cariani and his co-workers [Temporal Coding of Periodicity Pitch in

the Auditory System: An Overview [1999], and A Temporal Model for Pitch Multiplicity and Tonal

Consonance [2004]) reinforce this.”294

Prime Numbers, Primary Intervals, and Prime Limits

Another aspect of just intonation principles is the theory of prime limits, which is based on the concept

of prime numbers, or integers with only factors of one and themselves. This is an infinite series, but

musical significance is accorded to primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31. The primary intervals

of these eleven primes are 2/1, 3/2, 5/4, 7/4, 11/8, 13/8, 17/16, 19/16, 23/16, 29/16, and 31/16

(produced by the formula: p/2n=p [prime], with “2n being the greatest power of 2 less than p”), while

its subharmonic complements may be represented by 2/1, 4/3, 8/5, 8/7, 16/11, 16/13, 32/17, 32/19,

32/32, 32/29, and 32/31 (produced by the formula: 2n/p=p [prime], with “2n being the smallest power

of 2 less than p”). An example of this concept is ‘five limit just intonation’, which should be taken to

mean an intonation system based only on the primes 2, 3, and 5, with all intervals directly derived from

2/1, 3/2, and 5/4 (the octave, just perfect fifth, and just major third), or 2/1, 4/3, and 8/5 (octave, just

perfect fourth, and just minor sixth) from a subharmonic perspective.295

The Just Diatonic Scale

The construction of a just diatonic scale begins with the fundamental or unison (the frequency ratio 1/1),

and the establishment of a just major triad in the ratio of 1/1:5/4:3/2.296 The harmonic relationship of this

root position chord, which has the 1, 5, and 3 identities of the harmonic series, is also expressible as

4:5:6, or in the key of C major as the pitches C, E, and G. In first inversion, the triad is represented by the

294 Sethares, Tuning, Timbre, Spectrum, Scale 44.

295 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 28-30. 296 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters,” 108.

202 The Harmonic Consideration

ratio 5:6:8, and in second, by 3:4:5. This triad is beatless, made up of the first, fifth, and third partials of

the harmonic series, and contains the intervals of the unison (1/1), just major third (5/4), and just perfect

fifth (3/2). Analysis of this simultaneous sonority further reveals a 6/5 ratio (or just minor third) between

the third and the fifth. G4 (392.438Hz), or the just perfect fifth (3/2), presents 0.000 beats between the

third harmonic of C4 (1/1) and the second harmonic of G4 (784.877Hz), and 0.000 beats between the

sixth harmonic of C4 and the fourth harmonic of G4 (1569.753Hz); while E4 (327.032Hz), or the just

major third (5/4) then consistently also presents 0.000 beats between the fifth harmonic of C4 and the

fourth harmonic of E4 (1308.128Hz).

Table 125. The beating characteristics of the just major triad

C4

(PARTIAL)

FREQUENCY

(HERTZ)

E4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

G4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

5

6

– – – –

7

– – – –

8

261.626

– – – –

– – – –

523.251

– – – –

784.877

– – – –

1046.502

– – – –

1308.128

1569.753

– – – –

1831.379

– – – –

2093.005

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

– – – –

5

– – – –

6

– – – –

– – – –

327.032

– – – –

– – – –

654.064

– – – –

981.096

– – – –

– – – –

1308.128

– – – –

1635.160

– – – –

1962.192

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

4

– – – –

– – – –

5

– – – –

– – – –

– – – –

392.438

– – – –

– – – –

784.877

– – – –

– – – –

1177.315

– – – –

1569.753

– – – –

– – – –

1962.192

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

The next stage in the generation of a just diatonic scale requires simple mathematics, or the multiplication

of ratios, to deal with the addition of ratios. If one were to take the C major triad as an example, which

has a 6/5 ratio between 5/4 and 3/2, one can ascertain that adding 6/5 to 5/4 will produce the equation:

6/5 plus 5/4=(6Ï5):(5Ï4)=30/20=3/2. In this particular case, 3/2 represents the reduction of the

unnecessary larger 30/20 ratio. Subtraction of ratios is alternatively resolved via the multiplication of ratios

with the subtraction in question inverted. In other words, the subtraction of 6/5 from 3/2 is resolved

thus: 3/2 minus 6/5=(3Ï5):(2Ï6)=15/12=5/4. In cases where addition results in a figure that exceeds

the octave (2/1), or where the “numerator is greater than twice the denominator,” subtraction of an

appropriate number of 2/1s should be undertaken. For example, the equation of 3/2 plus

The Harmonic Consideration 203

3/2=(3Ï3):(2Ï2)=9/4 should be followed by 9/4 minus 2/1=(9Ï1):(4Ï2)=9/8. On the other hand,

in cases where subtraction results in a figure that exceeds the unison (1/1), or where the result is a “ratio

with a denominator greater than its numerator”, addition of an appropriate number of 2/1s should be

undertaken. For example, 9/8 minus 3/2=(9Ï2):(8Ï3)=18/24=3/4 should be followed by 3/4 plus

2/1=(3Ï2):(4Ï1)=6/4=3/2.297 The final stage in the establishment of the just diatonic scale involves the

further generation of just major triads on the fifth degree, 3/2 (or G) and its inversion, 4/3 (or F), which

produce the intervals 3/2, 15/8, and 9/8 (or the pitches G, B, and D), and 4/3, 5/3, and 1/1 (or the

pitches F, A, and C). The equations involved in obtaining 4:5:6 harmonic relationships (or appropriate

5/4s and 3/2s) for 1/1, 3/2, and 4/3 are:

i. C (unison) generated by the ratio 1/1

ii. E (just major third) by the equation 1/1 plus 5/4=(1Ï5):(1Ï4)=5/4

iii. G (just and Pythagorean perfect fifth) by 1/1 plus 3/2=(1Ï3):(1Ï2)=3/2

iv. G (just and Pythagorean perfect fifth) by the ratio 3/2

v. B (just diatonic major seventh) by the equation 3/2 plus 5/4=(3Ï5):(2Ï4)=15/8

vi D (just major tone) by 3/2 plus 3/2=(3Ï3):(2Ï2)=9/4; and 9/4 minus 2/1=(9Ï1):(4Ï2)=9/8

vii. F (just and Pythagorean perfect fourth) by the ratio 4/3

viii. A (just major sixth) by the equation 4/3 plus 5/4=(4Ï5):(3Ï4)=20/12=5/3

ix. C (unison) by 4/3 plus 3/2=(4Ï3):(3Ï2)=12/6=2/1; and 2/1 minus 2/1=(2Ï1):(1Ï2)=2/2=1/1

Ex. 86. Just major triads on 1/1, 3/2, and 4/3

The resulting scale consists of two dissimilar tetrachords separated by a just major tone, or the ratio 9/8,

which in relation to the fundamental, or unison (1/1), presents the intervals of the just major tone (9/8),

just major third (5/4), just perfect fourth (4/3), just perfect fifth (3/2), just major sixth (5/3), just diatonic

major seventh (15/8), and octave (2/1).

297 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 22-26.

204 The Harmonic Consideration

Ex. 87. The harmonic characteristics of the just diatonic scale

The Just Chromatic Scale The next stage involves the construction of a “secondary group of triads” with 4:5:6 harmonic

relationships to D, E, A, and B, or 9/8, 5/4, 5/3, and 15/8, which are produced via the following

equations:

i. D (just major tone) generated by the ratio 9/8

ii. F!( (just tritone, or augmented fourth) by the equation 9/8 plus 5/4=(9Ï5):(8Ï4)=45/32

iii. A( (Pythagorean major sixth) by 9/8 plus 3/2=(9Ï3):(8Ï2)=27/16

iv. E (just major third) by the ratio 5/4

v. G! (augmented fifth) by the equation 5/4 plus 5/4=(5Ï5):(4Ï4)=25/16

vi. B (just diatonic major seventh) by 5/4 plus 3/2=(5Ï3):(4Ï2)=15/8

vii. A (just major sixth) by the ratio 5/3

viii. C! (grave or small just chromatic semitone) by the equation 5/3 plus

5/4=(5Ï5):(3Ï4)=25/12; and 25/12 minus 2/1=(25Ï1):(12Ï2)=25:24

ix. E (just major third) by 5/3 plus 3/2=(5Ï3):(3Ï2)=15/6; and 15/6 minus

2/1=(15Ï1):(6Ï2)=15/12=5/4

x. B (just diatonic major seventh) by the ratio 15/8

xi. D! (augmented second) by 15/8 plus 5/4=(15Ï5):(8Ï4)=75/32; and 75/32 minus

2/1=(75Ï1):(32Ï2)=75/64

xii. F!( (just tritone, or augmented fourth) by 15/8 plus 3/2=(15Ï3):(8Ï2)=45/16; and 45/16

minus 2/1=(45Ï1):(16Ï2)=45/32

The Harmonic Consideration 205

Ex. 88. Just major triads on 9/8, 5/4, 5/3, and 15/8

Of particular note is the fact that the A major triad now presents the grave or small just chromatic

semitone (25/24), or what may be assumed to be the ‘sharp’ (!) ratio, and hence the ratio “applied to a

diatonic note to produce the sharpened form of that note.” Following from that, in order to generate the

correct ratios for the 4:5:6 triads built on D and B will require the raising of the 25/24 ratio by an

additional interval of a syntonic comma (81/80). Theorists generally refer to this interval (the sum of 25/24

and 81/80) as the large limma (135/128, or 92.179 cents), which may be mathematically expressed as:

25/24 plus 81/80=(25Ï81):(24Ï80)=2025/1920=135/128. 135/128, and its reciprocal, 128/135,

may also be theoretically referred to as the sharp, or greater limma, and flat, or hypolimma respectively.298

The generation of flats then utilizes Partch’s otonal and utonal theoretical basis to incorporate

pitches derived from the complement or mirror image of the harmonics series. The retrograde of a C

major triad – the initiating factor – is therefore utilized to produce an F minor triad, or what Fonville

describes as an inversion explained as a “subharmonic relationship or subharmonic series descending

from 1/1,” or in different terms; the intervallic complements of 1/1, 5/4, and 3/2, equal to 1/1, 8/5, and

4/3. The equations involved in obtaining 5:6:10 relationships (or appropriate 8/5s and 4/3s) for 1/1, 3/2,

and 4/3 are:

i. C (unison) generated by the ratio 1/1

ii. A" (just minor sixth) by the equation 1/1 minus 5/4=(1Ï4):(1Ï5)=4/5; and 4/5 plus

2/1=(4Ï2):(5Ï1)=8/5

iii. F (just and Pythagorean perfect fourth) by 1/1 minus 3/2=(1Ï2):(1Ï3)=2/3; and 2/3 plus

2/1=(2Ï2):(3Ï1)=4/3

iv. G (just and Pythagorean perfect fifth) by the ratio 3/2

v. E" (just minor third) by the equation 3/2 minus 5/4=(3Ï4):(2Ï5)=12/10=6/5

vi C (unison) by 3/2 minus 3/2=(3Ï2):(2Ï3)=6/6=1/1

vii. F (just and Pythagorean perfect fourth) by the ratio 4/3

298 Alexander J. Ellis, “On the Conditions, Extent, and Realization of a Perfect Musical Scale on Instruments with Fixed

Tones,” Proceedings of the Royal Society of London 13 (1863-64): 95.

206 The Harmonic Consideration

viii. D") (just diatonic semitone, or major half-tone) by the equation 4/3 minus

5/4=(4Ï4):(3Ï5)=16/15

ix. B") (Pythagorean minor seventh) by 4/3 minus 3/2=(4Ï2):(3Ï3)=8/9; and 8/9 plus

2/1=(8Ï2):(9Ï1)=16/9

Ex. 89. Just minor triads on 1/1, 3/2, and 4/3

The chromatic sonorities in the F and G minor triads now present the subtraction of a grave or small just

chromatic semitone (25/24) from a diatonic note, and hence, the introduction of the flat (") symbol. The

symbol could for example be applied to D, or 9/8 (or any other diatonic interval for that matter) to

produce the flattened form of that note or ratio simply by the subtraction of 25/24 from 9/8, or 9/8

minus 25/24=(9Ï24):(8Ï25)=216/200=27/25. Theorists generally refer to this interval as the great

limma, acute or large half-tone, which measures 133.238 cents. The correct ratios for the 5:6:10 triad

built on F require the lowering of the 9/8 and 9/5 ratios by an additional syntonic comma (81/80). For

example, D"), or the just diatonic semitone (16/15) is the result of the subtraction of 25/24 and 81/80

(equal to 135/128) from 9/8, or 9/8 minus 135/128=(9Ï128):(8Ï135)=1152/1080=16/15. The

otonal process produces the just minor scale (aeolian mode, natural minor, or descending melodic minor

scale). The scale consists of two dissimilar tetrachords separated by a just major tone, or the ratio 9/8,

which in relation to the fundamental, or unison (1/1) presents the intervals of just major tone (9/8), just

minor third (6/5), just perfect fourth (4/3), just perfect fifth (3/2), just minor sixth (8/5), acute or large

minor seventh (9/5), and octave (2/1).

Ex. 90. The just minor scale

The Harmonic Consideration 207

The process is then continued to include 5:6:10 relationships (or appropriate 8/5s and 4/3s) for 9/8, 5/4,

5/3, and 15/8, which are produced via the following equations:

i. D (just major tone) generated by the ratio 9/8

ii. B" (acute or large minor seventh) by the equation 9/8 minus 5/4=(9Ï4):(8Ï5)=36/40=9/5

iii. G (just and Pythagorean perfect fifth) by 9/8 minus 3/2=(9Ï2):(8Ï3)=18/24=3/4; and 3/4

plus 2/1=(3Ï2):(4Ï1)=6/4=3/2

iv. E (just major third) by the ratio 5/4

v. C (unison) by the equation 5/4 minus 5/4=(5Ï4):(4Ï5)=20/20=1/1

vi. A (just major sixth) by 5/4 minus 3/2=(5Ï2):(4Ï3)=10/12; and 10/12 plus

2/1=(10Ï2):(12Ï1)=20/12=5/3

vii. A (just major sixth) by the ratio 5/3

viii. F (just and Pythagorean perfect fourth) by the equation 5/3 minus 5/4=(5Ï4):(3Ï5)=20/15=4/3

ix. D) (just minor tone) by 5/3 minus 3/2=(5Ï2):(3Ï3)=10/9

x. B (just diatonic major seventh) by the ratio 15/8

xi. G (just and Pythagorean perfect fifth) by 15/8 minus 5/4=(15Ï4):(8Ï5)=60/40=3/2

xii. E (just major third) by 15/8 minus 3/2=(15Ï2):(8Ï3)=30/24=5/4

Ex. 91. Just minor triads on 9/8, 5/4, 5/3, and 15/8

Repeating the process of triadic construction to include all major and minor triads, as well as dominant

triads to all the basic just scales degrees (1/1, 16/15, 9/8, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 16/9, and 15/8)

results in a twenty-five-note just enharmonic scale.299

299 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters” 108-11.

208 The Harmonic Consideration

Ex. 92. The just enharmonic scale

Ben Johnston’s Fifty-Three-Tone Just Intonation Scale Johnston’s elaboration of the twenty-five-note just enharmonic scale – premised on a system of infinite

intervallic regeneration – firstly produces a twelve-note just scale, via the combination of the just diatonic

and just minor scales, and the further subdivision of the two just major tones of the structure; the process

resulting in the addition of D" and G" between C and D, and F and G respectively. The scale is then

developed into a nineteen-note just enharmonic scale, via the establishment of an enharmonic equivalent

of the previous twelve-note model, and therefore inclusive of C!, D!, F!, G!, and A!, as well as E! and F",

and B! and C"; and finally, expanded into a 2, 3, 5-limit just intonation fifty-three-tone enharmonic scale

that confers each of the nineteen pitches with a function as root, fifth, major third, and minor third – a

process limited to the overlapping of the “twelve chromatic regions of the octave.”

Each whole-tone, minor whole-tone, and diatonic semitone in Johnston’s just intonation fifty-three-

tone enharmonic scale features the identical pattern of adjacent intervals, which include either the

syntonic comma (81/80), diaskhisma (2048/2025),300 or grave or small diesis (3125/3072, or 29.614

cents). The grave or small diesis may be characterized as the difference between the grave or small just

chromatic semitone (25/24) and the great diesis (128/125).301

300 “The diaskhisma is a small interval composed of two major thirds down and four perfect fifths down

= 2411 532 ßß , also expressed as the 2,3,5-monzo [11ß4,ß2>. Its ratio is 2048/2025 [=0.20 semitones

=~19.55256881 or ~195/9 cents]. The term was used by Alexander Ellis in his English translation of Helmholtz, On the

Sensations of Tone (1875). It had been referred to earlier by Rameau (Traité de l’harmonie, 1722) as the ‘diminished

comma’. The standard epimoric approximation to the diaskhisma is the ratio 89:88 (=~19.56217479 cents), whose

2,3,5,11,89-monzo is ß3, 0, 0, ß1, 1>. Tuning treatises before c.1970 sometimes defined the diaskhisma as the 89:88

ratio without emphasis on the fact of its being an approximation, particularly in the German literature of c.1850-1950. A

good example is Helmholtz in On the Sensations of Tone.” For a further discussion, see Monzo, “Encyclopedia of

Microtonal Music Theory,” n. pag.

301 Ben Johnston, “Scalar Order as a Compositional Resource,” Perspectives of New Music 2.2 (Summer, 1964): 69-

73.

The Harmonic Consideration 209

Tabl

e 12

6. B

en J

ohns

ton’

s fif

ty-th

ree-

tone

just

into

natio

n sc

ale

Rela

tive

Pitc

h: A

4=44

0Hz

/ C4 (

mid

dle

C)=

261.

6255

654H

z

DEG

REE

NUM

BER

NO

TE

INTE

RVA

L RA

TIO

(F

RACT

ION

) RA

TIO

(D

ECIM

AL)

FR

EQUE

NCY

(H

ERTZ

) CE

NTS

TU

NIN

G

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÑÓ

ÑÔ

ÑÕ

ÑÖ

Ñ×

ÑØ

ÑÙ

ÒÐ

ÒÑ

ÒÒ

ÒÓ

ÒÔ

ÒÕ

ÒÖ

C C(

D$)

C!

C!(

D"

)

D"

C#(

D)

D D(

E$

D!

E")

E"

D#

E)

E E(

F"

E!

F)

F F(

G

$)

F!

uniso

n sy

nton

ic c

omm

a d

imin

ished

sec

ond

, or g

reat

die

sis

grav

e or

sm

all j

ust c

hrom

atic

sem

itone

, or m

inor

hal

f-ton

e Py

thag

orea

n ac

ute

or la

rge

limm

a ju

st d

iato

nic

sem

itone

, or m

ajor

hal

f-ton

e gr

eat l

imm

a, a

cute

or l

arge

hal

f-ton

e ac

ute

or la

rge

dou

ble

augm

ente

d o

ctav

e ju

st m

inor

tone

ju

st m

ajor

tone

(9t

h ha

rmon

ic)

acut

e or

larg

e to

ne

dim

inish

ed th

ird

augm

ente

d s

econ

d (7

5th

harm

onic

) Py

thag

orea

n m

inor

third

, or t

rihem

itone

ju

st m

inor

third

d

oubl

e au

gmen

ted

seco

nd

grav

e or

sm

all m

ajor

third

ju

st m

ajor

third

(5t

h ha

rmon

ic)

Pyth

agor

ean

maj

or th

ird, o

r dito

ne (8

1st h

arm

onic

) d

imin

ished

four

th

augm

ente

d th

ird

grav

e or

sm

all f

ourth

ju

st a

nd P

ytha

gore

an p

erfe

ct fo

urth

ac

ute

or la

rge

four

th

grav

e or

sm

all d

oubl

e d

imin

ished

fifth

gr

ave

or s

mal

l aug

men

ted

four

th

1/1

81/8

0 12

8/12

5 25

/24

135/

128

16/1

5 27

/25

1125

/102

4 10

/9

9/8

729/

640

144/

125

75/6

4 32

/27

6/5

625/

512

100/

81

5/4

81/6

4 32

/25

125/

96

320/

243

4/3

27/2

0 51

2/37

5 25

/18

1.00

0000

1.

0125

00

1.02

4000

1.

0416

67

1.05

4688

1.

0666

67

1.08

0000

1.

0986

33

1.11

1111

1.

1250

00

1.13

9063

1.

1520

00

1.17

1875

1.

1851

85

1.20

0000

1.

2207

03

1.23

4568

1.

2500

00

1.26

5625

1.

2800

00

1.30

2083

1.

3168

72

1.33

3333

1.

3500

00

1.36

5333

1.

3888

89

261.

626

264.

896

267.

905

272.

527

275.

933

279.

067

282.

556

287.

430

290.

695

294.

329

298.

008

301.

393

306.

592

310.

075

313.

951

319.

367

322.

995

327.

032

331.

120

334.

881

340.

658

344.

527

348.

834

353.

195

357.

206

363.

369

0.00

0 21

.506

41

.059

70

.672

92

.179

11

1.73

1 13

3.23

8 16

2.85

1 18

2.40

4 20

3.91

0 22

5.41

6 24

4.96

9 27

4.58

2 29

4.13

5 31

5.64

1 34

5.25

5 36

4.80

7 38

6.31

4 40

7.82

0 42

7.37

3 45

6.98

6 47

6.53

9 49

8.04

5 51

9.55

1 53

9.10

4 56

8.71

7

+00

+

22

+41

ß

29

ß08

+

12

+33

ß

37

ß18

+

04

+25

+

45

ß25

ß

06

+16

+

45

ß35

ß

14

+08

+

27

ß43

ß

23

ß02

+

20

+39

ß

31

210 The Harmonic Consideration

DEG

REE

NUM

BER

NO

TE

INTE

RVA

L RA

TIO

(F

RACT

ION

) RA

TIO

(D

ECIM

AL)

FR

EQUE

NCY

(H

ERTZ

) CE

NTS

TU

NIN

G

Ò×

ÒØ

ÒÙ

ÓÐ

ÓÑ

ÓÒ

ÓÓ

ÓÔ

ÓÕ

ÓÖ

Ó×

ÓØ

ÓÙ

ÔÐ

ÔÑ

ÔÒ

ÔÓ

ÔÔ

ÔÕ

ÔÖ

Ô×

ÔØ

ÔÙ

ÕÐ

ÕÑ

ÕÒ

ÕÓ ÐÑ

F!(

G")

G"

F#

(

G)

G G(

A$

G!

A")

A"

G#

A)

A A(

B$

)

A!

A!(

B"

)

B"

A#(

B)

B B(

C"

B!

C)

C

just

trito

ne, o

r aug

men

ted

four

th (

45th

har

mon

ic)

dim

inish

ed fi

fth

acut

e or

larg

e d

imin

ished

fifth

ac

ute

or la

rge

dou

ble

augm

ente

d fo

urth

gr

ave

or s

mal

l fift

h ju

st a

nd P

ytha

gore

an p

erfe

ct fi

fth (

3rd

har

mon

ic)

acut

e or

larg

e fif

th

dim

inish

ed s

ixth

au

gmen

ted

fifth

(25

th h

arm

onic

) Py

thag

orea

n m

inor

six

th

just

min

or s

ixth

d

oubl

e au

gmen

ted

fifth

gr

ave

or s

mal

l maj

or s

ixth

ju

st m

ajor

six

th

Pyth

agor

ean

maj

or s

ixth

(27t

h ha

rmon

ic)

dim

inish

ed s

even

th

augm

ente

d s

ixth

ac

ute

or la

rge

augm

ente

d s

ixth

Py

thag

orea

n m

inor

sev

enth

ac

ute

or la

rge

min

or s

even

th

acut

e or

larg

e d

oubl

e au

gmen

ted

six

th

grav

e or

sm

all m

ajor

sev

enth

ju

st d

iato

nic

maj

or s

even

th (

15th

har

mon

ic)

Pyth

agor

ean

maj

or s

even

th

dim

inish

ed o

ctav

e m

eant

one

augm

ente

d s

even

th (

125t

h ha

rmon

ic)

grav

e or

sm

all o

ctav

e

octa

ve

45/3

2 64

/45

36/2

5 37

5/25

6 40

/27

3/2

243/

160

192/

125

25/1

6 12

8/81

8/

5 62

5/38

4 40

0/24

3 5/

3 27

/16

128/

75

125/

72

225/

128

16/9

9/

5 18

75/1

024

50/2

7 15

/8

243/

128

48/2

5 12

5/64

16

0/81

2/1

1.40

6250

1.

4222

22

1.44

0000

1.

4648

44

1.48

1481

1.

5000

00

1.51

8750

1.

5360

00

1.56

2500

1.

5802

47

1.60

0000

1.

6276

04

1.64

6091

1.

6666

67

1.68

7500

1.

7066

67

1.73

6111

1.

7578

13

1.77

7778

1.

8000

00

1.83

1055

1.

8518

52

1.87

5000

1.

8984

38

1.92

0000

1.

9531

25

1.97

5309

2.00

0000

367.

911

372.

090

376.

741

383.

241

387.

593

392.

438

397.

344

401.

857

408.

790

413.

433

418.

601

425.

823

430.

659

436.

043

441.

493

446.

508

454.

211

459.

889

465.

112

470.

926

479.

051

484.

492

490.

548

496.

680

502.

321

510.

987

516.

791

52

3.25

1

590.

224

609.

776

631.

283

660.

896

680.

449

701.

955

723.

014

743.

014

772.

627

792.

180

813.

686

843.

300

862.

852

884.

359

905.

865

925.

418

955.

031

976.

537

996.

090

1017

.596

10

47.2

10

1066

.762

10

88.2

69

1109

.775

11

29.3

28

1158

.941

11

78.4

94

12

00.0

00

ß10

+

10

+31

ß

39

ß20

+

02

+23

+

43

ß27

ß

08

+14

+

43

ß37

ß

16

+06

+

25

ß45

ß

23

ß04

+

18

+47

ß

33

ß12

+

10

+29

ß

41

ß22

+00

The Harmonic Consideration 211

Harry Partch’s Forty-Three-Tone Just Intonation Scale Partch’s forty-three-tone just intonation scale, with its inclusion of eleven-limit intervals, represents a

harmonic expansion of Johnston’s 2, 3, 5-limit just intonation fifty-three-tone enharmonic scale. The

aesthetic rationale behind discontinuing beyond the seven-limit is illustrated by Partch within these

colourful remarks:

“The reasons why Monophony proceeds to the limit of 11 are basic and quite specific, as will be seen, but

the reason for resting at the limit of 11 is a purely personal and arbitrary one. When a hungry man has a

large table of aromatic and unusual viands spread before him he is unlikely to go tramping along the

seashore and in the woods for still another exotic fare. And however sceptical he is of the many warnings

regarding the unwholesomeness of his fare – like the ‘poison’ of the ‘love-apple’ tomato of a comparatively

few generations ago – he has no desire to provoke further alarums.”

According to Partch, the harmonic argument behind the abovementioned intonation system is based on

the premise that “the expansion of identities 1-3-5 through 7-9-11 provides a new and highly intriguing

triad, and immediately makes possible a wide variety in quality.” The identities 7-9-11 represent the

intervals 7/4, 9/8, and 11/8, and the ratio 7:9:11, which form an unusual variety of major triad consisting

of the septimal supermajor third (9/7, or 435.084 cents) and the undecimal augmented fifth (11/7, or

782.492 cents). Twenty unique triads, fifteen unique tetrads, as well as six unique pentads are now

made possible via the combination of the six identities of each individual otonality and utonality, and this

is excluding the further augmentation of harmonic resources effectuated via the inversion and extension

of simultaneous sonorities.

Ex. 93. Eleven-limit simultaneous sonorities i. The twenty eleven-limit triads

ii. The fifteen eleven-limit tetrads

212 The Harmonic Consideration

iii. The six eleven-limit pentads

With regards to the historical argument, Partch gives reference to Alexandrian astronomer, mathematician,

and geographer of the second century, Claudius Ptolemy (c.87-150).302

“In Ptolemy’s scales there is enough evidence to warrant the conclusion that his procedure was generally

governed by the principle of appropriating the smallest-number ratios permissible to the purpose of the

scale in question. In this light it is quite natural that he should have used all the ratios of the 11-limit as a

body.”

Partch’s forty-three-tone scale firstly adopts pairs of complementary five-limit intervals made available by

the first five partials of the harmonic series – the octave (2/1), unison (1/1), just perfect fifth (3/2), just

perfect fourth (4/3), just major third (5/4), and just minor third (6/5) – to then incorporate seven-limit and

eleven-limit intervallic ratios to construct a scale of twenty-nine degrees. To resolve some of the scalar

discontinuities of this unequal scale, the system is then expanded to include secondary ratios, which

result in a scale consisting of forty-three-tones. An example of one of these secondary ratios is 33/32

(undecimal comma, or 33rd harmonic), which is derived by calculating the 3/2 of 11/8 via the equation:

11/8 plus 3/2=(11Ï3):(8Ï2)=33/16; followed by 33/16 minus 2/1=(33Ï1):(16Ï2)=33/32. The final

set of intervals include four types of tritones: the undecimal superfourth (11/8), septimal subdiminished

fifth (7/5), and their complements; septimal tritone, or superaugmented fourth (10/7), and undecimal

subfifth (16/11); six types of thirds: the septimal subminor third (7/6), Pythagorean minor third (32/27),

just minor third (6/5), undecimal acute or large neutral third (11/9), just major third (5/4), septimal

supermajor third (9/7), and their complements; the septimal subminor sixth (14/9), just minor sixth (8/5),

undecimal grave or small neutral sixth (18/11), just major sixth (5/3), Pythagorean major sixth (27/16), and

septimal supermajor sixth (12/7); seven types of seconds: the septimal diatonic semitone (21/20), just

diatonic semitone (16/15), undecimal grave or small neutral second (12/11), undecimal acute or large

neutral second (11/10), just minor tone (10/9), just major tone (9/8), septimal supermajor second (8/7),

and their complements; the septimal subminor seventh (7/4), Pythagorean minor seventh (16/9), acute or

large minor seventh (9/5), undecimal grave or small neutral seventh (20/11), undecimal acute or large

neutral seventh (11/6), just diatonic major seventh (15/8), and septimal supermajor seventh (40/21).303

302 Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 123-35.

303 Bob Gilmore, “On Harry Partch’s Seventeen Lyrics by Li Po,” Perspectives of New Music 30.2 (Summer, 1992):

26-27.

The Harmonic Consideration 213

Tabl

e 12

7. H

arry

Par

tch’

s fo

rty-th

ree-

tone

just

into

natio

n sc

ale

Rela

tive

Pitc

h: A

4=44

0Hz

/ C4 (

mid

dle

C)=

261.

6255

654H

z

DEG

REE

NUM

BER

NO

TE

INTE

RVA

L RA

TIO

(FRA

CTIO

N)

RATI

O

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

TUN

ING

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÑÓ

ÑÔ

ÑÕ

ÑÖ

Ñ×

ÑØ

ÑÙ

ÒÐ

ÒÑ

C C (

C,

D ;

D")

D-

D E)

D )

D D *)

E ;

E ")

E "

E E)

E F K(

E *

F +(

F F (

F ,

uniso

n

synt

onic

com

ma

und

ecim

al c

omm

a (3

3rd

har

mon

ic)

sep

timal

chr

omat

ic s

emito

ne

just

dia

toni

c se

mito

ne, o

r maj

or h

alf-t

one

und

ecim

al g

rave

or s

mal

l neu

tral s

econ

d

und

ecim

al a

cute

or l

arge

neu

tral s

econ

d

just

min

or to

ne

just

maj

or to

ne (

9th

harm

onic

)

sep

timal

sup

erm

ajor

sec

ond

sep

timal

sub

min

or th

ird

Pyth

agor

ean

min

or th

ird, o

r trih

emito

ne

just

min

or th

ird

und

ecim

al a

cute

or l

arge

neu

tral t

hird

just

maj

or th

ird (

5th

harm

onic

)

und

ecim

al d

imin

ished

four

th

sep

timal

sup

erm

ajor

third

sep

timal

sub

four

th (

21st

har

mon

ic)

just

and

Pyt

hago

rean

per

fect

four

th

acut

e or

larg

e fo

urth

und

ecim

al s

uper

four

th (

11th

har

mon

ic)

1/1

81/8

0

33/3

2

21/2

0

16/1

5

12/1

1

11/1

0

10/9

9/8

8/7

7/6

32/2

7

6/5

11/9

5/4

14/1

1

9/7

21/1

6

4/3

27/2

0

11/8

1.00

0000

1.01

2500

1.03

1250

1.05

0000

1.06

6667

1.09

0909

1.10

0000

1.11

1111

1.12

5000

1.14

2857

1.16

6667

1.18

5185

1.20

0000

1.22

2222

1.25

0000

1.27

2727

1.28

5714

1.31

2500

1.33

3333

1.35

0000

1.37

5000

261.

626

264.

896

269.

801

274.

707

279.

067

285.

410

287.

788

290.

695

294.

329

299.

001

305.

230u

310.

075

313.

951

319.

765

327.

032

332.

978

336.

376

343.

384

348.

834

353.

195

359.

735

0.00

0

21.5

06

53.2

73

84.4

67

111.

731

150.

637

165.

004

182.

404

203.

910

231.

174

266.

871

294.

135

315.

641

347.

408

386.

314

417.

508

435.

084

470.

781

498.

045

519.

551

551.

318

+00

+22

ß47

ß16

+12

ß49

ß35

ß18

+04

+31

ß33

ß06

+16

+47

ß14

+18

+35

ß29

ß02

+20

ß49

214 The Harmonic Consideration

DEG

REE

NUM

BER

NO

TE

INTE

RVA

L RA

TIO

(FRA

CTIO

N)

RATI

O

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

TUN

ING

ÒÒ

ÒÓ

ÒÔ

ÒÕ

ÒÖ

Ò×

ÒØ

ÒÙ

ÓÐ

ÓÑ

ÓÒ

ÓÓ

ÓÔ

ÓÕ

ÓÖ

Ó×

ÓØ

ÓÙ

ÔÐ

ÔÑ

ÔÒ

ÔÓ ÐÑ

G;

F :

G-

G)

G

G*)

A;

GJ)

A"

A-(

A

A(

A*

B ;

B ")

B "

B -

B E)

B B *)

C -

C ) C

sep

timal

sub

dim

inish

ed fi

fth

sep

timal

trito

ne, o

r sup

erau

gmen

ted

four

th

und

ecim

al s

ubfif

th

grav

e or

sm

all f

ifth

just

and

Pyt

hago

rean

per

fect

fifth

(3r

d h

arm

onic

)

sep

timal

sup

erfif

th

sep

timal

sub

min

or s

ixth

und

ecim

al a

ugm

ente

d fi

fth

just

min

or s

ixth

und

ecim

al g

rave

or s

mal

l neu

tral s

ixth

just

maj

or s

ixth

Pyth

agor

ean

maj

or s

ixth

(27t

h ha

rmon

ic)

sep

timal

sup

erm

ajor

six

th

sep

timal

sub

min

or s

even

th (

7th

harm

onic

)

Pyth

agor

ean

min

or s

even

th

acut

e or

larg

e m

inor

sev

enth

und

ecim

al g

rave

or s

mal

l neu

tral s

even

th

und

ecim

al a

cute

or l

arge

neu

tral s

even

th

just

dia

toni

c m

ajor

sev

enth

(15

th h

arm

onic

)

sep

timal

sup

erm

ajor

sev

enth

und

ecim

al s

ubd

imin

ished

oct

ave

grav

e or

sm

all o

ctav

e

octa

ve

7/5

10/7

16/1

1

40/2

7

3/2

32/2

1

14/9

11/7

8/5

18/1

1

5/3

27/1

6

12/7

7/4

16/9

9/5

20/1

1

11/6

15/8

40/2

1

64/3

3

160/

81

2/1

1.40

0000

1.42

8571

1.45

4545

1.48

1481

1.50

0000

1.52

3810

1.55

5556

1.57

1429

1.60

0000

1.63

6364

1.66

6667

1.68

7500

1.71

4286

1.75

0000

1.77

7778

1.80

0000

1.81

8182

1.83

3333

1.87

5000

1.90

4762

1.93

9394

1.97

5309

2.00

0000

366.

276

373.

751

380.

546

387.

593

392.

438

398.

668

406.

973

411.

126

418.

601

428.

115

436.

043

441.

493

448.

501

457.

845

465.

112

470.

926

475.

683

479.

647

490.

548

498.

334

507.

395

516.

791

523.

251

582.

512

617.

488

648.

682

680.

449

701.

955

729.

219

764.

916

782.

492

813.

686

852.

592

884.

359

905.

865

933.

129

968.

826

996.

090

1017

.596

1034

.996

1049

.363

1088

.269

1115

.533

1146

.727

1178

.494

1200

.000

ß17

+17

+49

ß20

+02

+29

ß35

ß18

+14

ß47

ß16

+06

+33

ß31

ß04

+18

+35

+49

ß12

+16

+47

ß22

+00

The Harmonic Consideration 215

Adriaan Daniël Fokker’s Thirty-One-Tone Equally-Tempered Division of the Octave According to Wolf, Dutch physicist, music theorist, and composer Adriaan Daniël Fokker (1987-1972)304

“appears to be the first theorist to represent a 7-limit tuning system graphically with three implied axes or

dimensions of tonal space, the horizontal axis is assigned to fifths (3/2s), the vertical axis to major thirds

(5/4s), and an oblique axis, implying a third dimension, to tones generated by the 7/4 relationship.”

Fokker, inspired by the musical writings of Dutch mathematician, astronomer, and physicist Christian

Huygens (1629-95), became an ardent advocate of the tonal resources of seven-limit just intonation, and

their “practical realization” within the domain of thirty-one-tone equal temperament.305 Huygens, via the

recognition of the significance of the interval of the great diesis (128/125), or one-fifth of a whole-tone

( 589 ), which is calculated via the amount the octave (2/1) exceeds three consecutive just major thirds

(2/1÷125/64[5/4Ï5/4Ï5/4]=128/125, or 1200ß1158.941=41.059 cents), developed a theoretical

basis for the “octave partitioned into thirty-one steps.” Fokker’s extensive research into thirty-one-tone

equal temperament appears in his 1966 publication of New Music with 31 Notes.306

Notably, the thirty-one-tone equally-tempered division of the octave ( 312 ) produces the just

major third (5/4) and septimal subminor seventh (7/4), with a falsity of +0.783 and ß1.084 cents on

each count, which is a “faithful rendering” of the former, while a “nearly exact reproduction” of the latter.

The untrigesimal, or ‘tricesimoprimal’ equal temperament is therefore capable of approximating the

septimal dominant seventh tetrad (4:5:6:7) with the intervallic measurements of 387.097, 696.774, and

967.742 cents, which are extremely close to the ‘true’ seven-limit equivalents (386.314, 701.955, and

968.826 cents).307 “Thirty-one equal has an honourable history,” notes Rapoport, and “partly because it

has excellent approximations to the intervals represented by harmonics five and seven, and is the closed

regular expansion of quarter-comma meantone temperament.”308

304 “Fokker was born to Dutch parents in Indonesia. He received his doctorate in physics at Leiden in 1913,

pursued advanced work with Rutherford and Einstein (with whom he seems to have collaborated in research and

publication), and became a distinguished theoretical physicist and professor, with important publications on relativity

theory, atomic physics, acoustics, electricity, and magnetism.” For a further discussion, see Douglas Leedy, “Selected

Musical Compositions (1948-1972),” Notes 46.1 (Sep., 1989): 224. 305 Wolf, “Alternative Tunings, Alternative Tonalities,” Contemporary Music Review 8. 306 “New Music with 31 Notes appeared originally in German in 1966. It is a distillation of the ideas which have

punctuated his theoretical writings and make a fine introduction to the works of this seminal figure. The book is in two

parts, the first narrating the history of Fokker’s involvement in 31-tone temperament and the early fruits of this interest. The

longer second part is theoretical and deals with the aspects of temperament which have especially preoccupied Fokker.”

For a further discussion, see Joel Mandelbaum, “Toward the Expansion of Our Concepts of Intonation,” Perspectives of

New Music 13.1 (Autumn-Winter, 1974): 220.

307 A. D. Fokker, “Equal Temperament and the Thirty-One-Keyed Organ,” The Scientific Monthly 81.4 (Oct., 1955):

162-63.

308 Paul Rapoport, “Towards the Infinite Expansion of Tonal Resources,” Tempo 144 (Mar., 1983): 9.

216 The Harmonic Consideration

Tabl

e 12

8. A

dria

an D

anië

l Fok

ker’s

thirt

y-on

e-to

ne e

qua

lly-te

mp

ered

div

ision

of t

he o

ctav

e Re

lativ

e Pi

tch:

A4=

440H

z / C

4 (m

iddl

e C)

=26

1.62

5565

4Hz

DE

GRE

E N

UMBE

R N

OTE

INTE

RVA

LFA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY(H

ERTZ

) CE

NTS

TUN

ING

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÑÓ

ÑÔ

ÑÕ

ÑÖ

Ñ×

ÑØ

ÑÙ

ÒÐ

ÒÑ

ÒÒ

ÒÓ

ÒÔ

ÒÕ

ÒÖ

Ò×

ÒØ

ÒÙ

ÓÐ

ÓÑ ÐÑ

CCà

/ D$

C !

/ Dé

C è

/ D"

C #

/ Dá

D

D à /

E$

D ! /

Eé

D è /

E"

D # /

Eá

E E à

/ F"

E !

/ Fá

F

F à /

G$

F !

/ G

é F è

/ G

" F #

/ G

á G

G

à / A

$

G!

/ Aé

Gè

/ A"

G#

/ Aá

A

A

à / B

$

A!

/ Bé

Aè

/ B"

A#

/ Bá

B

B à /

C"

B ! /

Cá

C

uniso

n31

-et s

uper

octa

ve, o

r dim

inish

ed s

econ

d

31-e

t aug

men

ted

oct

ave,

or s

ubm

inor

sec

ond

31-e

t sup

erau

gmen

ted

oct

ave,

or m

inor

sec

ond

31

-et d

oubl

e au

gmen

ted

oct

ave,

or n

eutra

l sec

ond

31

-et m

ajor

tone

31

-et s

uper

maj

or s

econ

d, o

r dim

inish

ed th

ird

31-e

t aug

men

ted

sec

ond

, or s

ubm

inor

third

31

-et s

uper

augm

ente

d s

econ

d, o

r min

or th

ird

31-e

t dou

ble

augm

ente

d s

econ

d, o

r neu

tral t

hird

31

-et m

ajor

third

31

-et s

uper

maj

or th

ird, o

r dim

inish

ed fo

urth

31

-et a

ugm

ente

d th

ird, o

r sub

four

th

31-e

t per

fect

four

th

31-e

t sup

erfo

urth

, or d

imin

ished

fifth

31

-et a

ugm

ente

d fo

urth

, or s

ubd

imin

ished

fifth

31

-et s

uper

augm

ente

d fo

urth

, or d

imin

ished

fifth

31

-et d

oubl

e au

gmen

ted

four

th, o

r sub

fifth

31

-et p

erfe

ct fi

fth

31-e

t sup

erfif

th, o

r dim

inish

ed s

ixth

31

-et a

ugm

ente

d fi

fth, o

r sub

min

or s

ixth

31

-et s

uper

augm

ente

d fi

fth, o

r min

or s

ixth

31

-et d

oubl

e au

gmen

ted

fifth

, or n

eutra

l six

th

31-e

t maj

or s

ixth

31

-et s

uper

maj

or s

ixth

, or d

imin

ished

sev

enth

31

-et a

ugm

ente

d s

ixth

, or s

ubm

inor

sev

enth

31

-et s

uper

augm

ente

d s

ixth

, or m

inor

sev

enth

31

-et d

oubl

e au

gmen

ted

six

th, o

r neu

tral s

even

th

31-e

t maj

or s

even

th

31-e

t sup

erm

ajor

sev

enth

, or d

imin

ished

oct

ave

31-e

t aug

men

ted

sev

enth

, sub

octa

ve

oc

tave

1/1

312

231

)2

(

331

)2

(

431

)2

(

531

)2

(

631

)2

(

731

)2

(

831

)2

(

931

)2

(

1031

)2

(

1131

)2

(

1231

)2

(

1331

)2

(

1431

)2

(

1531

)2

(

1631

)2

(

1731

)2

(

1831

)2

(

1931

)2

(

2031

)2

(

2131

)2

(

2231

)2

(

2331

)2

(

2431

)2

(

2531

)2

(

2631

)2

(

2731

)2

(

2831

)2

(

2931

)2

(

3031

)2

(

2/

1

1.00

0000

1.02

2611

1.

0457

34

1.06

9380

1.

0935

60

1.11

8287

1.

1435

73

1.16

9431

1.

1958

73

1.22

2914

1.

2505

66

1.27

8843

1.

3077

59

1.33

7329

1.

3675

68

1.39

8491

1.

4301

13

1.46

2450

1.

4955

18

1.52

9334

1.

5639

14

1.59

9276

1.

6354

38

1.67

2418

1.

7102

34

1.74

8905

1.

7884

50

1.82

8889

1.

8702

43

1.91

2532

1.

9557

77

2.

0000

00

261.

626

267.

541

273.

591

279.

777

286.

103

292.

572

299.

188

305.

953

312.

871

319.

945

327.

180

334.

578

342.

143

349.

880

357.

791

365.

881

374.

154

382.

614

391.

266

400.

113

409.

160

418.

412

427.

872

437.

547

447.

441

457.

558

467.

904

478.

484

489.

303

500.

367

511.

681

52

3.25

1

0.00

038

.710

77

.419

11

6.12

9 15

4.83

9 19

3.54

8 23

2.25

8 27

0.96

8 30

9.67

7 34

8.38

7 38

7.09

7 42

5.80

6 46

4.51

6 50

3.22

6 54

1.93

5 58

0.64

5 61

9.35

5 65

8.06

5 69

6.77

4 73

5.48

4 77

4.19

4 81

2.90

3 85

1.61

3 89

0.32

3 92

9.03

2 96

7.74

2 10

06.4

52

1045

.161

10

83.8

71

1122

.581

11

61.2

90

12

00.0

00

+00

+

39

ß23

+

16

ß45

ß

06

+32

ß

29

+10

ß

52

ß13

+

26

ß35

+

03

+42

ß

19

+19

ß

42

ß03

+

35

ß26

+

13

ß48

ß

10

+29

ß

32

+06

ß

55

ß16

+

23

ß39

+00

The Harmonic Consideration 217

The notation of fifth tones, or the equal untrigesimal equal diesis ( 312 , or 38.710 cents) of thirty-one

equal temperament, was devised by Fokker, together with Dutch composer Henk Badings (1907-87),309

and is represented with the following nine symbols: semi-sharp (à), sharp (!), sesqui-sharp (è), double

sharp (#), semi-flat (á), flat ("), sesqui-flat (é), double flat ($), and natural ('). The selection of symbols

also represents the recommendations made by the International Musicological Society, following their

1967 meeting in Ljubljana, Slovenia, where it was agreed upon that the “best microtonal symbology”

should incorporate the “five standard accidental signs,” along with the semi-sharp and sesqui-sharp

symbols, as well as the semi-flat and sesqui-flat symbols introduced by Italian composer and violinist

Guiseppe Tartini (1692-1770) in 1754.310

La Homa Kanto for Harmonically Tuned Synthesizer Quartet La Homa Kanto, or ‘The Human Song’ in Esperanto,311 is a dedication to the late American composer Lou

Harrison (1917-2003),312 and its pitch material has been derived directly from Harrison’s five-tone scales,

presented in Lou Harrison’s Music Primer: Various Items About Music to 1970. Included are the first five

in the series, with the first (the diatonic or major pentatonic scale) acknowledged by Harrison as the

“prime pentatonic,” and “practically the Human Song.” According to Harrison, “These first five are the

most widespread, the core, the principal modes of Human Music. They also constitute the bone-work,

the firmest compositional basis for seven-tone music.”313

Ex. 94. Major Pentatonic Scale “The Human Song”

309 Richard Orton, “The 31-Note Organ,” The Musical Times 107.1478 (Apr., 1966): 342.

310 Read, 20th-Century Microtonal Notation 19-20. 311 Esperanto is “an artificial language invented as a means of international communication.” The origin of the name

derived from “Dr. Esperanto, a pen name of the inventor.” For a further discussion, see Catherine Soanes, ed., Oxford

Dictionary of Current English, 3rd ed. (Lodon: Oxford U. Press, 2001) 304. 312 “Lou Harrison, who celebrated his eightieth birthday in 1997, has often been cited as one of America’s most

original and influential composers. In addition to his prolific musical output, Harrison is also a skilled painter, calligrapher,

essayist, critic, poet, and instrument-builder. During his long and varied career, he has explored dance, Asian music,

tuning systems, and universal languages, and has actively championed political causes ranging from pacifism to gay rights.”

For a further discussion, see Miller, and Lieberman, Lou Harrison: Composing a World n.pag.

313 Lou Harrison, Lou Harrison’s Music Primer: Various Items About Music to 1970 (New York: C. F. Peters, 1971) 27-

29.

218 The Harmonic Consideration

In his music primer, Harrison also states that the “fullest musical bounty of any tradition is the mode,” and

expresses the following sentiments about five-tone modes:

“Time’s endless flowering of tone and tune here streams across to each; embodied matrix song; hearts gift

of ancestry and dreams. Behind the singer’s song stands quietly (or dancing) the strange remembered

body of the mode. Or is it fashioned piecewise (that body) from the rhymes and purlings of the tune?

Sure, scales are not. These sit, emotive, on systematic right. Perhaps modes meld from melodies

themselves.”314

Modes one, two, and five may be considered to be anhemitonic pentatonic forms (scales incorporating

half steps, or semitones), while three and four, hemitonic pentatonic forms (scales not incorporating half

steps, or semitones).315 Harrison’s five pentatonic scales include the diatonic or major pentatonic, minor

pentatonic (or fifth mode major pentatonic), Japanese hirajoshi, fifth mode Indonesian pélog, and

Indonesian sléndro (or second mode major pentatonic). The consideration of their complements

presents the third mode major pentatonic, fourth mode major pentatonic, third mode Indonesian pélog,

fourth mode Japanese hirajoshi, and Indonesian sléndro (or second mode major pentatonic).316

Ex. 95. Lou Harrison’s five pentatonic scales i. Lou Harrison pentatonic #2

ii. Lou Harrison pentatonic #2

314 Harrison, Lou Harrison’s Music Primer: Various Items About Music to 1970 27.

315 J. H. Kwabena Nketia, The Music of Africa (New York: W. W. Norton & Co., 1974) 118. 316 Persichetti, Twentieth-Century Harmony: Creative Aspects and Practice 50-51.

The Harmonic Consideration 219

ii. Lou Harrison pentatonic #3

iv. Lou Harrison pentatonic #4

v. Lou Harrison pentatonic #5

220 The Harmonic Consideration

The Harpsichord Sample The ‘1967 William Dowd French Double Harpsichord’ samples (recorded in a recital room space by

sound designer and programmer Peter Grech at the School of Music, Victorian College of the Arts,

Melbourne, Australia, utilizing an Audio-Technica AT4050/CM5 condenser microphone [with switchable

cardioid, omnidirectional, or figure-8 operation, and a frequency response from 20Hz-20kHz] and an

Akai S3000XL Midi Stereo Digital Sampler) include two sets of twenty (five-octave span) samples

(encompassing the upper and lower manuals of the harpsichord), as well as one corresponding set of

keyboard release clicks. All these sounds multi-sampled on an Akai S3000XL Midi Stereo Digital Sampler

– tuned firstly to standard A=440Hz twelve-tone equal temperament, and then modified within nineteen

patches collectively adhering to the tuning matrixes required to represent the ten unique systems of just

intonation utilized in the composition. In view of the superior aesthetic qualities of the upper manual,

and for the sake of harmonic clarity, the lower manual and keyboard release clicks are omitted from the

final 63-key ‘1967 William Dowd French Double Harpsichord’ sample. The technical requirements for the

realization of the live performance of the work is an Akai S3000XL Midi Stereo Digital Sampler, together

with four 61-key Midi Keyboards.

Fig. 7. Alex Pertout, 1967 William Dowd French Double Harpsichord.317

317 1967 William Dowd French Double Harpsichord, personal photograph of Alex Pertout, 3 Feb. 2007.

The Harmonic Consideration 221

Ben Johnston’s System of Notation

The system of notation utilized in the score is based on Johnston’s ‘extended just intonation’. It contains

twenty-three unique symbols, which may be combined in any manner – five identical to conventional

sharp (Ú), flat (Û), double sharp (#), double flat ($), and natural (Ö) accidentals (raising, lowering or

neutralizing a tone by 25/24 and 625/576, or 70.672 and 141.345 cents). Additional symbols include

(() and ()), raising or lowering a tone by 81/80 (one syntonic comma), or 21.506 cents, as well as the

following utonal and otonal sets of symbols for each partial chroma up to the thirty-first harmonic: (+) and

(*), lowering or raising a tone by 36/35 (one septimal comma), or 48.770 cents; (,) and (-), raising or

lowering a tone by 33/32 (one undecimal comma), or 53.273 cents; (.) and (/), raising or lowering a

tone by 65/64 (one tridecimal comma), or 26.841 cents; (0) and (1), raising or lowering a tone by

51/50 (one septendecimal comma), or 34.283 cents; (2) and (3), lowering or raising a tone by 96/95

(one nonadecimal comma), or 18.128 cents; (4) and (5), raising or lowering a tone by 46/45 (one

trivigesimal comma), or 38.051 cents; (6) and (7), raising or lowering a tone by 145/144 (one

nonavigesimal comma), or 11.981 cents; and finally, (8) and (9), raising or lowering a tone by 31/30

(one untrigesimal comma), or 56.767 cents.318 Johnston “infinitely expandable” system of notation for

extended just intonation categorizes commas above the seventh harmonic under the nomenclature of

‘chromas’. “These accidentals are used in combination for the more complex ratios,” notes Bob Gilmore,

with reference to Johnston’s not uncommon use of “three such symbols applied to one note.”319

Table 129. Ben Johnston’s notational symbols for just intonation commas and chromas through thirty-one

RAISE LOWER INTERVAL RATIO

(FRACTION)

CENTS AMOUNT BY

WHICH…

EXCEEDS

Ú ( * , . 0 3 4 6 8

Û ) + - / 1 2 5 7 9

chromatic semitone

syntonic comma

septimal comma

undecimal comma

tridecimal comma

septendecimal comma

nonadecimal comma

trivigesimal comma

nonavigesimal comma

untrigesimal comma

25/24

81/80

36/35

33/32

65/64

51/50

96/95

46/45

145/144

31/30

70.672

21.506

48.770

53.273

26.841

34.283

18.128

38.051

11.981

56.767

5/4

9/8

9/5

11/8

13/8

17/16

6/5

23/16

29/16

31/16

6/5

10/9

7/4

4/3

8/5

25/24

19/16

45/32

9/5

15/8

318 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters” 109-15.

319 Bob Gilmore, “Changing the Metaphor: Ratio Models of Musical Pitch in the Work of Harry Partch, Ben Johnston,

and James Tenney,” Perspectives of New Music 33.1/2 (Winter-Summer, 1995): 480.

222 The Harmonic Consideration

Compositional Strategy

Harrison offers the following commentary about the microtonal compositional process, and the utilization

of just intonation principles:

“After only a brief study of intervals it becomes clear that there are two ways of composing with them: 1)

arranging them into a fixed mode, or gamut, and then composing within that structure. This is Strict Style,

and is the vastly predominant world method. However, another way is possible – 2) to freely assemble, or

compose with whatever intervals one feels that he needs as he goes along. This is Free Style, and I used

this method first in my Simfony in Free Style.”320

The compositional strategy for the work involves the creation of two complimentary sets of melodic

material, consisting of what is essentially a four-bar melodic sentence based on Harrison’s first pentatonic

(or major pentatonic), with an additional four-bar variant based on its complement (or third mode major

pentatonic). In some aspects, this second sentence could be considered an almost comparable

inversion, although it does not represent an ‘authentic’ inversion in the serial sense of the word, but rather

an inversion of contour utilizing the inverted pitch material of the original major pentatonic. The third bar

of both versions represents the only literal inversion of intervals. A further observation reveals the

respective major and minor tonalities of the two four-bar melodic sentences.

Ex. 96. Two four-bar melodic sentences i. Four-bar melodic sentence

ii. Complimentary sentence

The work also features ten distinct tuning modulations: three-limit, five-limit, seven-limit, eleven-limit,

thirteen-limit, seventeen-limit, nineteen-limit, twenty-three-limit, twenty-nine-limit, and thirty-one-limit just

intonation systems, based on the third, fifth, seventh, eleventh, thirteenth, seventeenth, nineteenth,

twenty-third, twenty-ninth, and thirty-first partials of the harmonic series – each system adding its own

microtonal nuances to the recurring melodic material, which is further transformed via the introduction of

alternative scalar material, as well as via harmonic development pertinent to each individual just intonation

320 Harrison, Lou Harrison’s Music Primer: Various Items About Music to 1970 6.

The Harmonic Consideration 223

system. The structural framework of the composition is directly related to the primary motive of the four-

bar melodic sentence. In five-limit terms, the primary motive denotes the following intervals: 5/4, 3/2, and

9/8, and 5/4, and 3/2; and is expressed in the score as section one (rehearsal letters A and B –

incorporating Lou Harrison Pentatonic #1, and both three-limit and five-limit just intonation), which is

designated a 5/4 metrical structure; section 2 (rehearsal letters C and D – incorporating Lou Harrison

Pentatonic #2, and both seven-limit and eleven-limit just intonation), a 3/2 metrical structure; section 3

(rehearsal letters E and F – incorporating Lou Harrison Pentatonic #3, and both thirteen-limit and

seventeen-limit just intonation), a 9/8 metrical structure; section 4 (rehearsal letters G and H –

incorporating Lou Harrison Pentatonic #4, and both nineteen-limit and twenty-three-limit just intonation),

a 5/4 metrical structure; and section 5 (rehearsal letters I and J – incorporating Lou Harrison Pentatonic

#5, and both twenty-nine-limit and thirty-one-limit just intonation), a 12/8 metrical structure (equal to

3/2). A sequential series of metrical structures is therefore utilized in the work to reflect the primary

motive and its frequency ratios of 5/4, 3/2, 9/8, 5/4, and 3/2.

Composing With Melodicles

According to Harrison, composing with melodicles, or neumes “in some form is the oldest known

method of musical composition, probably deriving from Mesopotamia and Egypt. One makes a mosaic,

so to speak.” Acknowledging Henry Cowell as the source for this technique, Harrison proposes that

beginning with a selection of melodicles, the composer may: combine melodicles – essentially motivic

material – to form phrases; diatonically transpose inside, or chromatically transpose outside of a specific

mode; invert, retrograde, or retrograde-invert melodicles; while maintaining pitch integrity, alter rhythmic

design; as well as individually or collectively displace octaves.321 The abovementioned is no doubt

representative of a series of elementary suggestions later subjected to considerable development in the

compositional process. The technique is adopted in La Homa Kanto; theorized utilizing the first bar of

the work’s principal melodic sentence (a two-note motive), and remodelled within the following three

categories of motivic manipulation: melodic transformation of motive, rhythmic transformation of motive,

and harmonic transformation of motive.

Ex. 97. Original, complement, retrograde, complement retrograde of motive

321 Harrison, Lou Harrison’s Music Primer: Various Items About Music to 1970 1.

224 The Harmonic Consideration

The following series of examples represent a selection of the first category of motivic manipulation, or the

melodic transformation of motive.

Ex. 98. Melodic transformation of motive i. Motivic diatonic expansion (start)

ii. Motivic diatonic expansion (middle)

iii. Motivic diatonic expansion (end)

iv. Motivic diatonic expansion (upbeat)

v. Motivic chromatic expansion (start)

vi. Motivic chromatic expansion (middle)

vii. Motivic chromatic expansion (end)

The Harmonic Consideration 225

viii. Motivic chromatic expansion (upbeat)

ix. Motivic exclusion

x. Intervallic inclusion

xi. Motivic diatonic ornamentation

xii. Motivic chromatic ornamentation

xiii. Intervallic augmentation

xiv. Intervallic diminution

xv. Intonational reinterpretation

226 The Harmonic Consideration

xvi. Intervallic saturation

xvii. Intervallic redirection

xviii. Intervallic reordering

xix. Diatonic transposition

xx. Chromatic transposition

xxi. Contour transposition

xxii. Conjunct octave displacement

xxiii. Disjunct octave displacement

The Harmonic Consideration 227

The following series of examples represent a selection of the second category of motivic manipulation, or

the rhythmic transformation of motive.

Ex. 99. Rhythmic transformation of motive i. Pitch recapitulation

ii. Motivic recapitulation

iii. Motivic ostinato

iv. Rhythmic recapitulation

v. Rhythmic augmentation

vi. Rhythmic diminution

vii. Irregular rhythmic augmentation

228 The Harmonic Consideration

viii. Irregular rhythmic diminution

ix. Conjunct rhythmic displacement

x. Disjunct rhythmic displacement

xi. Rhythmic serialism

xii. Polyrhythmic representation

xiii. Time signature change

The following series of examples represent a selection of the third category of motivic manipulation, or

the harmonic transformation of motive.

Ex. 100. Harmonic transformation of motive i. Diatonic harmonic extension

The Harmonic Consideration 229

ii. Chromatic harmonic extension

iii. Two-part canon

iv. Two-part rhythmically displaced canon

v. Two-part inverted canon

vi. Two-part complementary canon

vii. Three-part canon

viii. Double canon

230 The Harmonic Consideration

ix. Augmentation canon

x. Diminution canon

Three-Limit Just Intonation

The primary interval of three-limit just intonation is the third harmonic, or just perfect fifth, which is

represented by the frequency ratio 3/2, and in relation to C (1/1), notated simply as G. The complement,

4/3, or just perfect fourth, is notated simply as F. The notational symbols introduced in three-limit just

intonation ([(] and [)], raising or lowering a tone by 81/80 [one syntonic comma], or 21.506 cents) are

in reference to the amount a series of four fifths minus a major third exceeds the unison

(3/2Ï3/2Ï3/2Ï3/2÷5/4=81/80).322

The essential three-limit pitch material is arrived at via a one-dimensional process, and the

generation of an ascending series of fifths from the fundamental C (1/1) to produce the six ‘otonal’

pitches: G, D, A(, E(, B(, and F!(( (3/2, 9/8, 27/16, 81/64, 243/128, and 729/512); while a descending

series to produce the other five ‘utonal’ pitches: F, B"), E"), A"), D")), and G")) (4/3, 16/9, 32/27,

128/81, 256/243, and 1024/729). The pitch material is then sequentially arranged as:

C D")) D E") E( F F!(( G")) G A") A( B") B

11

243256

89

2732

6481

34

512729

729

102423

81

128

1627

9

16

2140

The series of intervals presents the Pythagorean limma (256/243), just major tone (9/8), Pythagorean minor

third (32/27), Pythagorean major third (81/64), just perfect fourth (4/3), Pythagorean tritone, or

augmented fourth (729/512), Pythagorean diminished fifth (1024/729), just perfect fifth (3/2),

Pythagorean minor sixth (128/81), Pythagorean major sixth (27/16), Pythagorean minor seventh (16/9),

and Pythagorean major seventh (243/128). The following table depicts the essential tonal resources of

three-limit just intonation, indicating degree, notation, interval, ratio, and cents.

322 Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 452-53.

The Harmonic Consideration 231

Table 130. Essential three-limit pitch material

DEGREE

NUMBER

NOTE INTERVAL RATIO

(FRACTION)

RATIO

(DECIMAL)

CENTS

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÐÑ

C

D"))

D

E")

E(

F

F!((

G

A")

A(

B")

B(

C

unison

Pythagorean limma

just major tone (9th harmonic)

Pythagorean minor third, or trihemitone

Pythagorean major third, or ditone (81st harmonic)

just and Pythagorean perfect fourth

Pythagorean tritone, or augmented fourth

just and Pythagorean perfect fifth (3rd harmonic)

Pythagorean minor sixth

Pythagorean major sixth (27th harmonic)

Pythagorean minor seventh

Pythagorean major seventh

octave

1/1

256/243

9/8

32/27

81/64

4/3

729/512

3/2

128/81

27/16

16/9

243/128

2/1

1.000000

1.053498

1.125000

1.185185

1.265625

1.333333

1.423828

1.500000

1.580247

1.687500

1.777778

1.898438

2.000000

0.000

90.225

203.910

294.135

407.820

498.045

611.730

701.955

792.180

905.865

996.090

1109.775

1200.000

The adaptation of Harrison’s pentatonic #1 to three-limit intonation presents a collection of pitches that

include: C, D, E(, G, and A( (1/1, 9/8, 81/64, 3/2, and 27/16); as well as the complements: C, B"), A"),

F, and E") (2/1, 16/9, 128/81, 4/3, and 32/27). It must be noted that auxiliary pitches will be added to

the original pentatonic forms in order to beget a heptatonic scalar source.

Ex. 101. Lou Harrison pentatonic #1 (three-limit intonation)

232 The Harmonic Consideration

The following tuning matrix contains the data required to represent the system of intonation chromatically

within the program memory of the Akai S3000XL.

Table 131. 3-limit tuning matrix no. 1 ‘key of C’ (Program 01)

NOTE C D")) D E") E( F – – G A") A( B") B(

KEY C C! D D! E F F! G G! A A! B

CENTS +00 ß10 +04 ß06 +08 ß02 +00 +02 ß08 +06 ß04 +10

The first four bars of the composition serve to introduce the four-bar melodic sentence in three-limit

intonation, and what follows is essentially a two-part inverted canon highlighting the complement or

inversion of the principal melodic material. The end of the second system presents a D"("5/omit 3)

simultaneous sonority within a 7-1 suspension in the bass (the pitches D")), D")), and G, or the ratio

243:512:729), while the following harmony delineates D"("5)/F (the pitches F, D")) and G, or the ratio

324:512:729). In the latter example, D")) and G represents the interval of a Pythagorean tritone

(729/512), and F and D")), a Pythagorean minor sixth (128/81, or 792.180 cents). The third system now

introduces the four-bar variant based on the complement pentatonic scale. A hint of the C fundamental

highlights two characteristic intervals of three-limit just intonation: B"), or Pythagorean minor seventh

(16/9, or 996.090 cents), and A"), or Pythagorean minor sixth (128/81). Another two-part inverted

canon in the fourth system produces the succession: E") and B"), A") and D, and A") and E(, which

present the just perfect fifth (3/2), Pythagorean diminished fifth (1024/729, or 588.270 cents), and

Pythagorean diminished fourth (8192/6561, or 384.360 cents).

Another interesting simultaneous sonority presented in the fourth system is one proposed by

pitches E") and G, which pronounce the Pythagorean major third, or ditone (81/64, or 407.820 cents).

The nomenclature (ditone) is in direct reference to the resulting interval derived from the summation

process of two just major tones, or 9/8×9/8=81/64. It is important to note that whilst the three-limit

intonation system renders consonant just perfect fourths and fifths, as well as the just major tone (9/8), its

one-dimensional process of juxtaposed ascending fifths (series of 3/2s) and descending fourths (series

of 4/3s) presents thirds and sixths as complex and dissonant intervals. Kyle Gann offers the following

discussion with regards to the 81/64 interval:

“Before the advent of meantone tuning, the French academy at Notre Dame (13th and 14th centuries)

followed a medieval tradition since Boethius (4th century) in decreeing that only a series of perfect fifths

could make up a scale; their ratio was 3/2, and 3, after all, was the perfect number, connoting the Trinity

among other things. Thus the Pythagorean scale is a just intonation scale on a series of perfect fifths, all the

ratio numbers powers of either 3 or 2. This was an appropriate scale for a music in which perfect fifths

The Harmonic Consideration 233

and fourths were the overwhelmingly dominant simultaneous sonority, and in which the pitches CÚ, FÚ, and

GÚ hardly appeared if at all. Though used, the thirds were theoretical dissonances, and therefore avoided

at final cadences.”323

The root position three-limit major triad is represented by the ratio 64:81:96, identities 1-81-3, intervals

1/1, 81/64, and 3/2, and the pitches C, E(, and G; and presents the just major third (5/4) and just perfect

fifth (3/2) with a falsity of +21.506 and +0.000 cents on each count, which is a deviation of a syntonic

comma (81/80) for the major third. G4, or the just perfect fifth (3/2), presents 0.000 beats between the

third harmonic of C4 (1/1) and the second harmonic of G4, and 0.000 beats between the sixth harmonic

of C4 and the fourth harmonic of G4; while E(4 (331.120Hz), or the Pythagorean major third (81/64),

presents 16.352 beats between the fifth harmonic of C4 and the fourth harmonic of E(4 (1324.479Hz).

Table 132. The beating characteristics of the three-limit major triad

C4

(PARTIAL)

FREQUENCY

(HERTZ)

E(4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

G4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

5

6

– – – –

7

– – – –

8

261.626

– – – –

– – – –

523.251

– – – –

784.877

– – – –

1046.502

– – – –

1308.128

1569.753

– – – –

1831.379

– – – –

2093.005

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

– – – –

5

– – – –

6

– – – –

– – – –

331.120

– – – –

– – – –

662.240

– – – –

993.360

– – – –

– – – –

1324.479

– – – –

1655.599

– – – –

1896.719

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

16.352

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

4

– – – –

– – – –

5

– – – –

– – – –

– – – –

392.438

– – – –

– – – –

784.877

– – – –

– – – –

1177.315

– – – –

1569.753

– – – –

– – – –

1962.192

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

Doty makes the following observation with regards to the intervallic properties of Pythagorean intonation

and its application during the European Middle Ages:

“Although three-limit scales are poorly suited to chordal harmony, they work well melodically and are

eminently suitable for polyphony based on fourths and fifths, such as the motets and organa of the

323 Kyle Gann, “An Introduction to Historical Tunings,” Kyle Gann’s Home Page, 1997, 15 December 2005,

<http://www.kylegann.com/histune.html>.

234 The Harmonic Consideration

European Ars Antiqua period (c. 1110-1300 C.E.). Pythagorean scales were the sole theoretical basis for

intonation throughout the European Middle Ages. The compositional practices of this period, which treat

thirds and sixths as dissonances, are understandable in light of the properties of Pythagorean tuning.”324

The root position three-limit minor triad is represented by the ratio 54:64:81, identities 27-1-81, intervals

27/16, 1/1, and 81/64, and the pitches A(, C, and E(; and presents the just minor third (6/5) and just

perfect fifth (3/2) with a falsity of ß21.506 and +0.000 cents on each count, which is a deviation of a

syntonic comma (81/80) for the major third. E(5 (662.240Hz), or the Pythagorean major third (81/64),

presents 0.000 beats between the third harmonic of A(4 (1324.479Hz), or the Pythagorean major sixth

(27/16), and the second harmonic of E(5 (1324.479Hz), and 0.000 beats between the sixth harmonic

of A(4 (2648.959Hz) and the fourth harmonic of D5 (2648.959Hz); while C5 (523.251Hz), or the octave

(2/1), 32.703 beats between the sixth harmonic of A(4 and the fifth harmonic of C5 (2616.256Hz).

Table 133. The beating characteristics of the three-limit minor triad

A(4

(PARTIAL)

FREQUENCY

(HERTZ)

C5

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

E(5

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

– – – –

5

6

7

– – – –

– – – –

8

441.493

– – – –

– – – –

882.986

– – – –

1324.479

– – – –

1765.973

– – – –

– – – –

2207.466

2648.959

3090.452

– – – –

– – – –

3531.945

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

– – – –

5

– – – –

6

– – – –

– – – –

– – – –

523.251

– – – –

– – – –

1046.502

– – – –

1569.753

– – – –

– – – –

2093.005

– – – –

2616.256

– – – –

3139.507

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

32.703

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

– – – –

4

– – – –

– – – –

5

– – – –

– – – –

– – – –

662.240

– – – –

– – – –

1324.479

– – – –

– – – –

1986.719

– – – –

– – – –

2648.959

– – – –

– – – –

3311.199

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

The concluding sonorities of the fourth system include a G("5)/B triad resolving to a C tonal centre within

two successive 2-3 and 7-1 suspensions in the bass. This harmony presents the pitches B(, G, and

324 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 37-38.

The Harmonic Consideration 235

D")), and within the ratio 59049:93312:131072, characterizes the Pythagorean minor sixth (128/81),

and Pythagorean diminished fifth (1024/729).

Five-Limit Just Intonation

The primary interval of five-limit just intonation is the fifth harmonic, or just major third, which is

represented by the frequency ratio 5/4, and in relation to C (1/1), notated simply as E. The complement,

8/5, or just minor sixth, is notated simply as A". The essential five-limit pitch material is arrived at via a now

two-dimensional process, which requires a two-dimensional 2, 3, 5-limit pitch lattice. Pitch lattices allow

for the graphic representation of pitch relationships, with one dimension in this case represented by

horizontal columns defining just major thirds (the sequential addition or subtraction of 5/4s to the right or

left of any particular pitch), while the other dimension; vertical rows defining just perfect fifths (the

sequential addition or subtraction of 3/2s to the top or bottom of any particular pitch). In different terms,

north and south vertical links (0.00° and 180.00°) within the lattice produce 3/2 and 4/3 relationships;

east and west horizontal links (90.00° and 270.00°), 5/4 and 8/5; northeast and southwest diagonal links

(45.00° and 225.00°), 15/8 and 16/15; while northwest and southeast diagonal links (315.00° and

135.00°), 6/5 and 5/3.325

A five-limit scale is constructed via the establishment of just major triads (4:5:6) on the fundamental

C (1/1), G (3/2), and D (9/8), which generate the six otonal pitches: D, E, F, G, A, and B (9/8, 5/4, 4/3,

3/2, 5/3, and 15/8); as well as the establishment of complement minor triads (5:6:10), which generate

the six utonal pitches: B"), A", G, F, E", and D") (16/9, 8/5, 3/2, 4/3, 6/5, and 16/15).

Fig. 8. Five-limit otonal pitch generation

Fig. 9. Five-limit utonal pitch generation

325 Von Gunden, The Music of Ben Johnston 60.

G (3/2)

C (1/1) E (5/4)

D (9/8)

G (3/2) B (15/8) F (4/3) A (5/3)

C (1/1)

C (1/1) A" (8/5)

F (4/3)

F (3/2) D") (16/15)

B") (16/9)

E" (6/5)

C (1/1)

G (3/2)

236 The Harmonic Consideration

F!(, or the just tritone, or augmented fourth (45/32) is adopted to represent the interval of an augmented

fourth, along with the complement, G"), or the diminished fifth (64/45, or 609.776). The pitch material is

then sequentially arranged as:

C D") D E" E F F!( G") G A" A B") B

11

1516

89

56

45

34

3245

4564

23

58

35

9

16

815

The series of intervals presents the just diatonic semitone (16/15), just major tone (9/8), just minor third

(6/5), just major third (5/4), just perfect fourth (4/3), just tritone, or augmented fourth (45/32), diminished

fifth (64/45), just perfect fifth (3/2), just minor sixth (8/5), just major sixth (5/3), Pythagorean minor seventh

(16/9), and just diatonic major seventh (15/8). The prime number five is a prerequisite for consonant

thirds and sixths, and together with their respective complements, 5/4, 6/5, 5/3, and 8/5 represent the

“principle consonances of the five-limit.” The dissonant Pythagorean major third (81/64) is now replaced

by the consonant just major third with a superparticular vibrating ratio of 5/4. “This interval is the serene

consonance we expect a major third to be,” notes Doty.326 The following table depicts the essential

tonal resources of five-limit just intonation, indicating degree, notation, interval, ratio, and cents.

Table 134. Essential five-limit pitch material

DEGREE

NUMBER

NOTE INTERVAL RATIO

(FRACTION)

RATIO

(DECIMAL)

CENTS

ÐÑ ÐÒ ÐÓ ÐÔ ÐÕ ÐÖ Ð× ÐØ ÐÙ ÑÐ ÑÑ ÑÒ

ÐÑ

C

D")

D E"

E

F

F!(

G

A"

A

B")

B

C

unison

just diatonic semitone, or major half-tone

just major tone (9th harmonic)

just minor third

just major third (5th harmonic)

just and Pythagorean perfect fourth

just tritone, or augmented fourth (45th harmonic)

just and Pythagorean perfect fifth (3rd harmonic)

just minor sixth

just major sixth

Pythagorean minor seventh

just diatonic major seventh (15th harmonic)

octave

1/1

16/15

9/8

6/5

5/4

4/3

45/32

3/2

8/5

5/3

16/9

15/8

2/1

1.000000

1.066667

1.125000

1.200000

1.250000

1.333333

1.406250

1.500000

1.600000

1.666667

1.777778

1.875000

2.000000

0.000

111.731

203.910

315.641

386.314

498.045

590.224

701.955

813.686

884.359

996.090

1088.269

1200.000

326 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 38-39.

The Harmonic Consideration 237

Fig. 1

0. 2

, 3, 5

-lim

it p

itch

latti

ce (

nota

tion,

ratio

and

cen

t val

ues)

238 The Harmonic Consideration

Five-limit intonation presents an intonational reinterpretation of Harrison’s pentatonic #1 within the

pitches C, D, E, G, and A (1/1, 9/8, 5/4, 3/2, and 5/3); and the complements: C, B"), A", F, and E" (2/1,

16/9, 8/5, 4/3, and 6/5).

Ex. 102. Lou Harrison pentatonic #1 (five-limit intonation)

The following two tuning matrixes contain the data required to represent the system of intonation

chromatically within the program memory of the Akai S3000XL.

Table 135. 5-limit tuning matrix no. 1 ‘key of C’ (Program 02)

NOTE C D") D E" E F – – G A" A B") B

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +12 +04 +16 ß14 ß02 +00 +02 +14 ß16 ß04 ß12

Table 136. 5-limit tuning matrix no. 2 ‘key of C’ (Program 03)

NOTE C D") D E" E F – – G A" A B" B

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +12 +04 +16 ß14 ß02 +00 +02 +14 ß16 +18 ß12

As well as expressing a diatonic harmonic extension of the four-bar melodic sentence, the first

simultaneous sonority of section B represents the 1, 3, and 5 identities of the harmonic series that form

the consonant or ‘pure’ major triad of just intonation (4:5:6). “The five-limit major triad conforms in all

aspects to our definition of a consonant chord in just intonation,” notes Doty. “It is free from disturbing

interference beats (assuming a harmonic timbre), and is accompanied by first-order difference tones that

The Harmonic Consideration 239

reinforce the identity of the series from which the chord derives.” The first-order difference tones that

Doty refers to include C2 and C3 for the root position five-limit major triad (4:5:6), which in combination

with the fundamental tones essentially form the pitch series C2, C3, C4, E4, and G4, or the first, second, fourth, fifth, and sixth partials of the harmonic series. The inversions of the triad on the other hand include

C2, , C3, and G3 for the first inversion five-limit major triad (5:6:8); and C3 and C4 for the second inversion

five-limit major triad (3:4:5). This chord is obtained via the two-dimensional five-limit process that now

incorporates not only 3/2 vertical derivatives of the C fundamental, but also horizontal 5/4 relationships

that produce the just major third (5/4), as well as the just perfect fifth (3/2) of three-limit just intonation;

not to mention the complements – the 8/5 relationships that produce the just major sixth (5/3), as well as

the just perfect fourth (4/3) of three-limit just intonation. “In root position, the fifth harmonic of the root

(1 identity) coincides with the fourth harmonic of the major third (5 identity), and the sixth harmonic of

the major third coincides with the fifth harmonic of the perfect fifth (3 identity),” explains Doty. The

following example illustrates the consonant nature of the five-limit major triad. Differential tones are

represented in the bass clef by filled noteheads, while periodicity pitch, by triangular noteheads.

Ex. 103. Differential tones and periodicity pitches produced by the five-limit major triad

The second most important chord of five-limit intonation is the five-limit minor triad (10:12:15), which

according to Doty is more consonant than the three-limit Pythagorean minor triad, yet nevertheless less

consonant than the five-limit major triad due to the fact of possessing identities 3, 5, and 15, but no 1

identity. Analysis of this simultaneous sonority further reveals a 6/5 ratio (or just minor third) between the

fifth and the seventh. B4 (490.548Hz), or the just diatonic major seventh (15/8), presents 0.000 beats

between the third harmonic of E4 (981.096Hz), or the just major third (5/4), and the second harmonic of

B4 (981.096Hz), and 0.000 beats between the sixth harmonic of E4 (1962.192Hz) and the fourth

harmonic of B4 (1962.192Hz); while G4, or the just perfect fifth (3/2), then consistently also presents

0.000 beats between the sixth harmonic of E4 and the fifth harmonic of G4 (1962.192Hz).327

327 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 46-47.

240 The Harmonic Consideration

Table 137. The beating characteristics of the just minor triad

E4

(PARTIAL)

FREQUENCY

(HERTZ)

G4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

B4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

– – – –

5

6

7

– – – –

– – – –

8

327.032

– – – –

– – – –

654.064

– – – –

981.096

– – – –

1308.128

– – – –

– – – –

1635.160

1962.192

2289.224

– – – –

– – – –

2616.256

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

– – – –

5

– – – –

6

– – – –

– – – –

– – – –

392.438

– – – –

– – – –

784.877

– – – –

1177.315

– – – –

– – – –

1569.753

– – – –

1962.192

– – – –

2354.630

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

– – – –

4

– – – –

– – – –

5

– – – –

– – – –

– – – –

490.548

– – – –

– – – –

981.096

– – – –

– – – –

1471.644

– – – –

– – – –

1962.192

– – – –

– – – –

2452.740

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

The beating characteristics of the just minor triad display equal consonance in comparison to the just

major triad, but upon analysis of the inherent differential tones and periodicity pitch it becomes evident

that in the case of the three inversions (10:12:15, 12:15:20, and 15:20:24), one or more tones alien to

the fundamental triad are introduced. These tones include A" and B", or 8/5 and 9/5, which have no

relevance to the conventional musical application of the minor triad. Although, according to Partch’s

theories, these two triads may be stated as being equally consonant, for they form the theoretical basis of

‘otonalities’ and ‘utonalities’ – the major and minor tonalities of the harmonics and subharmonic series.

Partch makes the following observations:

“In utonality (‘minor’) the conception is somewhat different (to otonality [‘major’]), since the series of

identities descends in pitch from its unity, though the practical results are exactly the same; the unity is here

the ‘fifth of the chord’. The long controversy as to the correct location of the ‘root’ of the ‘minor’ triad is

rhetoric, so far as creative music goes, since the composer needs no greater authority than his fancy to put

the ‘root’ wherever he wants to put it.” 328

328 Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 110-12.

The Harmonic Consideration 241

In opposition to the subharmonic argument for the minor triad, Doty states that, “the Partchian view,

however, ignores the implications of difference tones and periodicity pitches. These phenomena will

always reinforce a harmonic interpretation of a chord.”329

Ex. 104. Differential tones and periodicity pitches produced by the five-limit minor triad

Other simultaneous sonorities of interest in five-limit just intonation include the root position major seventh

tetrad (8:10:12:15) with identities 1, 5, 3, and 15; minor seventh tetrad (10:12:15:18) with identities 5, 3,

15, and 9, but no 1 identity; major ninth pentad (8:10:12:15:18) with identities 1, 5, 3, 15, and 9; and

minor ninth pentad (20:24:30:36:45) with identities 5, 3, 15, 9, and 45, and no 1 identity.

Ex. 105. Five-limit chords

The second system features a two-part inverted canon, but with intervallic augmentation and saturation

applied to the complement or inversion of the principal melodic material. The result, is the melodic line’s

transformation from the sequence A" and F, into A", G, and E. Also, the fact that the transformed motive

now occupies the upper position of the harmony (soprano and alto) means that the final sonority in the

system is now D"("5) – the pitches D"), G, and F, or the ratio 32:45:80 (as opposed to the three-limit first

inversion D"("5)/F sonority). In the third system, a minor alteration in the tuning scheme is required, due to

the resulting dissonant Cmin7(omit 5) sonority – the pitches C, E", and B"), or the ratio 135:162:200.

Analysis of the harmonic properties of the simultaneous sonority reveals an appropriate just minor third

(6/5) between the bottom two sonorities, yet a grave or small fifth (40/27, or 680.449 cents) between

the upper two sonorities. In order to adhere to the correct intervallic properties of the five-limit minor

seventh tetrad (10:12:15:18, with identities 5-3-15-9), it is then simply a matter of retuning the minor

seventh from a B"), or Pythagorean minor seventh (16/9), to a B", or acute or large minor seventh (9/5),

329 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 47.

242 The Harmonic Consideration

which essentially forms a 3/2 relationship between the upper two sonorities, to produce a Cmin7(omit 5)

sonority with the ratio 10:12:15. Interesting sonorities presented in the fourth system – which also

features a two-part inverted canon, with the complement or inversion of the principal melodic material

occupying the upper position of the harmony – include the final sonorities at the end of the system (a

double inverted canon) that facilitate a modulation to G major via an E" augmented triad – the pitches E",

G, and B, or the ratio 16:20:25, moving to a second inversion G major triad – the pitches D, G, and B, or

the ratio 3:4:5.

Seven-Limit Just Intonation

The primary interval of seven-limit just intonation is the seventh harmonic, or septimal subminor seventh,

which is represented by the frequency ratio 7/4 (968.826 cents), and notated as B;. The complement,

8/7 (231.174 cents), or septimal supermajor second, is notated as D*). According to Gayle Young, “this

interval is closer to a major second than to a minor third,” yet Partch reports that it was accepted as a

“consonance by Mersenne in the seventh century.”330 The notational symbols for the partial chroma ([+]

and [*], lowering or raising a tone by 36/35 [one septimal comma], or 48.770 cents) are in reference to

“the amount the acute or large minor seventh (9/5) exceeds the seventh harmonic” (7/4Ï36/35=9/5).331

7/4 is 31.174 cents flat from the equal minor seventh ( 56 ]2[ ), while 8/7, 31.174 sharp from the equal

major second ( 6 2 ). Doty states the following about the 7/4 ratio: “7/4 is a powerful consonance,

whereas tempered, three-limit, and five-limit minor sevenths are all quite dissonant. Thus the harmonic

use of 7/4 demands a departure from common practice. 7/4 is a necessary constituent of consonant

dominant seventh chords and diminished triads.”332

Seven-limit intonation now requires a three-dimensional 2, 3, 5, 7-limit pitch lattice, capable of

displaying relationships not just within seven-limit intonation, but also front and back interrelationships

between the four specified limits. According to Heidi Von Gunden:

“Johnston began designing scales using three, four, and even five prime numbers. As the number of

generating ratios increases, certain complications result. One is the inability to show the network of

relationships on a two-dimensional plane, such as a piece of paper. A system using four generating ratios

needs to be represented with a three-dimensional design. Notice that this (2, 3, 5, 7-limit pitch) lattice

maintains the 3/2 ratios on the vertical axes, the 5/4 ratios on the horizontal axes, and the 7/4 ratios are the

third dimensions seen as axes behind (for the seventh above) and in front (for the seventh below) of the

3/2 axes. This lattice produces natural dominant seventh chords if one reads vertically one block north (for

330 Gayle Young, “The Pitch Organization of Harmonium for James Tenney,” Perspectives of New Music 26.2

(Summer, 1988): 205. 331 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters” 113.

332 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 51.

The Harmonic Consideration 243

the 3/2 ratio), one block horizontally east (for the 5/4 ratio), and one block to the rear of the 1/1 ratio (for

the 7/4 ratio).”333

The process adopted for the creation of seven-limit intonation pitch material involves the establishment of

just major triads (4:5:6) from the seventh harmonics of the fundamental C (1/1), G (3/2), and D (9/8),

which generate the six otonal pitches: D;, E;, F+(, G;, A;, and B; (21/20, 7/6, 21/16, 7/5, 14/9, and

7/4); as well as the establishment of complement minor triads (5:6:10), which generate the six utonal

pitches: B*), A*, G*), F:, E*, and D*) (40/21, 12/7, 32/21, 10/7, 9/7, and 8/7). It must be noted that

F+( (21/16) and G*) (32/21) will purely serve a theoretical purpose, as the work adopts the three-limit

perfect fourth (4/3) and fifth (3/2) in actual scale formation.

Fig. 11. Seven-limit otonal pitch generation

Fig. 12. Seven-limit utonal pitch generation

The pitch material is then sequentially arranged as:

C D; D*) E; E* F F: G; G A; A* B; B*)

11

2021

78

67

79

34

7

10

57

23

9

14

712

47

2140

F+( G*)

1621

2132

The series of intervals presents the septimal diatonic semitone (21/20), septimal supermajor second (8/7),

septimal subminor third (7/6), septimal supermajor third (9/7), just perfect fourth (4/3), septimal tritone, or

superaugmented fourth (10/7), septimal subdiminished fifth (7/5), septimal subminor sixth (14/9),

septimal supermajor sixth (12/7), septimal subminor seventh (7/4), and septimal supermajor seventh

333 Von Gunden, The Music of Ben Johnston 128-29.

A* (12/7)

D*) (8/7) F: (10/7)

D*) (8/7)

G*) (32/21) B*) (40/21) A* (12/7) C: (15/14)

E* (9/7)

B; (7/4) G; (7/5)

E; (7/6)

F+( (21/16) D; (21/20)

B; (7/4)

C; (28/15)

A; (14/9)

E; (7/6)

244 The Harmonic Consideration

(40/21); as well as the non-essential tones septimal subfourth (21/16) and septimal superfifth (32/21).

The following table depicts the essential tonal resources of seven-limit just intonation, indicating degree,

notation, interval, ratio, and cents.

Table 138. Essential seven-limit pitch material

DEGREE

NUMBER

NOTE INTERVAL RATIO

(FRACTION)

RATIO

(DECIMAL)

CENTS

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÐÑ

C

D;

D*)

E;

E*

F+(

F:

G*)

A;

A*

B;

B*)

C

unison

septimal diatonic semitone

septimal supermajor second

septimal subminor third

septimal supermajor third

septimal subfourth (21st harmonic)

septimal tritone, or superaugmented fourth

septimal superfifth

septimal subminor sixth

septimal supermajor sixth

septimal subminor seventh (7th harmonic)

septimal supermajor seventh

octave

1/1

21/20

8/7

7/6

9/7

21/16

10/7

32/21

14/9

12/7

7/4

40/21

2/1

1.000000

1.050000

1.142857

1.166667

1.285714

1.312500

1.428571

1.523810

1.555556

1.714286

1.750000

1.904762

2.000000

0.000

84.467

231.174

266.871

435.084

470.781

617.488

729.219

764.916

933.129

968.826

1115.533

1200.000

The adaptation of Harrison’s pentatonic #2 to seven-limit intonation presents a collection of pitches that

include: C, E;, F, G, and B; (1/1, 7/6, 4/3, 3/2, and 7/4); as well as the complements: C, A*, G, F, and

D*) (2/1, 12/7, 3/2, 4/3, and 8/7).

Ex. 106. Lou Harrison pentatonic #2 (seven-limit intonation)

The Harmonic Consideration 245

Fig. 1

3. 2

, 3, 5

, 7-li

mit

pitc

h la

ttice

(no

tatio

n, ra

tio a

nd c

ent v

alue

s)

246 The Harmonic Consideration

The modulation to G (presented in the tuning matrixes) effectively results in the following transposition of

the pitch series:

G A;( A* B; B* C C: D; D E; E* F+( F:

23

4063

7

12

47

1427

11

1415

2021

89

67

79

1621

7

10

C7( D*)

3263

78

The following two tuning matrixes contain the data required to represent the system of intonation

chromatically within the program memory of the Akai S3000XL.

Table 139. 7-limit tuning matrix no. 1 ‘key of G’ (Program 04)

NOTE C C: D E; E* F+( F: G A;( A* B; B*

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +19 +04 ß33 +35 ß29 +17 +02 ß14 +33 ß31 +37

Table 140. 7-limit tuning matrix no. 2 ‘key of G’ (Program 05)

NOTE C C: D E; E* F+( F: G A;( A* B; B

KEY C C! D D! E F F! G G! A A! B

CENTS +00 +19 +04 ß33 +35 ß29 +17 +02 ß14 +33 ß31 ß12

The first simultaneous sonorities of section C represent the 1, 5, 3, and 7 identities that form the septimal

dominant seventh tetrad (4:5:6:7). “The most important chord in the seven-limit is unquestionably the

dominant seventh chord,” notes Doty. “This is the most consonant possible chord consisting of four

unique identities. The difference tones of the just dominant seventh chord strongly reinforce the 1 and 3

identities of the series from which the chord derives, adding to its stability and clarity.”

The Harmonic Consideration 247

Ex. 107. Differential tones produced by the seven-limit septimal dominant seventh tetrad

The septimal dominant seventh tetrad is represented within the work in the forms of G7(omit 3), or the

ratio 4:6:7; and G7/D, or the ratio 3:4:5:6:7. Norden offers the following discussion with regards to the

disparity between theory and performance practice in the Western classical music tradition, and its

misconception of the seven-limit septimal dominant seventh tetrad:

“We hear just intonation many times in fine performances of choral, chamber, and orchestral music, but we

have not recognized it theoretically. Theory has differed widely from practice. The dominant seventh

chord with the harmonic seventh (ratios 4:5:6:7), frequently sounded in performance uninfluenced by

fixed-pitch instruments, is completely ignored in theory books. This is a beautiful chord, easily tuned. The

chord we theorize about is the diatonic dominant-seventh chord (ratios 36:45:54:64), which in equal

temperament is distorted from (386, 316, and 294 cents) to (400, 300, and 300 cents).”334

Another interesting seven-limit chord is the septimal dominant ninth pentad (with the ratio 4:5:6:7:9, and

identities 1-5-3-7-9), which adds the major ninth (9/4) to the tetrad. Doty makes the following

observations:

“Depending on the register in which the major ninth is sounded, beating may occur between some of the

lower harmonics of a pair of tones in this relation, adding roughness to the interval. When added to the

dominant seventh chord, the major ninth spawns another relatively dissonant interval, the acute minor

seventh, 9/5. Nevertheless, the overall impression created by the just dominant ninth chord, at least in root

position, is one of stability.”

It is important to note that the septimal dominant seventh and ninth chords additional present a series of

subsets, which include the septimal diminished triad (5:6:7, with identities 5-3-7), septimal half-

diminished seventh tetrad (5:6:7:9, with identities 5-3-7-9), “incomplete dominant seventh” triad (4:6:7,

with identities 1-3-7), and “added-second” tetrad (8:9:10:12, with identities 1-9-5-3). The septimal

diminished triad is the second most consonant chord in seven-limit just intonation (preceded by the

334 Norden, “A New Theory of Untempered Music: A Few Important Features with Special Reference to ‘A Capella’

Music,” The Musical Quarterly 232.

248 The Harmonic Consideration

seven-limit septimal major triad), and due to there being no 5 identity, and hence no tritone, the sonority

has little in common with the traditional function of a dominant.

Ex. 108. Differential tones produced by the seven-limit septimal dominant ninth pentad

The first system of section C also features the septimal major and minor triads (14:18:21 and 6:7:9, with

identities 7-9-21 and 3-7-9). According to Doty, the subminor septimal minor triad is “quite distinct from

the five-limit minor triad (10:12:15), and is, in the opinion of some listeners, more consonant.” It is

interesting to note that the differential tones produced by the root position septimal minor triad imply “a

fundamental a 3/2 below the root”, as opposed to the five-limit, “5/4 below the root.”335

Ex. 109. Differential tones produced by the seven-limit septimal major triad

The root position seven-limit major triad is represented by the ratio 14:18:21, identities 7-9-21, intervals

7/4, 9/8, and 21/16, and the pitches B;, D, and F+(; and presents the just major third (5/4) and just

perfect fifth (3/2) with a falsity of +48.770 and +0.000 cents on each count, which is a deviation of a

septimal comma (36/35) for the major third. F+(5 (686.767Hz), or the septimal subfourth (21/16, or

470.781 cents), presents 0.000 beats between the third harmonic of B;4 (1373.534Hz), or the septimal

subminor seventh (7/4), and the second harmonic of F+(5 (1373.534Hz), and 0.000 beats between the

sixth harmonic of B;4 (2747.068Hz) and the fourth harmonic of F+(5 (2747.068Hz); while D5

(588.658Hz), or the just major tone (9/8), 65.406 beats between the fifth harmonic of B;4 (2289.224Hz) and the fourth harmonic of D5 (2354.630Hz).

335 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 56-7.

The Harmonic Consideration 249

Table 141. The beating characteristics of the seven-limit major triad

B;4

(PARTIAL)

FREQUENCY

(HERTZ)

D5

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

F+(5

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

5

6

– – – –

7

– – – –

8

457.845

– – – –

– – – –

915.689

– – – –

1373.534

– – – –

1831.379

– – – –

2289.224

2747.068

– – – –

3204.913

– – – –

3662.758

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

5

– – – –

– – – –

6

– – – –

– – – –

588.658

– – – –

– – – –

1177.315

– – – –

1765.973

– – – –

– – – –

2354.630

– – – –

2943.288

– – – –

3531.945

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

65.406

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

4

– – – –

– – – –

5

– – – –

– – – –

– – – –

686.767

– – – –

– – – –

1373.534

– – – –

– – – –

2060.301

– – – –

2747.068

– – – –

– – – –

3433.836

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

Ex. 110. Differential tones produced by the seven-limit septimal minor triad

The root position seven-limit minor triad is represented by the ratio 6:7:9, identities 3-7-9, intervals 3/2,

7/4, and 9/8, and the pitches G, B;, and D; and presents the just minor third (6/5) and just perfect fifth

(3/2) with a falsity of ß48.770 and +0.000 cents on each count, which is a deviation of a septimal

comma (36/35) for the minor third. D5, or the just major tone (9/8), presents 0.000 beats between the

third harmonic of G4 (1177.315Hz), or the just perfect fifth (3/2), and the second harmonic of D5

(1177.315Hz), and 0.000 beats between the sixth harmonic of G4 (2354.630Hz) and the fourth

harmonic of D5; while B;4 (457.845Hz), or the septimal subminor seventh (7/4), 65.406 beats between

the sixth harmonic of G4 and the fifth harmonic of B;4, and 0.000 beats between the seventh harmonic

of G4 (2747.068Hz) and the sixth harmonic of B;4.

250 The Harmonic Consideration

Table 142. The beating characteristics of the seven-limit minor triad

G4

(PARTIAL)

FREQUENCY

(HERTZ)

B;4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

D5

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

– – – –

5

6

7

– – – –

8

392.438

– – – –

– – – –

784.877

– – – –

1177.315

– – – –

1569.753

– – – –

– – – –

1962.192

2354.630

2747.068

– – – –

3139.507

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

– – – –

5

6

– – – –

– – – –

– – – –

457.845

– – – –

– – – –

915.690

– – – –

1373.534

– – – –

– – – –

1831.379

– – – –

2289.224

2747.068

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

65.406

0.000

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

– – – –

4

– – – –

5

– – – –

– – – –

– – – –

588.658

– – – –

– – – –

1177.315

– – – –

– – – –

1765.973

– – – –

– – – –

2354.630

– – – –

2943.288

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

The principal melodic material is developed further via utonal and otonal intonational reinterpretations

(from quintal to septimal major and minor chromatic harmonic extensions), as well as via pitch and

rhythmic recapitulation. The beginning of the second system features motivic chromatic extension

applied to both the start and end of the two-part inverted canon.

The sonorities at the end of the second system (bar 40) imply a Gmin9 within the pitches G, F+(,

G, A*, and D (28:49:56:64:84). Although a minor third is omitted in literal terms, it does make a

presence two beats earlier in the same bar. In the third system, presented are Gmin7(omit 5) within the

pitches G, B;, and F+( (12:14:21); and Edim/G within the pitches G, B;, and E* (192:224:329); with the

entries of the complimentary sentence transformed via intervallic diminution and motivic diatonic

ornamentation on each count. In the fourth and final system, compositional techniques introduced

include intervallic inclusion and rhythmic diminution. The concluding sonorities of the fourth system (bar

48) include a Dmin7(add 11)/F chord resolving to a G tonal centre – featuring a G5 chord (2:3:4:8) –

within two successive 2-3 and 7-1 suspensions, and the pitches F+(, G, D, A*, and C

(147:168:252:384:448).

The Harmonic Consideration 251

Eleven-Limit Just Intonation The primary interval of eleven-limit just intonation is the eleventh harmonic, or undecimal superfourth,

which is represented by the frequency ratio 11/8 (551.318 cents), and notated as F,. The complement,

16/11 (648.682 cents), or undecimal subfifth, is notated as G-. “The frequency of a sound lying exactly

midway between the equal tempered F that lies just below and the equal tempered FÚ that lies just

above our 11th partial can be found by multiplying (523.251) by the 11th power of the 24th root of 2

(523.251Ï 1124 ]2[ =718.923Hz),” notes Wilford W. Berard.336 The accuracy of this numerical

approximation may be stated as being within two cents of the ‘true’ frequency of 11/8. The notational

symbols for the partial chroma ([,] and [-], raising or lowering a tone by 33/32 [one undecimal comma],

or 53.273 cents) are in reference to the “amount the eleventh harmonic exceeds the just perfect fourth”

(4/3Ï33/32=11/8). 11/8 is 51.318 cents sharp from the equal perfect fifth ( 712 ]2[ ), while 16/11,

48.682 flat from the equal perfect fifth ( 712 ]2[ ).337

The process adopted for the creation of eleven-limit intonation pitch material involves the

establishment of just minor triads (5:6:10) from the eleventh harmonics of the fundamental C (1/1), G

(3/2), and F (4/3), which generate the six otonal pitches: DE), EE), F,, GE), AE, and BE) (11/10, 11/9,

11/8, 22/15, 33/20, and 11/6); as well as the establishment of complement major triads (4:5:6), which

generate the six utonal pitches: B-, A-(, G-, FD(,E-, and D- (20/11, 18/11, 16/11, 15/11, 40/33, and

12/11).

Fig. 14. Eleven-limit otonal pitch generation

Fig. 15. Eleven-limit utonal pitch generation

336 Wilford W. Berard, “The Eleventh and Thirteenth Partials,” Journal of Music Theory 5.1 (Spring, 1961): 96-7. 337 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters” 114.

D- (12/11)

G- (16/11) B- (20/11)

G- (16/11)

C- (64/33) E- (40/33) D- (12/11) FD( (15/11)

A-( (18/11)

F, (11/8) DE) (11/10)

BE) (11/6)

C, (33/32) AE (33/20)

F, (11/8)

GE) (22/15)

EE) (11/9)

BE) (11/6)

252 The Harmonic Consideration

The pitch material is then sequentially arranged as:

C DE) D- EE) E- F FD( GE) G AE A-( BE) B-

11

1011

1112

911

3340

34

1115

1522

23

2033

1118

611

1120

F, G-

811

1116

The series of intervals presents the undecimal acute or large neutral second (11/10), undecimal grave or

small neutral second (12/11), undecimal acute or large neutral third (11/9), undecimal grave or small

neutral third (40/33), just perfect fourth (4/3), undecimal tritone, or augmented fourth (15/11), undecimal

subdiminished fifth (22/15), just perfect fifth (3/2), undecimal acute or large neutral sixth (33/20),

undecimal grave or small neutral sixth (18/11), undecimal acute or large neutral seventh (11/6), undecimal

grave or small neutral seventh (20/11); as well as the non-essential tones undecimal superfourth (11/8)

and undecimal subfifth (16/11). The following table depicts the essential tonal resources of eleven-limit

just intonation, indicating degree, notation, interval, ratio, and cents.

Table 143. Essential eleven-limit pitch material

DEGREE

NUMBER

NOTE INTERVAL RATIO

(FRACTION)

RATIO

(DECIMAL)

CENTS

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÐÑ

C

DE)

D-

EE)

E-

F,

FD(

G-

AE

A-(

BE)

B-

C

unison

undecimal acute or large neutral second

undecimal grave or small neutral second

undecimal acute or large neutral third

undecimal grave or small neutral third

undecimal superfourth (11th harmonic)

undecimal tritone, or augmented fourth

undecimal subfifth

undecimal acute or large neutral sixth

undecimal grave or small neutral sixth

undecimal acute or large neutral seventh

undecimal grave or small neutral seventh

octave

1/1

11/10

12/11

11/9

40/33

11/8

15/11

16/11

33/20

18/11

11/6

20/11

2/1

1.000000

1.100000

1.090909

1.222222

1.212121

1.375000

1.363636

1.454545

1.650000

1.636364

1.833333

1.818181

2.000000

0.000

165.004

150.637

347.408

333.041

551.318

536.951

648.682

866.959

852.592

1049.363

1034.996

1200.000

The Harmonic Consideration 253

Fig. 1

6. 2

, 3, 5

, 11-

limit

pitc

h la

ttice

(no

tatio

n, ra

tio a

nd c

ent v

alue

s)

254 The Harmonic Consideration

Eleven-limit intonation presents an intonational reinterpretation of Harrison’s pentatonic #2 within the

pitches C, EE), F, G, and BE) (1/1, 11/9, 4/3, 3/2, and 11/6); and the complements: C, A-(, G, F, and D-

(2/1, 18/11, 3/2, 4/3, and 12/11).

Ex. 111. Lou Harrison pentatonic #2 (eleven-limit intonation)

The modulation to G (presented in the tuning matrixes) effectively results in the following transposition of

the pitch series:

G AE A-) BE) B- C CD( DE) D EE E-( F, FD(

23

2033

1118

611

1120

11

4445

1011

89

8099

2227

811

1115

C, D-

3233

1112

The following four tuning matrixes contain the data required to represent the system of intonation

chromatically within the program memory of the Akai S3000XL.

Table 144. 11-limit tuning matrix no. 1 ‘key of G’ (Program 06)

NOTE C C, D – – E-( FD( F, G – – A-( BE) – –

KEY C C! D D! E F F! G G! A A! B

CENTS +00 ß47 +04 +00 ß45 +37 ß49 +02 +00 ß47 +49 +00

The Harmonic Consideration 255

Table 145. 11-limit tuning matrix no. 2 ‘key of G’ (Program 07)

NOTE C C, D – – E-( F+( F, G – – A( BE) B

KEY C C! D D! E F F! G G! A A! B

CENTS +00 ß47 +04 +00 ß45 ß29 ß49 +02 +00 +06 +49 ß12

Table 146. 11-limit tuning matrix no. 3 ‘key of G’ (Program 08)

NOTE CD( – – D – – EE FD( F, G – – A-( B- – –

KEY C C! D D! E F F! G G! A A! B

CENTS +39 +00 +04 +00 ß31 +37 ß49 +02 +00 ß47 +35 +00

Table 147. 11-limit tuning matrix no. 4 ‘key of G’ (Program 09)

NOTE C C, D – – E-( FD( F, G – – AE BE) – –

KEY C C! D D! E F F! G G! A A! B

CENTS +00 ß47 +04 +00 ß45 +37 ß49 +02 +00 ß33 +49 +00

The first simultaneous sonorities of section D represent the 1, 5, 3, 7, 9, and 11 identities that form the

undecimal dominant eleventh hexad (4:5:6:7:9:11). “In root position and in a reasonably high register,

the addition of the 11 identity to the dominant-ninth chord doesn’t seriously injure its stability or

consonance, although it does add an indescribable, piquant quality to the sonority,” notes Doty.338

Ex. 112. Differential tones produced by the eleven-limit undecimal dominant eleventh hexad

338 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 62.

256 The Harmonic Consideration

The undecimal dominant eleventh hexad is represented within the work in the forms of G7(add 11/omit 3),

or the ratio 4:6:7:11; and G11/D, or the ratio 3:4:5:6:7:8:11. Also introduced in the first system of

section D are undecimal major and minor triads. The root position eleven-limit major triad is represented

by the ratio 66:80:99, identities 33-5-99, intervals 33/32, 5/4, and 99/64, and the pitches C,, E, and G,;

and presents the just major third (5/4) and just perfect fifth (3/2) with a falsity of ß53.273 and +0.000

cents on each count, which is a deviation of an undecimal comma (33/32) for the major third. G,4

(404.702Hz), or the undecimal superfifth (99/64, or 755.228 cents), presents 0.000 beats between the

third harmonic of C,4 (809.404Hz), or the undecimal comma (33/32), and the second harmonic of G,4

(809.404Hz), and 0.000 beats between the sixth harmonic of C,4 (1618.808Hz) and the fourth

harmonic of G,4 (1618.808Hz); while E4, or the just major third (5/4), 40.879 beats between the fifth

harmonic of C,4 (1349.007Hz) and the fourth harmonic of E4 (1308.128Hz).

Table 148. The beating characteristics of the eleven-limit major triad

C,4

(PARTIAL)

FREQUENCY

(HERTZ)

E4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

G,4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

5

6

– – – –

7

– – – –

8

269.801

– – – –

– – – –

539.603

– – – –

809.404

– – – –

1079.205

– – – –

1349.007

1618.808

– – – –

1888.610

– – – –

2158.411

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

– – – –

5

– – – –

6

– – – –

– – – –

327.032

– – – –

– – – –

654.064

– – – –

981.096

– – – –

– – – –

1308.128

– – – –

1635.160

– – – –

1962.192

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

40.879

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

4

– – – –

– – – –

5

– – – –

– – – –

– – – –

404.702

– – – –

– – – –

809.404

– – – –

– – – –

1214.106

– – – –

1618.808

– – – –

– – – –

2023.510

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

The root position eleven-limit minor triad is represented by the ratio 18:22:27, identities 9-11-27, intervals

9/8, 11/8, and 27/16, and the pitches D, F,, and A(; and presents the just minor third (6/5) and just

perfect fifth (3/2) with a falsity of +53.273 and +0.000 cents on each count, which is a deviation of an

undecimal comma (33/32) for the minor third. A(4 (441.493Hz), or the Pythagorean major sixth (27/16),

presents 0.000 beats between the third harmonic of D4 (882.986Hz), or the just major tone (9/8), and

The Harmonic Consideration 257

the second harmonic of A(4 (882.986Hz), and 0.000 beats between the sixth harmonic of D4

(1765.973Hz) and the fourth harmonic of A(4 (1765.973Hz); while F,4 (359.735Hz), or the undecimal

superfourth (11/8), 32.703 beats between the sixth harmonic of D4 and the fifth harmonic of F,4

(1798.676Hz).

Table 149. The beating characteristics of the eleven-limit minor triad

D4

(PARTIAL)

FREQUENCY

(HERTZ)

F,4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

A(4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

– – – –

5

6

7

– – – –

– – – –

8

294.329

– – – –

– – – –

588.658

– – – –

882.986

– – – –

1177.315

– – – –

– – – –

1471.644

1765.973

2060.301

– – – –

– – – –

2354.630

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

– – – –

5

– – – –

6

– – – –

– – – –

– – – –

359.735

– – – –

– – – –

719.470

– – – –

1079.205

– – – –

– – – –

1438.941

– – – –

1798.676

– – – –

2158.411

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

32.703

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

– – – –

4

– – – –

– – – –

5

– – – –

– – – –

– – – –

441.493

– – – –

– – – –

882.986

– – – –

– – – –

1324.479

– – – –

– – – –

1765.973

– – – –

– – – –

2207.466

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

The sonorities at the end of the second system (bar 56) imply a Gm9 within the pitches G, D, BE), A-(,

and F, (198:297:484:864:1089); and in the third system, Gmin7(add11/omit 5) within the pitches G, BE),

F,, and C, (36:44:66:99); and Edim/G within the pitches G, BE), and E-( (180:220:297). The

concluding sonorities of the fourth system (bar 65) facilitate a modulation to D major via a B" augmented

triad – the pitches BE), D, and FD(, or the ratio 110:135:162, moving to a second inversion D major triad

– the pitches A-(, D, and FD(, or the ratio 40:55:66.

258 The Harmonic Consideration

Thirteen-Limit Just Intonation

The primary interval of thirteen-limit just intonation is the thirteenth harmonic, or tridecimal grave or small

neutral, or overtone sixth, which is represented by the frequency ratio 13/8 (840.528 cents), and

notated as AO. The complement, 16/13 (359.472 cents), or tridecimal acute or large neutral third, is

notated as E/. The notational symbols for the partial chroma ([.] and [/], raising or lowering a tone by

65/64 [one tridecimal comma], or 26.841 cents) is in reference to the “amount the thirteenth harmonic

exceeds the just minor sixth” (8/5Ï65/43=13/8).339 13/8 is 40.528 cents sharp from the equal minor

sixth ( 23 ]2[ ), while 16/13, 40.528 flat from the equal major third ( 3 2 ).

The process adopted for the creation of thirteen-limit intonation pitch material involves the

establishment of just minor triads (5:6:10) from the thirteen harmonics of the fundamental C (1/1), G

(3/2), and F (4/3), which generate the four otonal pitches: DO), EO, GO), and AO (13/12, 39/32, 13/9,

and 13/8); as well as the establishment of complement major triads (4:5:6), which generate the four

utonal pitches: B/, A/, FN(, and E/ (24/13, 64/39, 18/13, and 16/13). In this particular case, the

exercise omits necessary major second, perfect fourth, perfect fifth, and minor seventh pitch material.

Fig. 17. Thirteen-limit otonal pitch generation

Fig. 18. Thirteen-limit utonal pitch generation

The necessary additional pitch material is obtained via a secondary process that involves the calculation

of 5/4 relationships with DO) (13/12) and GO) (13/9), which generate the otonal pitches F. (65/48)

and BO) (65/36), and utonal complements G/ (96/65) and D/ (72/65). The pitch material is then

sequentially arranged as:

339 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters” 115.

B/ (24/13)

E/ (16/13) GN (20/13)

E/ (16/13)

A/ (64/39) CN (40/39) B/ (24/13) DN (15/13)

FN( (18/13)

AO (13/8) FO (13/10)

DO) (13/12)

EO (39/32) CO (39/20)

AO (13/8)

Bg) (26/15)

GO) (13/9)

DO) (13/12)

The Harmonic Consideration 259

C DO) D/ EO E/ F FN( GO) G AO A/ BO) B/

11

1213

6572

3239

1316

34

1318

913

23

8

13

3964

3665

1324

F. G/

4865

6596

The series of intervals presents the tridecimal grave or small neutral second (13/12), tridecimal acute or

large neutral second (72/65), tridecimal grave or small neutral third (39/32), tridecimal acute or large

neutral third (16/13), just perfect fourth (4/3), tridecimal tritone, or augmented fourth (18/13), tridecimal

subdiminished fifth (13/9), just perfect fifth (3/2), tridecimal grave or small neutral, or overtone sixth

(13/8), tridecimal acute or large neutral sixth (64/39), tridecimal grave or small neutral seventh (65/36),

tridecimal acute or large neutral seventh (24/13); as well as the non-essential tones tridecimal superfourth

(65/48) and tridecimal subfifth (96/65). The following table depicts the essential tonal resources of

thirteen-limit just intonation, indicating degree, notation, interval, ratio, and cents.

Table 150. Essential thirteen-limit pitch material

DEGREE

NUMBER

NOTE INTERVAL RATIO

(FRACTION)

RATIO

(DECIMAL)

CENTS

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÐÑ

C

DO)

D/

EO

E/

F.

FN(

G/

AO

A/

BO)

B/

C

unison

tridecimal grave or small neutral second

tridecimal acute or large neutral second

tridecimal grave or small neutral third (39th harmonic)

tridecimal acute or large neutral third

tridecimal superfourth

tridecimal tritone, or augmented fourth

tridecimal subfifth

tridecimal grave or small neutral, or overtone sixth (13th harmonic)

tridecimal acute or large neutral sixth

tridecimal grave or small neutral seventh

tridecimal acute or large neutral seventh

octave

1/1

13/12

72/65

39/32

16/13

65/48

18/13

96/65

13/8

64/39

65/36

24/13

2/1

1.000000

1.083333

1.107692

1.218750

1.230769

1.354167

1.384615

1.476923

1.625000

1.641026

1.805556

1.846154

2.000000

0.000

138.573

177.069

342.483

359.472

524.886

563.382

675.114

840.528

857.517

1022.931

1061.427

1200.000

260 The Harmonic Consideration

Fig. 1

9. 2

, 3, 5

, 13-

limit

pitc

h la

ttice

(no

tatio

n, ra

tio a

nd c

ent v

alue

s)

The Harmonic Consideration 261

The adaptation of Harrison’s pentatonic #3 to thirteen-limit intonation presents a collection of pitches

that include: C, D/, EO, G, and AO (1/1, 72/65, 39/32, 3/2, and 13/8); as well as the complements: C,

BO), A/, F, and E/ (2/1, 65/36, 64/39, 4/3, and 16/13).

Ex. 113. Lou Harrison pentatonic #3 (thirteen-limit intonation)

The modulation to D (presented in the tuning matrixes) effectively results in the following transposition of

the pitch series:

D EO E/( F.( FN( G GN( AO A( BO B/ C. CN(

89

3239

6581

256351

1318

23

5281

8

13

1627

64

117

1324

6465

2627

G. A/(

128195

65

108

The following two tuning matrixes contain the data required to represent the system of intonation

chromatically within the program memory of the Akai S3000XL.

Table 151. 13-limit tuning matrix no. 1 ‘key of D’ (Program 10)

NOTE C. CN( D EO E/( F.( FN( G GN( A( BO B/

KEY C C! D D! E F F! G G! A A! B

CENTS +27 ß35 +04 +42 ß19 +46 ß37 +02 ß33 +06 +44 ß39

262 The Harmonic Consideration

Table 152. 13-limit tuning matrix no. 2 ‘key of D’ (Program 11)

NOTE C. CN( D EO E/( F.( FN( G AO A( BO B/

KEY C C! D D! E F F! G G! A A! B

CENTS +27 ß35 +04 +42 ß19 +46 ß37 +02 +41 +06 +44 ß39

Introduced in the first system of section E (bars 65-66), are tridecimal major and minor triads. The root

position thirteen-limit major triad is represented by the ratio 26:32:39, identities 13-1-39, intervals 13/8,

1/1, and 39/32, and the pitches AO, C, and EO; and presents the just major third (5/4) and just perfect

fifth (3/2) with a falsity of +26.841 and +0.000 cents on each count, which is a deviation of a tridecimal

comma (65/64) for the major third. EO5 (637.712Hz), or the tridecimal grave or small neutral third

(39/32, or 342.483 cents), presents 0.000 beats between the third harmonic of AO4 (1275.425Hz), or

the tridecimal grave or small neutral sixth (13/8), and the second harmonic of EO5 (1275.425Hz), and

0.000 beats between the sixth harmonic of AO4 (2550.849Hz) and the fourth harmonic of EO5

(2550.849Hz); while C5, or the octave (2/1), 32.703 beats between the fifth harmonic of AO4

(2125.708Hz) and the fourth harmonic of C5 (2093.005Hz).

Table 153. The beating characteristics of the thirteen-limit major triad

AO 4

(PARTIAL)

FREQUENCY

(HERTZ)

C5

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

EO5

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

5

6

– – – –

7

– – – –

8

425.142

– – – –

– – – –

850.283

– – – –

1275.425

– – – –

1700.566

– – – –

2125.708

2550.849

– – – –

2975.991

– – – –

3401.132

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

– – – –

5

– – – –

6

– – – –

– – – –

523.251

– – – –

– – – –

1046.502

– – – –

1569.753

– – – –

– – – –

2093.005

– – – –

2616.256

– – – –

3139.507

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

32.703

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

4

– – – –

– – – –

5

– – – –

– – – –

– – – –

637.712

– – – –

– – – –

1275.425

– – – –

– – – –

1913.137

– – – –

2550.849

– – – –

– – – –

3188.562

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

The Harmonic Consideration 263

The root position thirteen-limit minor triad is represented by the ratio 32:39:48, identities 1-39-3, intervals

1/1, 39/32, and 3/2, and the pitches C, EO, and G; and presents the just minor third (6/5) and just

perfect fifth (3/2) with a falsity of ß26.841 and +0.000 cents on each count, which is a deviation of a

tridecimal comma (65/64) for the minor third. G4, or the just perfect fifth (3/2), presents 0.000 beats

between the third harmonic of C4 (784.877Hz), or the unison (1/1), and the second harmonic of G4,

and 0.000 beats between the sixth harmonic of C4 (1569.753Hz) and the fourth harmonic of G4; while

EO4 (318.856Hz), or the tridecimal grave or small neutral third (39/32), 24.527 beats between the sixth

harmonic of C4 and the fifth harmonic of EO4 (1594.281Hz).

Table 154. The beating characteristics of the thirteen-limit minor triad

C4

(PARTIAL)

FREQUENCY

(HERTZ)

EO4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

G4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

– – – –

5

6

7

– – – –

– – – –

8

261.626

– – – –

– – – –

523.251

– – – –

784.877

– – – –

1046.502

– – – –

– – – –

1308.128

1569.753

1831.379

– – – –

– – – –

2093.005

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

– – – –

5

– – – –

6

– – – –

– – – –

– – – –

– – – –

318.856

– – – –

– – – –

637.712

– – – –

956.568

– – – –

– – – –

1275.425

– – – –

1594.281

– – – –

1913.137

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

24.527

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

– – – –

4

– – – –

– – – –

5

– – – –

– – – –

– – – –

392.438

– – – –

– – – –

784.877

– – – –

– – – –

1177.315

– – – –

– – – –

1569.753

– – – –

– – – –

1962.192

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

The sonorities at the beginning of the second system (bars 69-70) present intonational reinterpretations

of an implied thirteen-limit tridecimal dominant thirteenth heptad via the simultaneous sonorities

D9(!11)/F!, featuring the pitches D, FN(, A(, C., E/(, and GN(

(30420:37440:45630:54925:67392:84240); and Dmin9(!11)/F, featuring the pitches D, F.(, A(, C.,

E/(, and GN( (56160:68445:84240:101400:124416:155520). In striking contrast, the ‘harmonic’ root

position thirteen-limit tridecimal dominant thirteenth heptad has a ratio of 4:5:6:7:9:11:13, and features

264 The Harmonic Consideration

the just major third (5/4), just perfect fifth (3/2), septimal subminor seventh (7/4), undecimal superfourth

(11/8), and tridecimal grave or small neutral sixth (13/8).

Ex. 114. Differential tones produced by the thirteen-limit tridecimal dominant thirteenth heptad

The simultaneous sonorities in the fourth system (bars 77-78) include a B"7/D tetrad, with the pitches D,

BO, F.(, and AO (144:234:351:416).

Numerous compositional techniques are simultaneously utilized in the development of the four-bar

melodic sentence and complimentary sentence in thirteen-limit just intonation. Notary amongst these

includes the time signature change (from quadruple, or double duple time [4/4] to compound triple time

[9/8]) applied to section E (bar 65), which significantly alters the rhythmic proportions of the motivic

material. Intervallic redirection and utonal and otonal harmonic extension (bars 65 and 66 essentially

outlining a major/minor tonality shift) is also applied to the two-note motive. A motivic ostinato featuring

chromatic expansion and rhythmic recapitulation is additionally introduced, which generates a sense of

moto perpetuo, or perpetual motion.340

340 “Perpetuum mobile [Lat., perpetual motion; It. moto perpetuo]. A composition in which rhythmic motion, often

in a single-note-value at rapid tempo, is continuous from beginning to end. Among composers who have used the term

as title for such a piece are Paganini (op. 11), Weber (Piano Sonata op. 24, last movement), Mendelssohn (op. 119), and

Johann Strauss, Jr. (op. 257). The technique is also encountered in some Chopin etudes.” For a further discussion, see

Randel, ed., The New Harvard Dictionary of Music 628.

The Harmonic Consideration 265

Seventeen-Limit Just Intonation

The primary interval of seventeen-limit just intonation is the seventeenth harmonic, or septendecimal

chromatic semitone, which is represented by the frequency ratio 17/16 (104.955 cents), and notated as

CP. The complement, 32/17 (1095.045 cents), or septendecimal superdiminished octave, is notated as

CQ. The notational symbols for the partial chroma ([0] and [1], raising or lowering a tone by 51/50 [one

septendecimal comma], or 34.283 cents) is in reference to the “amount the seventeenth harmonic

exceeds the grave or small just chromatic semitone” (25/24Ï51/50=17/16). 17/16 is 4.955 cents sharp

from the equal semitone ( 12 2 ), while 32/17, 4.955 flat from the equal diminished octave ( 1112 ]2[ ).341

The process adopted for the creation of seventeen-limit intonation pitch material involves the

establishment of just minor triads (5:6:10) from the seventeenth harmonics of the fundamental C (1/1), G

(3/2), and F (4/3), which generate the five otonal pitches: D0), E0, FP, A0, and B0) (17/15, 51/40,

17/12, 17/10, and 17/9); as well as the establishment of complement major triads (4:5:6), which

generate the five utonal pitches: BQ, AQ, GQ, EQ, and DQ (30/17, 80/51, 24/17, 20/17, and 18/17). In

this particular case, the exercise omits necessary perfect fourth and perfect fifth pitch material.

Fig. 20. Seventeen-limit otonal pitch generation

Fig. 21. Seventeen-limit utonal pitch generation

The necessary additional pitch material is obtained via a secondary process that involves the calculation

of an 8/5 relationship with B0) (17/9), which generates the otonal pitch G0) (68/45), and utonal

complement F1( (45/34). A minor adjustment is also made with regards to E0 (51/40), and due to the

fact this pitch does not represent the simplest ratio available within the seventeen-limit intonation scheme.

E0) (34/27) and AQ( (27/17) are adopted as a consequence. The pitch material is then sequentially

arranged as:

341 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters” 115.

GQ (24/17)

CQ (32/17) EQ (20/17)

CQ (32/17)

FQ (64/51) AQ (80/51) GQ (24/17) BQ (30/17)

DQ (18/17)

CP (17/16) A0 (17/10)

FP (17/12)

GP (51/32) E0 (51/40)

CP (17/16)

D0) (17/15)

B0) (17/9)

FP (17/12)

266 The Harmonic Consideration

C DQ D0) EQ E0) F FP GQ G AQ( A0 BQ B0)

11

1718

1517

1720

2734

34

1217

1724

23

1727

1017

1730

9

17

F1( G0)

3445

4568

The series of intervals presents the septendecimal diatonic semitone (18/17), septendecimal supermajor

second (17/15), septendecimal subminor third (20/17), septendecimal supermajor third (34/27), just

perfect fourth (4/3), septendecimal tritone, or superaugmented fourth (17/12), just perfect fifth (3/2),

septendecimal subdiminished fifth (24/17), septendecimal subminor sixth (27/17), septendecimal

supermajor sixth (17/10), septendecimal subminor seventh (30/17), and septendecimal supermajor

seventh (17/9); as well as the non-essential tones septendecimal subfourth (45/34) and septendecimal

superfifth (68/45). The following table depicts the essential tonal resources of seventeen-limit just

intonation, indicating degree, notation, interval, ratio, and cents.

Table 155. Essential seventeen-limit pitch material

DEGREE

NUMBER

NOTE INTERVAL RATIO

(FRACTION)

RATIO

(DECIMAL)

CENTS

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÐÑ

C

DQ

D0)

EQ

E0)

F1(

FP

G0)

AQ(

A0

BQ

B0)

C

unison

septendecimal diatonic semitone

septendecimal supermajor second

septendecimal subminor third

septendecimal supermajor third

septendecimal subfourth

septendecimal tritone, or superaugmented fourth

septendecimal superfifth

septendecimal subminor sixth

septendecimal supermajor sixth

septendecimal subminor seventh

septendecimal supermajor seventh

octave

1/1

18/17

17/15

20/17

34/27

45/34

17/12

68/45

27/17

17/10

30/17

17/9

2/1

1.000000

1.058824

1.133333

1.176471

1.259259

1.323529

1.416667

1.511111

1.588235

1.700000

1.764706

1.888889

2.000000

0.000

98.955

216.687

281.358

399.090

485.268

603.000

714.732

800.910

918.642

983.313

1101.045

1200.000

Seventeen-limit intonation presents an intonational reinterpretation of Harrison’s pentatonic #3 within the

pitches C, D0), EQ, G, and AQ( (1/1, 17/15, 20/17, 3/2, and 27/17); and the complements: C, BQ, A0,

F, and E0) (2/1, 30/17, 17/10, 4/3, and 34/27).

The Harmonic Consideration 267

Fig. 2

2. 2

, 3, 5

, 17-

limit

pitc

h la

ttice

(no

tatio

n, ra

tio a

nd c

ent v

alue

s)

268 The Harmonic Consideration

Ex. 115. Lou Harrison pentatonic #3 (seventeen-limit intonation)

The modulation to D (presented in the tuning matrixes) effectively results in the following transposition of

the pitch series:

D EQ( E0 F1( FP G GP AQ( A( BQ( B0 C1( CP

89

6881

4051

3445

1217

23

3251

1727

1627

136243

80153

68

135

1617

G1( A0

272405

1017

The following two tuning matrixes contain the data required to represent the system of intonation

chromatically within the program memory of the Akai S3000XL.

Table 156. 17-limit tuning matrix no. 1 ‘key of D’ (Program 12)

NOTE C1( CP D EQ( E0 F1( FP G GP A( BQ( B0

KEY C C! D D! E F F! G G! A A! B

CENTS ß13 +05 +04 +03 +21 ß15 +03 +02 +07 +06 +05 +23

Table 157. 17-limit tuning matrix no. 2 ‘key of D’ (Program 13)

NOTE C1( DQ D EQ( E0 F1( FP G AQ( A( BQ( B0

KEY C C! D D! E F F! G G! A A! B

CENTS ß13 ß01 +04 +03 +21 ß15 +03 +02 +01 +06 +05 +23

The Harmonic Consideration 269

Introduced in the first system of section F (bars 81-82), are septendecimal major and minor triads. The

root position seventeen-limit major triad is represented by the ratio 54:68:81, identities 27-17-81,

intervals 27/16, 17/16, and 81/64, and the pitches A(, CP, and E(; and presents the just major third

(5/4) and just perfect fifth (3/2) with a falsity of +34.283 and +0.000 cents on each count, which is a

deviation of a septendecimal comma (51/50) for the major third. E(5, or the Pythagorean major third

(81/64), presents 0.000 beats between the third harmonic of A(4, or the Pythagorean major sixth

(27/16), and the second harmonic of E(5, and 0.000 beats between the sixth harmonic of A(4 and the

fourth harmonic of E(5 (2648.959Hz); while CP5 (555.954Hz), or the septendecimal chromatic

semitone (17/16), 16.352 beats between the fifth harmonic of A(4 (2207.466Hz) and the fourth

harmonic of CP5 (2223.817Hz).

Table 158. The beating characteristics of the seventeen-limit major triad

A(4

(PARTIAL)

FREQUENCY

(HERTZ)

CP5

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

E(5

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

5

– – – –

6

7

– – – –

8

441.493

– – – –

– – – –

882.986

– – – –

1324.479

– – – –

1765.973

– – – –

2207.466

– – – –

2648.959

3090.452

– – – –

3531.945

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

5

– – – –

– – – –

6

– – – –

– – – –

555.954

– – – –

– – – –

1111.909

– – – –

1667.863

– – – –

– – – –

2223.817

2616.256

– – – –

– – – –

3335.726

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

16.352

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

– – – –

4

– – – –

5

– – – –

– – – –

– – – –

662.240

– – – –

– – – –

1324.479

– – – –

– – – –

1986.719

– – – –

– – – –

2648.959

– – – –

3311.199

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

The root position seventeen-limit minor triad is represented by the ratio 34:40:51, identities 17-5-51,

intervals 17/16, 5/4, and 51/32, and the pitches CP, E, and GP; and presents the just minor third (6/5)

and just perfect fifth (3/2) with a falsity of ß34.283 and +0.000 cents on each count, which is a

deviation of a septendecimal comma (51/50) for the minor third. GP4 (416.966Hz), or the

septendecimal superaugmented fifth (51/32, or 806.910 cents), presents 0.000 beats between the third

harmonic of CP4 (833.931Hz), or the septendecimal chromatic semitone (17/16), and the second

270 The Harmonic Consideration

harmonic of GP4 (833.931Hz), and 0.000 beats between the sixth harmonic of CP4 (1667.863Hz) and

the fourth harmonic of GP4 (1667.863Hz); while E4 (327.032Hz), or the just major third (5/4), 32.703

beats between the sixth harmonic of CP4 and the fifth harmonic of E4 (1635.160Hz).

Table 159. The beating characteristics of the seventeen-limit minor triad

CP4

(PARTIAL)

FREQUENCY

(HERTZ)

E4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

GP4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

– – – –

5

6

7

– – – –

– – – –

8

277.977

– – – –

– – – –

555.954

– – – –

833.931

– – – –

1111.909

– – – –

– – – –

1389.886

1667.863

1945.840

– – – –

– – – –

2223.817

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

– – – –

5

– – – –

6

– – – –

– – – –

– – – –

327.032

– – – –

– – – –

654.064

– – – –

981.096

– – – –

– – – –

1308.128

– – – –

1635.160

– – – –

1962.192

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

32.703

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

– – – –

4

– – – –

– – – –

5

– – – –

– – – –

– – – –

416.966

– – – –

– – – –

833.931

– – – –

– – – –

1250.897

– – – –

– – – –

1667.863

– – – –

– – – –

2084.829

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

The sonorities at the beginning of the second system (bars 85-86) present an intonational reinterpretation

of the simultaneous sonorities presented in the previous E section, which include hexads D9(!11)/F!,

featuring the pitches D, FP, A(, C1(, E0, and GP (4590:5780:6885:8100:10404:13005); and

Dmin9(!11)/F, featuring the pitches D, F1(, A(,C1(, E0, and GP (4590:5400:6885:8100:10404:13005).

The motivic ostinato introduced in section E (thirteen-limit just intonation) featuring chromatic expansion

and rhythmic recapitulation is now further developed in section F (seventeen-limit just intonation), with

chromatic ornamentation, as well as transformation via intonational reinterpretation. The simultaneous

sonorities in the fourth system (bars 93-94) include a B"7/D tetrad with the pitches D, BQ(, F1(, and AQ(

(170:270:405:486). The concluding sonorities of the fourth system (bar 96) facilitate a modulation to A

major (in section G) via an E"(!11) hexad – the pitches EQ(, G, BQ(,DQ, F1( and FD(, or the ratio

54:68:81:96:120:153.

The Harmonic Consideration 271

Nineteen-Limit Just Intonation

The primary interval of nineteen-limit just intonation is the nineteenth harmonic, or nonadecimal subminor,

or overtone minor third, which is represented by the frequency ratio 19/16 (297.513 cents), and

notated as EW. The complement, 32/19 (902.487 cents), or nonadecimal supermajor sixth, is notated as

A3. The notational symbols for the partial chroma ([2] and [3], lowering or raising a tone by 96/95 [one

nonadecimal comma], or 18.128 cents) is in reference to the “amount the just minor third (6/5) exceeds

the nineteenth harmonic” (19/16Ï96/95=6/5).342 19/16 is 2.487 cents flat from the equal minor third

( 4 2 ), while 32/19, 2.487 sharp from the equal major sixth ( 34 ]2[ ).

The process adopted for the creation of nineteen-limit intonation pitch material involves the

establishment of just minor triads (5:6:10) from the nineteenth harmonics of C (1/1), G (3/2), and F (4/3),

which generate the five otonal pitches: DW), EW, GW, AW, and BW (19/18, 19/16, 57/40, 19/12, and

57/32); as well as the establishment of the complement major triads (4:5:6), which generate five utonal

pitches: B3, A3, FV, E3, and D3) (36/19, 32/19, 80/57, 24/19, and 64/57). In this particular case, the

exercise omits necessary perfect fourth and perfect fifth pitch material.

Fig. 23. Nineteen-limit otonal pitch generation

Fig. 24. Nineteen-limit utonal pitch generation

The necessary additional pitch material is obtained via a secondary process that involves the calculation

of a 5/4 relationship with DW) (19/18), which generates the otonal pitch F2 (95/72) and utonal

complement G3 (144/95). A minor adjustment is also made with regards to GW (57/40), and due to the

fact this pitch does not represent the simplest ratio available within the nineteen-limit intonation scheme.

GW) (38/27) and FV( (27/19) are adopted as a consequence. The pitch material is then sequentially

arranged as:

342 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters” 115.

E3 (24/19)

A3 (32/19) CV (20/19)

A3 (32/19)

D3) (64/57) FV (80/57) E3 (24/19) FV (30/19)

B3 (36/19)

EW (19/16) CW (19/10)

AW (19/12)

BW (57/32) GW (57/40)

EW (19/16)

FW (19/15)

DW) (19/18)

AW (19/12)

272 The Harmonic Consideration

C DW) D3) EW E3 F FV( GW) G AW A3 BW B3

11

1819

5764

1619

1924

34

1927

2738

23

1219

1932

3257

1936

F2 G3

7295

95

144

The series of intervals presents the nonadecimal diatonic semitone (19/18), nonadecimal supermajor

second (64/57), nonadecimal subminor, or overtone minor third (19/16), nonadecimal supermajor third

(24/19), just perfect fourth (4/3), nonadecimal tritone, or superaugmented fourth (27/19), just major fifth

(3/2), nonadecimal subdiminished fifth (38/27), nonadecimal subminor sixth (19/12), nonadecimal

supermajor sixth (32/19), nonadecimal subminor seventh (57/32), nonadecimal supermajor seventh

(36/19); as well as the non-essential tones nonadecimal subfourth (95/72) and nonadecimal superfifth

(144/95). The following table depicts the essential tonal resources of nineteen-limit just intonation,

indicating degree, notation, interval, ratio, and cents.

Table 160. Essential nineteen-limit pitch material

DEGREE

NUMBER

NOTE INTERVAL RATIO

(FRACTION)

RATIO

(DECIMAL)

CENTS

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÐÑ

C

DW)

D3)

EW

E3

F2

FV(

G3

AW

A3

BW

B3

C

unison

nonadecimal diatonic semitone

nonadecimal supermajor second

nonadecimal subminor, or overtone minor third (19th harmonic)

nonadecimal supermajor third

nonadecimal subfourth

nonadecimal tritone, or superaugmented fourth

nonadecimal superfifth

nonadecimal subminor sixth

nonadecimal supermajor sixth

nonadecimal subminor seventh (57th harmonic)

nonadecimal supermajor seventh

octave

1/1

19/18

64/57

19/16

24/19

95/72

27/19

144/95

19/12

32/19

57/32

36/19

2/1

1.000000

1.055556

1.122807

1.187500

1.263158

1.319444

1.421053

1.515789

1.583333

1.684211

1.781250

1.894737

2.000000

0.000

93.603

200.532

297.513

404.442

479.917

608.352

720.083

795.558

902.487

999.468

1106.397

1200.000

The Harmonic Consideration 273

Fig. 2

5. 2

, 3, 5

, 19-

limit

pitc

h la

ttice

(no

tatio

n, ra

tio a

nd c

ent v

alue

s)

274 The Harmonic Consideration

The adaptation of Harrison’s pentatonic #4 to nineteen-limit intonation presents a collection of pitches

that include: C, E3, F, G, and B3 (1/1, 24/19, 4/3, 3/2, and 36/19); as well as the complements: C, AW,

G, F, and DW) (2/1, 19/12, 3/2, 4/3, and 19/18).

Ex. 116. Lou Harrison pentatonic #4 (nineteen-limit intonation)

The modulation to A( (presented in the tuning matrix) effectively results in the following transposition of

the pitch series:

A( BW B3 C2( CV( D DV( EW E( F2( FV( G2( GV(

1627

3257

1936

512513

7681

89

608729

1619

6481

128171

1927

10241539

152243

D2 E3(

256285

190243

The following tuning matrix contains the data required to represent the system of intonation chromatically

within the program memory of the Akai S3000XL.

Table 161. 19-limit tuning matrix no. 1 ‘key of A’ (Program 14)

NOTE C2( CV( D DV( E( F2( FV( G2( GV( A( BW B3

KEY C C! D D! E F F! G G! A A! B

CENTS +13 +10 +04 +14 +08 +01 +08 +05 +12 +06 ß01 +06

The root position nineteen-limit major triad is represented by the ratio 38:48:57, identities 19-3-57,

intervals 19/16, 3/2, and 57/32, and the pitches EW, G, and BW; and presents the just major third (5/4)

and just perfect fifth (3/2) with a falsity of ß18.128 and +0.000 cents on each count, which is a

The Harmonic Consideration 275

deviation of a nonadecimal comma (96/95) for the major third. BW4 (466.021Hz), or the nonadecimal

subminor seventh (57/32, or 999.468 cents), presents 0.000 beats between the third harmonic of EW4

(932.041Hz), or the nonadecimal subminor third (19/16), and the second harmonic of BW4

(932.041Hz), and 0.000 beats between the sixth harmonic of EW4 (1864.082Hz) and the fourth

harmonic of BW4 (1864.082Hz); while G4, or the just perfect fifth (3/2), 16.352 beats between the fifth

harmonic of EW4 (1553.402Hz) and the fourth harmonic of G4.

Table 162. The beating characteristics of the nineteen-limit major triad

EW4

(PARTIAL)

FREQUENCY

(HERTZ)

G4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

BW4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

5

6

– – – –

7

– – – –

8

310.680

– – – –

– – – –

621.361

– – – –

932.041

– – – –

1242.721

– – – –

1553.402

1864.082

– – – –

2174.763

– – – –

2485.443

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

– – – –

5

– – – –

6

– – – –

– – – –

392.438

– – – –

– – – –

784.877

– – – –

1177.315

– – – –

– – – –

1569.753

– – – –

1962.192

– – – –

2354.630

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

16.352

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

4

– – – –

– – – –

5

– – – –

– – – –

– – – –

466.021

– – – –

– – – –

932.041

– – – –

– – – –

1398.062

– – – –

1864.082

– – – –

– – – –

2330.103

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

The root position nineteen-limit minor triad is represented by the ratio 16:19:24, identities 1-19-3,

intervals 1/1, 19/16, and 3/2, and the pitches C, EW, and G; and presents the just minor third (6/5) and

just perfect fifth (3/2) with a falsity of +18.128 and +0.000 cents on each count, which is a deviation of

a nonadecimal comma (96/95) for the minor third. The nonadecimal subminor third (19/16) has been

proposed by a number of theorists as a possible substitute for the just minor third (6/5) in the five-limit

minor triad (10:12:15), which yields identities 3, 5, and 15, but no 1 identity. The alternative septimal

subminor third (7/6) in the seven-limit minor triad (6:7:9) presents identities 3, 7, and 9, and therefore

also lacks 1. In striking contrast, the harmonic structure of the nineteen-limit minor triad (16:19:24) allows

for the representation of the 1 identity, which coincides with the root the chord.

276 The Harmonic Consideration

Doty offers the following discussion with regards to the musical significance of the nonadecimal subminor

third:

“However, 19/16 is definitely not a consonance, no more so than is the tempered minor third, and

therefore does not yield a consonant triad. In any case, it appears that a group of tones with relative

frequencies higher than 8 or 9 is likely to produce an ambiguous sensation of periodicity pitch, so that

nothing of practical value is achieved making the root of the chord the fundamental of the harmonic

series.”343

G4, or the just perfect fifth (3/2), presents 0.000 beats between the third harmonic of C4, or the unison

(1/1), and the second harmonic of G4, and 0.000 beats between the sixth harmonic of C4 and the fourth

harmonic of G4; while EW4 (310.680Hz), or the nonadecimal subminor third (19/16), 16.352 beats

between the sixth harmonic of C4 and the fifth harmonic of EW4.

Table 163. The beating characteristics of the nineteen-limit minor triad

C4

(PARTIAL)

FREQUENCY

(HERTZ)

EW4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

G4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

– – – –

5

6

7

– – – –

– – – –

8

261.626

– – – –

– – – –

523.251

– – – –

784.877

– – – –

1046.502

– – – –

– – – –

1308.128

1569.753

1831.379

– – – –

– – – –

2093.005

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

– – – –

5

– – – –

6

– – – –

– – – –

– – – –

310.680

– – – –

– – – –

621.361

– – – –

932.041

– – – –

– – – –

1242.721

– – – –

1553.402

– – – –

1864.082

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

16.352

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

– – – –

4

– – – –

– – – –

5

– – – –

– – – –

– – – –

392.438

– – – –

– – – –

784.877

– – – –

– – – –

1177.315

– – – –

– – – –

1569.753

– – – –

– – – –

1962.192

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

The end of the second system in section G presents a B"("5/omit 3) simultaneous sonority within a 7-1

suspension in the bass – the pitches BW, BW, and E(, or the ratio 19:38:54, with the following harmony

343 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 64-65.

The Harmonic Consideration 277

delineating the triad B"("5)/D – the pitches D, BW, and E(, or the ratio 12:19:27. The concluding

sonorities of the fourth system include a E("5)/G! triad (171:216:304) resolving to an A tonal centre within

a chord inversion and a 7-1 suspension in the bass.

Twenty-Three-Limit Just Intonation

The primary interval of twenty-three-limit just intonation is the twenty-third harmonic, or trivigesimal tritone,

or superaugmented fourth, which is represented by the frequency ratio 23/16 (628.274 cents), and

notated as FX(. The complement, 32/23 (571.726 cents), or trivigesimal subdiminished fifth, is notated

as GY). The notational symbols for the partial chroma ([4] and [5], raising or lowering a tone by 46/45

[one trivigesimal comma], or 38.051 cents) is in reference to the “amount the twenty-third harmonic

exceeds the just tritone” (45/32Ï46/45=23/16). 23/16 is 28.274 cents sharp from the equal

augmented fourth ( 2 2 ), while 32/23, 28.274 flat from the equal diminished fifth ( 2 2 ).344

The process adopted for the creation of twenty-three-limit intonation pitch material involves the

establishment of just minor triads (5:6:10) from the twenty-third harmonics of the fundamental C (1/1), G

(3/2), and F (4/3), which generate the six otonal pitches: D4, E4, FX(, G4, A4(, and B4 (23/20, 23/18,

23/16, 23/15, 69/40, and 23/12); as well as the establishment of complement major triads (4:5:6), which

generate the six utonal pitches: BY), AY, GY), F5, EY), and DY) (40/23, 36/23, 32/23, 30/23, 80/69,

and 24/23).

Fig. 26. Twenty-three-limit otonal pitch generation

Fig. 27. Twenty-three-limit utonal pitch generation

344 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters” 115.

DY) (24/23)

GY) (32/23) BY) (40/23)

GY) (32/23)

CY) (128/69) EY) (80/69) DY) (24/23) F5 (30/23)

AY (36/23)

FX( (23/16) D4 (23/20)

B4 (23/12)

CX( (69/64)A4( (69/40)

FX( (23/16)

G4 (23/15)

E4 (23/18)

B4 (23/12)

278 The Harmonic Consideration

A minor adjustment is also made with regards to A4( (69/40), and due to the fact this pitch does not

represent the simplest ratio available within the twenty-three-limit intonation scheme. A4 (46/27) and EY

(27/23) is adopted as a consequence. The pitch material is then sequentially arranged as:

C DY) D4 EY E4 F FX( GY) G AY A4 BY) B4

11

2324

2023

2327

1823

34

1623

2332

23

2336

2746

2340

1223

F5 G4

2330

1523

The series of intervals presents the trivigesimal diatonic semitone (24/23), trivigesimal supermajor second

(23/20), trivigesimal subminor third (27/23), trivigesimal supermajor third (23/18), just perfect fourth (4/3),

trivigesimal tritone, or superaugmented fourth (23/16), just major fifth (3/2), trivigesimal subdiminished fifth

(32/23), trivigesimal subminor sixth (36/23), trivigesimal supermajor sixth (46/27), trivigesimal subminor

seventh (40/23), trivigesimal supermajor seventh (23/12); as well as the non-essential tones trivigesimal

subfourth (30/23) and trivigesimal subminor sixth (36/23). The following table depicts the essential tonal

resources of twenty-three-limit just intonation, indicating degree, notation, interval, ratio, and cents.

Table 164. Essential twenty-three-limit pitch material

DEGREE

NUMBER

NOTE INTERVAL RATIO

(FRACTION)

RATIO

(DECIMAL)

CENTS

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÐÑ

C

DY)

D4

EY

E4

F5

FX(

G4

AY

A4

BY)

B4

C

unison

trivigesimal diatonic semitone

trivigesimal supermajor second

trivigesimal subminor third

trivigesimal supermajor third

trivigesimal subfourth

trivigesimal tritone, or superaugmented fourth (23rd harmonic)

trivigesimal superfifth

trivigesimal subminor sixth

trivigesimal supermajor sixth

trivigesimal subminor seventh

trivigesimal supermajor seventh

octave

1/1

24/23

23/20

27/23

23/18

30/23

23/16

23/15

36/23

46/27

40/23

23/12

2/1

1.000000

1.043478

1.150000

1.173913

1.277778

1.304348

1.437500

1.533333

1.565217

1.703704

1.739130

1.916667

2.000000

0.000

73.681

241.961

277.591

424.364

459.994

628.274

740.006

775.636

922.409

958.039

1126.319

1200.000

The Harmonic Consideration 279

Fig. 2

8. 2

, 3, 5

, 23-

limit

pitc

h la

ttice

(no

tatio

n, ra

tio a

nd c

ent v

alue

s)

280 The Harmonic Consideration

Twenty-three-limit intonation presents an intonational reinterpretation of Harrison’s pentatonic #4 within

the pitches C, E4, F, G, and B4 (1/1, 23/18, 4/3, 3/2, and 23/12); and the complements: C, BQ, G, F, and

DY) (2/1, 30/17, 3/2, 4/3, and 24/23).

Ex. 117. Lou Harrison pentatonic #4 (twenty-three-limit intonation)

The modulation to A( (presented in the tuning matrixes) effectively results in the following transposition

of the pitch series:

A( BY B4( C5( CX( D DX( EY E( F5( FX( G5 GX(

1627

4681

320621

368729

6469

89

512621

2327

6481

184243

1623

92

135

128207

D5 E4(

368405

160207

The following two tuning matrixes contain the data required to represent the system of intonation

chromatically within the program memory of the Akai S3000XL.

Table 165. 23-limit tuning matrix no. 1 ‘key of A’ (Program 15)

NOTE C5( CX( D DX( E( F5( FX( G5 GX( A( BY B4(

KEY C C! D D! E F F! G G! A A! B

CENTS ß17 +30 +04 +34 +08 ß18 +28 ß36 +32 +06 ß20 +48

The Harmonic Consideration 281

Table 166. 23-limit tuning matrix no. 2 ‘key of A’ (Program 16)

NOTE C5( CX( D DX( E( F5( FX( G5( GX( A( BY B4(

KEY C C! D D! E F F! G G! A A! B

CENTS ß17 +30 +04 +34 +08 ß18 +28 ß15 +32 +06 ß20 +48

The first system of section H features the trivigesimal major triad. The root position twenty-three-limit

major triad is represented by the ratio 18:23:27, identities 9-23-27, intervals 9/8, 23/16, and 27/16, and

the pitches D, FX(, and A(; and presents the just major third (5/4) and just perfect fifth (3/2) with a falsity

of +38.051 and +0.000 cents on each count, which is a deviation of a trivigesimal comma (46/45) for

the major third. A(4, or the Pythagorean major sixth (27/16), presents 0.000 beats between the third

harmonic of D4, or the just major tone (9/8), and the second harmonic of A(4, and 0.000 beats between

the sixth harmonic of D4 and the fourth harmonic of A(4; while FX(4 (376.087Hz), or the trivigesimal

tritone, or superaugmented fourth (23/16), 32.703 beats between the fifth harmonic of D4 (1471.644Hz)

and the fourth harmonic of FX(4 (1504.347Hz).

Table 167. The beating characteristics of the twenty-three-limit major triad

D4

(PARTIAL)

FREQUENCY

(HERTZ)

FX(4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

A(4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

5

6

– – – –

7

– – – –

8

294.329

– – – –

– – – –

588.658

– – – –

882.986

– – – –

1177.315

– – – –

1471.644

1765.973

– – – –

2060.301

– – – –

2354.630

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

– – – –

5

– – – –

6

– – – –

– – – –

376.087

– – – –

– – – –

752.174

– – – –

1128.260

– – – –

– – – –

1504.347

– – – –

1880.434

– – – –

2256.521

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

32.703

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

4

– – – –

– – – –

5

– – – –

– – – –

– – – –

441.493

– – – –

– – – –

882.986

– – – –

– – – –

1324.479

– – – –

1765.973

– – – –

– – – –

2207.466

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

282 The Harmonic Consideration

The root position twenty-three-limit minor triad is represented by the ratio 46:54:69, identities 23-27-69,

intervals 23/16, 27/16, and 69/64, and the pitches FX(, A(, and CX(; and presents the just minor third

(6/5) and just perfect fifth (3/2) with a falsity of ß38.051 and +0.000 cents on each count, which is a

deviation of a trivigesimal comma (46/45) for the minor third. CX(5 (564.130Hz), or the trivigesimal

chromatic semitone (69/64, or 130.229 cents), presents 0.000 beats between the third harmonic of

FX(4 (1128.260Hz), or the trivigesimal superaugmented fourth (23/16), and the second harmonic of

CX(5 (1128.260Hz) and 0.000 beats between the sixth harmonic of FX(4 (2256.521Hz) and the fourth

harmonic of CX(5 (2256.521Hz); while A(4, or the Pythagorean major sixth (27/16), 49.059 beats

between the sixth harmonic of FX(4 and the fifth harmonic of A(4.

Table 168. The beating characteristics of the twenty-three-limit minor triad

FX(4

(PARTIAL)

FREQUENCY

(HERTZ)

A(4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

CX(5

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

– – – –

5

6

7

– – – –

– – – –

8

376.087

– – – –

– – – –

752.174

– – – –

1128.260

– – – –

1504.347

– – – –

– – – –

1880.434

2256.521

2632.607

– – – –

– – – –

3008.694

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

– – – –

5

– – – –

6

– – – –

– – – –

– – – –

441.493

– – – –

– – – –

882.986

– – – –

1324.479

– – – –

– – – –

1765.973

– – – –

2207.466

– – – –

2648.959

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

49.059

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

– – – –

4

– – – –

– – – –

5

– – – –

– – – –

– – – –

564.130

– – – –

– – – –

1128.260

– – – –

– – – –

1692.390

– – – –

– – – –

2256.521

– – – –

– – – –

2820.651

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

The final chord in the second system is the triad B"("5), featuring the pitches BY, E(, and D, or the ratio

144:207:368. The third system requires a minor alteration in the tuning scheme, due to the resulting

dissonant Amin7(omit 5) simultaneous sonority – the pitches A(, C5(, and G5, or the ratio 23:27:40.

Analysis of the harmonic properties of the simultaneous sonority reveals the appropriate trivigesimal

subminor third (27/23) between the bottom two sonorities, yet a grave or small fifth (40/27) between

the upper two sonorities. In order to adhere to the more aesthetically pleasing intervallic properties of

the twenty-three-limit minor seventh tetrad it is then simply a matter of retuning G5 (135/92) up a

The Harmonic Consideration 283

syntonic comma (81/80) to G5( (2187/1472), which essentially forms a 3/2 relationship between the

upper two sonorities, and generates a Amin7(omit 5) simultaneous sonority with the ratio 46:54:81.

Interesting sonorities presented in the fourth system include the final sonorities at the end of the system,

which facilitate a modulation to E major via an Emaj7/D! tetrad – the pitches DX(,E(, GX(, and B4(, or

the ratio 345:360:460:552, moving to a second inversion E major triad – the pitches B4(,E(, and GX(,

or the ratio 207:270:345.

Twenty-Nine-Limit Just Intonation

The primary interval of twenty-nine-limit just intonation is the twenty-ninth harmonic, or grave or small

nonavigesimal neutral seventh, which is represented by the frequency ratio 29/16 (1029.577 cents), and

notated as B_. The complement, 32/29 (170.423 cents), or nonavigesimal acute or large neutral second,

is notated as D7). The notational symbols for the partial chroma ([6] and [7], raising or lowering a tone

by 145/144 [one nonavigesimal comma], or 11.981 cents) is in reference to the “amount the twenty-

ninth harmonic exceeds the acute or large minor seventh” (9/5Ï145/144=29/16). 29/16 is 29.577

cents sharp from the equal minor seventh ( 56 ]2[ ), while 32/29, 29.577 flat from the equal major second

( 12 2 ).345

The process adopted for the creation of twenty-nine-limit intonation pitch material involves the

establishment of just minor triads (5:6:10) from the twenty-ninth harmonics of the fundamental C (1/1), G

(3/2), and F (4/3), which generate the six otonal pitches: D_, E_, F6(, G_, A_, and B_ (87/80, 29/24,

87/64, 29/20, 29/18, and 29/16); as well as the establishment of complement major triads (4:5:6), which

generate the six utonal pitches: B7), A7, G7), F^, E7, and D7) (160/87, 48/28, 128/87, 40/29, 36/29,

and 32/29).

Fig. 29. Twenty-nine-limit otonal pitch generation

Fig. 30. Twenty-nine-limit utonal pitch generation

345 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters” 115.

A7 (48/29)

D7) (32/29) F^ (40/29)

D7) (32/29)

G7) (128/87) B7) (160/87) A7 (48/29) C^ (30/29)

E7 (36/29)

B_ (29/16) G_ (29/20)

E_ (29/24)

F6( (87/64) D_ (87/80)

B_ (29/16)

C_ (29/15)

A_ (29/18)

E_ (29/24)

284 The Harmonic Consideration

A minor adjustment is also made with regards to D_ (87/80), and due to the fact this pitch does not

represent the simplest ratio available within the twenty-nine-limit intonation scheme. D_) (29/27) and

B7 (54/29) is adopted as a consequence. The pitch material is sequentially arranged as:

C D_) D7) E_ E7 F F^ G_ G A_ A7 B_ B7

11

2729

2932

2429

2936

34

2940

2029

23

1829

2948

1629

2954

F6( G7)

6487

87

128

The series of intervals presents the nonavigesimal grave or small neutral second (29/27), nonavigesimal

acute or large neutral second (32/29), nonavigesimal grave or small neutral third (29/24), nonavigesimal

acute or large neutral third (36/29), just perfect fourth (4/3), nonavigesimal tritone, or augmented fourth

(40/29), just perfect fifth (3/2), nonavigesimal subdiminished fifth (29/20), nonavigesimal grave or small

neutral sixth (29/18), nonavigesimal acute or large neutral sixth (48/29), nonavigesimal grave or small

neutral seventh (29/16), nonavigesimal acute or large neutral seventh (54/29); as well as the non-essential

tones nonavigesimal superfourth (87/64) and nonavigesimal subfifth (128/87). The following table

depicts the essential tonal resources of twenty-nine-limit just intonation, indicating degree, notation,

interval, ratio, and cents.

Table 169. Essential twenty-nine-limit pitch material

DEGREE

NUMBER

NOTE INTERVAL RATIO

(FRACTION)

RATIO

(DECIMAL)

CENTS

ÐÑ ÐÒ ÐÓ ÐÔ ÐÕ ÐÖ Ð× ÐØ ÐÙ ÑÐ ÑÑ ÑÒ

ÐÑ

C

D_)

D7)

E_

E7

F6(

F^

G7)

A_

A7

B_

B7

C

unison

nonavigesimal grave or small neutral second

nonavigesimal acute or large neutral second

nonavigesimal grave or small neutral third

nonavigesimal acute or large neutral third

nonavigesimal superfourth (87th harmonic)

nonavigesimal tritone, or augmented fourth

nonavigesimal subfifth

nonavigesimal grave or small neutral sixth

nonavigesimal acute or large neutral sixth

nonavigesimal grave or small neutral seventh (29th harmonic)

nonavigesimal acute or large neutral seventh

octave

1/1

29/27

32/29

29/24

36/29

87/64

40/29

128/87

29/18

48/29

29/16

54/29

2/1

1.000000

1.074074

1.103448

1.208333

1.241379

1.359375

1.379310

1.471264

1.611111

1.655172

1.812500

1.862069

2.000000

0.000

123.712

170.423

327.622

374.333

531.532

556.737

668.468

825.667

872.378

1029.577

1076.288

1200.000

The Harmonic Consideration 285

Fig. 3

1. 2

, 3, 5

, 29-

limit

pitc

h la

ttice

(no

tatio

n, ra

tio a

nd c

ent v

alue

s)

286 The Harmonic Consideration

The adaptation of Harrison’s pentatonic #5 to twenty-nine-limit intonation presents a collection of

pitches that include: C, D7), F, G, and B_ (1/1, 32/29, 4/3, 3/2, and 29/16); as well as the complements:

C, B_, G, F, and D7) (2/1, 29/16, 3/2, 4/3, and 32/29).

Ex. 118. Lou Harrison pentatonic #5 (twenty-nine-limit intonation)

The modulation to E( (presented in the tuning matrixes) effectively results in the following transposition of

the pitch series:

E( F6( F^( G6( G^( A( A^( B_( B( C6( C^( D6( D^(

6481

6487

5881

512783

464729

1627

232405

12802349

128243

256261

232243

20482349

18562187

A6(( B7

40967047

2954

The following tuning matrix contains the data required to represent the system of intonation chromatically

within the program memory of the Akai S3000XL.

Table 170. 29-limit tuning matrix no. 1 ‘key of E’ (Program 17)

NOTE C6( C^( D6( D^( E( F6( F^( G6( G^( A( A^( B(

KEY C C! D D! E F F! G G! A A! B

CENTS +33 ß20 +37 ß16 +08 +32 ß22 +35 ß18 +06 ß35 +10

The first system of section I (bars 128-29) features the nonavigesimal minor and major triads. The root

position twenty-nine-limit major triad is represented by the ratio 58:72:87, identities 29-9-87, intervals

29/16, 9/8, and 87/64, and the pitches B_, D, and F6(; and presents the just major third (5/4) and just

The Harmonic Consideration 287

perfect fifth (3/2) with a falsity of +11.981 and +0.000 cents on each count, which is a deviation of a

nonavigesimal comma (145/144) for the major third. F6(5 (711.295Hz), or the nonavigesimal

superfourth (87/64, or 531.532 cents), presents 0.000 beats between the third harmonic of B_4

(1422.589Hz), or the nonavigesimal grave or small neutral seventh (29/16), and the second harmonic of

F6(5 (1422.589Hz), and 0.000 beats between the sixth harmonic of B_4 (2845.178Hz) and the fourth

harmonic of F6(5 (2845.178Hz); while D5, or the just major tone (9/8), 16.352 beats between the fifth

harmonic of B_4 (2370.982Hz) and the fourth harmonic of D5.

Table 171. The beating characteristics of the twenty-nine-limit major triad

B_4

(PARTIAL)

FREQUENCY

(HERTZ)

D5

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

F6(5

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

5

6

– – – –

7

– – – –

8

474.196

– – – –

– – – –

948.393

– – – –

1422.589

– – – –

1896.785

– – – –

2370.982

2845.178

– – – –

3319.374

– – – –

3793.571

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

– – – –

5

– – – –

6

– – – –

– – – –

588.658

– – – –

– – – –

1177.315

– – – –

1765.973

– – – –

– – – –

2354.630

– – – –

2943.288

– – – –

3531.945

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

16.352

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

4

– – – –

– – – –

5

– – – –

– – – –

– – – –

711.295

– – – –

– – – –

1422.589

– – – –

– – – –

2133.884

– – – –

2845.178

– – – –

– – – –

3556.473

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

The root position twenty-nine-limit minor triad is represented by the ratio 24:29:36, identities 3-29-9,

intervals 3/2, 29/16, and 9/8, and the pitches G, B_, and D; and presents the just minor third (6/5) and

just perfect fifth (3/2) with a falsity of ß11.981 and +0.000 cents on each count, which is a deviation of

a nonavigesimal comma (145/144) for the minor third. D5, or the just major tone (9/8), presents 0.000

beats between the third harmonic of G4, or the just perfect fifth (3/2), and the second harmonic of D5,

and 0.000 beats between the sixth harmonic of G4 and the fourth harmonic of D5; while B_4

(474.196Hz), or the nonavigesimal grave or small neutral seventh (29/16), 16.352 beats between the

sixth harmonic of G4 and the fifth harmonic of B_4.

288 The Harmonic Consideration

Table 172. The beating characteristics of the twenty-nine-limit minor triad

G4

(PARTIAL)

FREQUENCY

(HERTZ)

B_4

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

D5

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

– – – –

5

6

7

– – – –

8

392.438

– – – –

– – – –

784.877

– – – –

1177.315

– – – –

1569.753

– – – –

– – – –

1962.192

2354.630

2747.068

– – – –

3139.507

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

– – – –

5

– – – –

6

– – – –

– – – –

474.196

– – – –

– – – –

948.393

– – – –

1422.589

– – – –

– – – –

1896.785

– – – –

2370.982

– – – –

2845.178

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

16.352

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

– – – –

4

– – – –

5

– – – –

– – – –

– – – –

588.658

– – – –

– – – –

1177.315

– – – –

– – – –

1765.973

– – – –

– – – –

2354.630

– – – –

2943.288

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

The sonorities at the beginning of the second system (bars 133-34) present hexads Emin9(!11)/G,

featuring the pitches E(, G6(, B(, D6(, F^(, and A^( (1392:1682:2088:2523:3072:3840); and

E9(!11)/G!, featuring the pitches E(, G^(, B(, D6(, F^(, and A^( (464:576:696:841:1024:1280). The

simultaneous sonorities in the fourth and fifth systems (bars 141-42) include an Emin9/D pentad with the

pitches E(, B(, D6(, F^(, and G6( (1392:2088:2523:3072:3364).

Time signature change (from quadruple, or double duple time [4/4] to compound quadruple time

[12/8]) is now applied to section I (bar 129); significantly altering not only the rhythmic proportions of the

motivic material yet again, but also expanding the motivic ostinato introduced in section E (thirteen-limit

just intonation. The technique of otonal and utonal harmonic extension (bars 129 and 130 essentially

now outline a minor/major tonality shift) is additionally applied to the two-note motive. Thirty-One-Limit Just Intonation The primary interval of thirty-one-limit just intonation is the thirty-first harmonic, or untrigesimal supermajor

seventh, which is represented by the frequency ratio 31/16 (1145.036 cents), and notated as B8. The

complement, 32/31 (54.964 cents), or untrigesimal diatonic semitone, or Greek enharmonic quarter-tone

is notated as Da). The notational symbols for the partial chroma ([8] and [9], raising or lowering a tone

The Harmonic Consideration 289

by 31/30 [one untrigesimal comma], or 56.767 cents) is in reference to the “amount the thirty-first

harmonic exceeds the just diatonic major seventh” (15/8Ï31/30=31/16). 31/16 is 45.036 cents sharp

from the equal major seventh ( 1112 ]2[ , or approximately 967/512,346 while 32/31, 45.036 cents flat from

the equal minor second ( 12 2 ).347

The process adopted for the creation of thirty-one-limit intonation pitch material involves the

establishment of just minor triads (5:6:10) from the thirty-first harmonics of the fundamental C (1/1), G

(3/2), and F (4/3), which generate the six otonal pitches: D8, E8, F`(, G8, A8, and B8 (93/80, 31/24,

93/64, 31/20, 31/18, and 31/16); as well as the establishment of complement major triads (4:5:6), which

generate the six utonal pitches: Ba), Aa, Ga), F9, Ea, and Da) (160/93, 48/31, 128/93, 40/31, 36/31,

and 32/31).

Fig. 32. Thirty-one-limit otonal pitch generation

Fig. 33. Thirty-one-limit utonal pitch generation

A minor adjustment is also made with regards to D8 (93/80), and due to the fact this pitch does not

represent the simplest ratio available within the thirty-one-limit intonation scheme. D8) (31/27) and Ba)

(54/31) are adopted as a consequence. The pitch material is then sequentially arranged as:

C Da) D8) Ea E8 F F`( Ga) G Aa A8 Ba B8

11

3132

2731

3136

2431

34

6493

93

128

23

3148

1831

3154

1631

F9 G8

3140

2031

346 Daniélou, Tableau Comparatif des Intervalles Musicaux 29.

347 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters” 115.

Aa (48/31)

Da) (32/31) F9 (40/31)

Da) (32/31)

Ga) (128/93) Ba) (160/93) Aa (48/31) C9 (60/31)

Ea (36/31)

B8 (31/16) G8 (31/20)

E8 (31/24)

F`( (93/64) D8 (93/80)

B8 (31/16)

C8 (31/30)

A8 (31/18)

E8 (31/24)

290 The Harmonic Consideration

The series of intervals presents the nonavigesimal grave or small neutral second (29/27), nonavigesimal

acute or large neutral second (32/29), nonavigesimal grave or small neutral third (29/24), nonavigesimal

acute or large neutral third (36/29), just perfect fourth (4/3), nonavigesimal tritone, or augmented fourth

(40/29), just perfect fifth (3/2), nonavigesimal subdiminished fifth (29/20), nonavigesimal grave or small

neutral sixth (29/18), nonavigesimal acute or large neutral sixth (48/29), nonavigesimal grave or small

neutral seventh (29/16), nonavigesimal acute or large neutral seventh (54/29); as well as the non-essential

tones nonavigesimal superfourth (87/64) and nonavigesimal subfifth (128/87). The following table

depicts the essential tonal resources of thirty-one-limit just intonation, indicating degree, notation, interval,

ratio, and cents.

Table 173. Essential thirty-one-limit pitch material

DEGREE

NUMBER

NOTE INTERVAL RATIO

(FRACTION)

RATIO

(DECIMAL)

CENTS

ÐÑ ÐÒ ÐÓ ÐÔ ÐÕ ÐÖ Ð× ÐØ ÐÙ ÑÐ ÑÑ ÑÒ

ÐÑ

C

Da)

D8)

Ea

E8

F9

F`(

G8

Aa

A8

Ba

B8

C

unison

untrigesimal diatonic semitone, or Greek enharmonic quarter-tone

untrigesimal supermajor second

untrigesimal subminor third

untrigesimal supermajor third

untrigesimal subfourth

untrigesimal tritone, or superaugmented fourth (93rd harmonic)

untrigesimal superfifth

untrigesimal subminor sixth

untrigesimal supermajor sixth

untrigesimal subminor seventh

untrigesimal supermajor seventh (31st harmonic)

octave

1/1

32/31

31/27

36/31

31/24

40/31

93/64

31/20

48/31

31/18

54/31

31/16

2/1

1.000000

1.032258

1.148148

1.161290

1.291667

1.290323

1.453125

1.550000

1.548387

1.722222

1.741935

1.937500

2.00000

0.000

54.964

239.171

258.874

443.081

441.278

646.991

758.722

756.919

941.126

960.829

1145.036

1200.000

The thirty-one limit is the source of one of La Monte Young’s tunings for the ‘dream chord’ – a sonority

utilized in a number of his ‘dream’ compositions, such as The Four Dreams of China, and The Second

Dream of the High Tension Line Step-Down Transformer. The essence of the ‘dream chord’ is

represented by the ratio 6:8:9, and identities 3-1-9, which is then extended via the inclusion of an

additional tone generated by the division of the 9/8 interval. Three varieties of the ‘dream chord’ include

the tetrads 24:32:35:36 (3-1-35-9), 42:56:62:63 (21-7-31-63), and 12:16:17:18 (3-1-17-9). The

second example features the untrigesimal ratio of 31/28 (equal to 176.210 cents).348

348 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 64.

The Harmonic Consideration 291

Fig. 3

4. 2

, 3, 5

, 31-

limit

pitc

h la

ttice

(no

tatio

n, ra

tio a

nd c

ent v

alue

s)

292 The Harmonic Consideration

Thirty-one-limit intonation presents an intonational reinterpretation of Harrison’s pentatonic #4 within the

pitches C, D8), F, G, and Ba (1/1, 31/27, 4/3, 3/2, and 54/31); and the complements: C, Ba, A0, F, and

D8) (2/1, 54/31, 3/2, 4/3, and 31/27).

Ex. 119. Lou Harrison pentatonic #5 (thirty-one-limit intonation)

The modulation to E( (presented in the tuning matrixes) effectively results in the following transposition of

the pitch series:

E( F9( F`( G9( G`( A( A`(( Ba B( C9( C`( D9( D`(

6481

6281

6493

496729

512837

1627

40967533

3154

128243

124243

256279

19842187

20482511

A9( B8(

248405

12802511

The following two tuning matrixes contain the data required to represent the system of intonation

chromatically within the program memory of the Akai S3000XL.

Table 174. 31-limit tuning matrix no. 1 ‘key of E’ (Program 18)

NOTE C9( C`( D9( D`( E( F9( F`( G9( G`( A( – – B(

KEY C C! D D! E F F! G G! A A! B

CENTS ß35 +49 ß31 +53 +08 ß37 +47 ß33 +51 +06 +00 +10

The Harmonic Consideration 293

Table 175. 31-limit tuning matrix no. 2 ‘key of E’ (Program 19)

NOTE C9( C`( D9( D`( E( F9( F`( G9( G`( A( – – A`((

KEY C C! D D! E F F! G G! A A! B

CENTS ß35 +49 ß31 +53 +08 ß37 +47 ß33 +51 +06 +00 ß45

The first system of section J (bars 145-46) features the untrigesimal minor and major triads. The root

position thirty-one-limit major triad is represented by the ratio 24:31:36, identities 3-31-9, intervals 3/2,

31/16, and 9/8, and the pitches G, B8, and D; and presents the just major third (5/4) and just perfect fifth

(3/2) with a falsity of +56.767 and +0.000 cents on each count, which is a deviation of a untrigesimal

comma (31/30) for the major third. D5, or the just major tone (9/8), presents 0.000 beats between the

third harmonic of G4, or the just perfect fifth (3/2), and the second harmonic of D5, and 0.000 beats

between the sixth harmonic of G4 and the fourth harmonic of D5; while B84, or the untrigesimal

supermajor seventh (31/16), 65.406 beats between the fifth harmonic of G4 and the fourth harmonic of

B84 (2027.598Hz).

Table 176. The beating characteristics of the thirty-one-limit major triad

G4

(PARTIAL)

FREQUENCY

(HERTZ)

B84

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

D5

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

5

6

– – – –

7

– – – –

8

392.438

– – – –

– – – –

784.877

– – – –

1177.315

– – – –

1569.753

– – – –

1962.192

2354.630

– – – –

2747.068

– – – –

3139.507

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

– – – –

5

– – – –

6

– – – –

– – – –

506.900

– – – –

– – – –

1013.799

– – – –

1520.699

– – – –

– – – –

2027.598

– – – –

2534.498

– – – –

3531.945

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

65.406

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

4

– – – –

– – – –

5

– – – –

– – – –

– – – –

588.658

– – – –

– – – –

1177.315

– – – –

– – – –

1765.973

– – – –

2354.630

– – – –

– – – –

2943.288

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

294 The Harmonic Consideration

The root position thirty-one-limit minor triad is represented by the ratio 62:72:93, identities 31-9-93,

intervals 31/16, 9/8, and 93/64, and the pitches B8, D, and F`(; and presents the just minor third (6/5)

and just perfect fifth (3/2) with a falsity of ß56.767 and +0.000 cents on each count, which is a

deviation of a untrigesimal comma (31/30) for the minor third. F`(5 (760.349Hz), or the untrigesimal

tritone, or superaugmented fourth (93/64), presents 0.000 beats between the third harmonic of B84

(1520.699Hz), or the untrigesimal supermajor seventh (31/16), and the second harmonic of F`(5

(1520.699Hz), and 0.000 beats between the sixth harmonic of B84 (3041.397Hz) and the fourth

harmonic of F`(5 (3041.397Hz); while D5, or the just major tone (9/8), 98.110 beats between the sixth

harmonic of B84 and the fifth harmonic of D5.

Table 177. The beating characteristics of the thirty-one-limit minor triad

B84

(PARTIAL)

FREQUENCY

(HERTZ)

D5

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

F`(5

(PARTIAL)

FREQUENCY

(HERTZ)

BEATS

(HERTZ)

1

– – – –

– – – –

2

– – – –

3

– – – –

4

– – – –

– – – –

5

6

– – – –

7

– – – –

8

506.900

– – – –

– – – –

1013.799

– – – –

1520.699

– – – –

2027.598

– – – –

– – – –

2534.498

3041.397

– – – –

3548.297

– – – –

4055.196

– – – –

1

– – – –

– – – –

2

– – – –

3

– – – –

– – – –

4

– – – –

5

6

– – – –

– – – –

– – – –

– – – –

588.658

– – – –

– – – –

1177.315

– – – –

1765.973

– – – –

– – – –

2354.630

– – – –

2943.288

3531.945

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

98.110

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

1

– – – –

– – – –

2

– – – –

– – – –

3

– – – –

– – – –

4

– – – –

– – – –

5

– – – –

– – – –

– – – –

760.349

– – – –

– – – –

1520.699

– – – –

– – – –

2281.048

– – – –

– – – –

3041.397

– – – –

– – – –

3801.746

– – – –

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

– – – –

0.000

– – – –

– – – –

– – – –

– – – –

The sonorities in the second system (bars 149-50) present hexads Emin9(!11)/G, featuring the pitches E(,

G9(, B(, D9(, F`(, and A`(( (26784:31104:40176:46656:61504:77841); and E9(!11)/G!, featuring

the pitches E(, G`(, B(, D9(, F`(, and A`(( (26784:34596:40176:46656:61504:77841). The

simultaneous sonorities in the beginning of the fifth system (bars 157-58) include an Emin9/B hexad with

the pitches B(, D9(, E(, B(, F`(, and G9( (2511:2916:3348:5022:7688:7776). The work is

concluded with an Emin7 tetrad, with the pitches E(, B(, D9(, and G9( (62:93:108:144, with identities

31-93-27-9), which essentially outline an untrigesimal minor seventh simultaneous sonority harmonically

The Harmonic Consideration 295

constructed from the tonic (1/1), untrigesimal subminor third (36/31), just perfect fifth (3/2), and

untrigesimal subminor seventh (54/31).

Johnston’s Dictum

Johnston offers the following summary of some the characteristics of extended just intonation principles,

and ‘the harmonic consideration’:

“Definitely the affect is unique with each tuning. Each overtone is a unique rasa. The third partial, which

generates perfect fifths and fourths, contributes stability and strength. The fifth partial, which is the third and

sixth, contributes warmth of emotion; ordinary human warmth. The seventh partial creates sensuality, for

example in vernacular music like the blues. The eleventh partial introduces ambiguity, because the intervals

of 12/11 and 11/10, which are the overtone intervals surrounding the eleventh partial, are neutral seconds,

squarely in between major and minor. The 11/9 is a neutral third. The 11/8 is in between a perfect fourth

and augmented fourth. The thirteenth partial has a melancholy, dark quality. Nearly every time I’ve used it, it

has something to do with death, which would square with the meaning of thirteen in numerology. The

seventeenth and nineteenth don’t really bring anything new, because seventeen is almost exactly a

tempered half-step and nineteen is close to a tempered minor third.”349

349 Keislar, Blackwood, Eaton, Harrison, Johnston, Mandelbaum, and Schottstaedt, “Six American Composers on

Nonstandard Tunings,” Perspectives of New Music 198.

296 The Harmonic Consideration

Conclusion

‘Manual’ of Microtonal Composition The thesis, via its presentation of an articulated exposition of three ‘original’ and unique microtonal

composition models individually exploring the expanded tonal resources of Pythagorean intonation,

equal temperament, and just intonation was not only able to demonstrate the uniqueness, as well as

musical potential of such systems, but also mark some directions for further exploration. The works –

Àzàdeh for santñr and tape, Exposiciones for sampled microtonal Schoenhut toy piano, and La Homa

Kanto for harmonically tuned synthesizer quartet – together with the classification of 724 intervals in the

octave presented at the close (the culmination of research directly related to the compositions), stand as

a testament to the limited potential of the vast resources of the microtonal paradigm; merely three grains

of sand in a gargantuan ocean. The study will nevertheless have the capacity to serve as one possible

‘manual’ of microtonal composition, and hopefully inspire the composer of tomorrow to adopt the

notion, if not some of the principles. A Vast Universe of Subtle Intervallic Relationships In conclusion, it may be stated that the virtues of the microtonal paradigm are best summed up by some

of it proponents. What follows is a collection of observations that characterize the general appeal of

microtonal composition, at the same time disclosing the rationale of some composers for the total

abandonment of twelve-tone equally-tempered composition. In a Perspectives of New Music article

entitled Six American Composers on Nonstandard Tunings, Douglas Keislar asks a selection of American

composers actively involved in the application of “microtonal scales or other non-standard tunings” to

describe what they find most interesting about composing with nonstandard tunings. The replies of

Easley Blackwood, John Eaton, Lou Harrison, Ben Johnston, Joel Mandelbaum, and William Schottstaedt

follow:

Blackwood: “The aspect that intrigues me most is finding conventional harmonic progressions, or at least

coherent progressions found by extension of their analogues in the more familiar tunings.”

Eaton: “Microtones permit a greater variety of harmonic and melodic motion, which in opera helps

delineate and define character. My interest in microtones came from three directions. First, I wrote some

of the very first pieces that involved woodwind multiphonics in the early 1960s, and I was intrigued by the

‘out-of-tuneness’ of the multiphonics Secondly, I was interested in cluster music. After a while, though, it

seemed like a lot of sound and fury signifying absolutely nothing. But by changing the tuning between or

within clusters, I could again generate harmonic and melodic motion and have events of some significance

occur. Finally, during this period of my life I was making a living as a jazz musician. With jazz I could get

involved immediately with microtonal intervals.”

Harrison: “The reason for my interest (in nonstandard tunings) is very simple. Real intervals – the ones with

whole number ratios – grab you; they’re beautiful; they draw you into the music; whereas fake intervals like

those of equal temperament don’t do much.”

Johnston: “I love extending my vocabulary and trying to imagine unfamiliar sounds. You can generally

imagine a melodic line, but it’s very difficult to imagine what combination of strange intervals will sound

like.”

Mandelbaum: “I find that extending the consonance to the seventh partial provides a fascinating means of

enrichment. It retains traditional consonance and dissonance, unlike atonal music, which has to abolish the

old in order to arrive at the new.”

Schottstaedt: “I like the unusual sounds, the intense dissonances in particular, such as the squeezed minor

seconds and stretched fourth in Dinosaur Music. I’ve never been much interested in getting cleaner

consonances; beats don’t offend me.”350

Doty offers an interesting discussion on the ‘golden age’ of Western music, and the direct benefits that

may be attributed to the adoption of equal temperament in the common practice era, which unlike the

previous meantone system of tuning, facilitated unrestricted modulation, and hence the development of

complex chromatic harmony. Although in spite of its benevolent influences on harmonic music, twelve-

tone equal temperament is also acknowledged for ultimately leading to its “demise as a vital

compositional style.” Doty offers the following conclusions:

“Twelve-tone equal temperament is a limited and closed system. Once you have modulated around the

so-called circle of fifths, through its twelve major and twelve minor keys, and once you have stacked up

every combination of tones that can reasonably be considered a chord, there is nowhere left to go in

search of new resources. This is essentially where Western composers found themselves at the beginning

of the twentieth century. Everything that could be done with the equally-tempered scale and the

principles of tonal harmony had been tried, and the system was breaking down. This situation led many

composers to the erroneous conclusion that consonance, tonality, and even pitch had been exhausted as

organizing principles. What was really exhausted were merely the very limited resources of the tempered

scale. By substituting twelve equally spaced tones for a vast universe of subtle intervallic relationships, the

composers and theorists of the eighteenth and nineteenth centuries effectively painted Western music into

a corner from which it has not, as yet, extradited itself.”351

350 Keislar, Blackwood, Eaton, Harrison, Johnston, Mandelbaum, and Schottstaedt, “Six American Composers on

Nonstandard Tunings,” Perspectives of New Music 184-85.

351 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 5.

298 Conclusion

In the final report presented at the end of his residency at the University of Wisconsin in Madison

between May 1944 and May 1947 (the setting for the completion of Genesis of a Music – published in

1949 by University of Wisconsin Press), Partch paints an even bleaker picture of the twelve-tone equally-

tempered continuum when he speaks of “Western world’s current three-hundred years of twelve-tone

paralysis.” Ronald V. Wiecki makes the following statement about the legacy of Partch:

“Partch’s work may remain a curiosity, a rare and esoteric treasure, even if the present glimmer of interest in

microtonality grows into a significant aspect of musical practice. But his aesthetic theories, both in their

origins and their expression, provide an interesting look into the sense of failure present in American music

in the early 1920s, a period in which Partch’s attitudes must have taken shape. That Partch found it

necessary to react in such an extreme manner to this sense of crisis is eloquent testimony to its

pervasiveness. How conscious Partch was of this situation needs to be examined further, but it can hardly

be disputed that Partch’s music is passionate, that it contains truths distilled from the American experience,

and that it successfully addresses the essential problem of all American composers of the time – that of

establishing a uniquely American musical practice, one not slavishly or unthinkingly based on an imported

model. Partch’s response to this problem simply attempted to penetrate much more deeply into the

historical roots of the problem, and his solution will remain to inspire others.”352

352 Ronald V. Wiecki, “Relieving ‘12-Tone Paralysis’: Harry Partch in Madison, Wisconsin, 1944-1947” American Music

9.1 (Spring, 1991): 43-60.

Conclusion 299

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Appendix A

Comparative Table of Musical Intervals

Tabl

e 17

8. C

omp

arat

ive

tabl

e of

mus

ical

inte

rval

s Re

lativ

e Pi

tch:

A4=

440H

z / C

4 (m

idd

le C

)=26

1.62

5565

4Hz

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

ÐÑ

ÐÒ

ÐÓ

ÐÔ

ÐÕ

ÐÖ

Ð×

ÐØ

ÐÙ

ÑÐ

ÑÑ

ÑÒ

ÑÓ

ÑÔ

ÑÕ

ÑÖ

Ñ×

ÑØ

ÑÙ

ÒÐ

ÒÑ

uniso

n (1

st h

arm

onic

)

equa

l hun

dre

dth

-sem

itone

one-

elev

enth

syn

toni

c co

mm

a, o

r skh

isma

one-

sixth

syn

toni

c co

mm

a

cycl

ic o

ctav

e (A

) LII

I

one-

fifth

syn

toni

c co

mm

a

two-

nint

h sy

nton

ic c

omm

a

one-

qua

rter s

ynto

nic

com

ma

two-

seve

nth

synt

onic

com

ma

one-

third

syn

toni

c co

mm

a

one-

half

synt

onic

com

ma

nona

vige

simal

com

ma

equa

l six

teen

th-to

ne

thre

e-q

uarte

r syn

toni

c co

mm

a

equa

l tw

elfth

-tone

nona

dec

imal

com

ma

subd

imin

ished

sec

ond

, or d

iask

hism

a

synt

onic

com

ma

53-e

t syn

toni

c co

mm

a

Pyth

agor

ean

com

ma

(A)

XII

equa

l eig

hth-

tone

1/1

1200

2, o

r ap

pro

xim

atel

y 17

31/1

730

118081

, or 3

2805

/327

68

68081

3õó/2

øô 5

8081

4.5

8081

48081

3.5

8081

38081

28081

145/

144

962

1.33

3333

8081

722

96/9

5

2048

/202

5

81/8

0 53

2

3ñò/2

ñù, o

r 531

441/

5242

88

482

1.00

0000

1.00

0578

1.00

1130

1.00

2073

1.00

2090

1.00

2488

1.00

2764

1.00

3110

1.00

3556

1.00

4149

1.00

6231

1.00

6944

1.00

7246

1.00

9360

1.00

9674

1.01

0526

1.01

1358

1.01

2500

1.01

3164

1.01

3643

1.01

4545

261.

626

261.

777

261.

921

262.

168

262.

172

262.

276

262.

349

262.

439

262.

556

262.

711

263.

256

263.

442

263.

521

264.

074

264.

156

264.

380

264.

597

264.

896

265.

070

265.

195

265.

431

0.00

0

1.00

0

1.95

5

3.58

4

3.61

5

4.30

1

4.77

9

5.37

7

6.14

5

7.16

9

10.7

53

11.9

81

12.5

00

16.1

30

16.6

67

18.1

28

19.5

53

21.5

06

22.6

42

23.4

60

25.0

00

312 Appendices

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

ÒÒ

ÒÓ

ÒÔ

ÒÕ

ÒÖ

Ò×

ÒØ

ÒÙ

ÓÐ

ÓÑ

ÓÒ

ÓÓ

ÓÔ

ÓÕ

ÓÖ

Ó×

ÓØ

ÓÙ

ÔÐ

ÔÑ

ÔÒ

ÔÓ

trid

ecim

al c

omm

a (6

5th

harm

onic

)

one

and

one

-qua

rter s

ynto

nic

com

ma

min

imal

die

sis

43-e

t dim

inish

ed s

econ

d

grav

e or

sm

all d

iesis

one

and

one

-hal

f syn

toni

c co

mm

a

equa

l six

th-to

ne

sep

tend

ecim

al c

omm

a

one

and

thre

e-q

uarte

r syn

toni

c co

mm

a

trivi

gesim

al c

omm

a

31-e

t sup

eroc

tave

, or d

imin

ished

sec

ond

und

ecim

al g

rave

or s

mal

l chr

omat

ic s

emito

ne

equa

l fift

h-to

ne

dim

inish

ed s

econ

d, o

r gre

at d

iesis

two

synt

onic

com

mas

, or M

athi

eu s

uper

die

sis

53-e

t gre

at d

iesis

grea

t die

sis (A

) XX

IV

two

and

one

-qua

rter s

ynto

nic

com

ma

sep

timal

com

ma

equa

l qua

rter-t

one

23-e

t Gre

ek e

nhar

mon

ic o

r sep

timal

qua

rter-t

one

und

ecim

al c

omm

a (3

3rd

har

mon

ic)

65/6

4

0.8

8081

2000

0/19

683

432

3125

/307

2

0.66

6667

8081

362

51/5

0

0.57

1429

8081

46/4

5

312

45/4

4

302

128/

125

0.5

8081, o

r 656

1/64

00

253

)2

(

3 òô/2

óø 0.

4444

448081

36/3

5

242

, or a

ppro

xim

atel

y 52

7/51

2 23

2

33/3

2

1.01

5625

1.01

5649

1.01

6105

1.01

6250

1.01

7253

1.01

8808

1.01

9441

1.02

0000

1.02

1977

1.02

2222

1.02

2611

1.02

2727

1.02

3374

1.02

4000

1.02

5156

1.02

6502

1.02

7473

1.02

8345

1.02

8571

1.02

9302

1.03

0596

1.03

1250

265.

713

265.

720

265.

839

265.

877

266.

139

266.

546

266.

712

266.

858

267.

375

267.

439

267.

541

267.

572

267.

741

267.

905

268.

207

268.

559

268.

813

269.

041

269.

101

269.

292

269.

630

269.

801

26.8

41

26.8

83

27.6

60

27.9

07

29.6

14

32.2

59

33.3

33

34.2

83

37.6

36

38.0

51

38.7

10

38.9

06

40.0

00

41.0

59

43.0

13

45.2

83

46.9

20

48.3

89

48.7

70

50.0

00

52.1

74

53.2

73

Appendices 313

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

ÔÔ

ÔÕ

ÔÖ

Ô×

ÔØ

ÔÙ

ÕÐ

ÕÑ

ÕÒ

ÕÓ

ÕÔ

ÕÕ

ÕÖ

Õ×

ÕØ

ÕÙ

ÖÐ

ÖÑ

ÖÒ

ÖÓ

ÖÔ

ÖÕ

two

and

one

-hal

f syn

toni

c co

mm

a

22-e

t Gre

ek e

nhar

mon

ic o

r sep

timal

qua

rter-t

one

untri

gesim

al d

iato

nic

sem

itone

, or G

reek

enh

arm

onic

qua

rter-t

one

43-e

t dou

ble

augm

ente

d s

even

th

Gre

ek e

nhar

mon

ic q

uarte

r-ton

e, o

r unt

riges

imal

com

ma

21-e

t Gre

ek e

nhar

mon

ic o

r sep

timal

qua

rter-t

one

two

and

thre

e-q

uarte

r syn

toni

c co

mm

a

20-e

t Gre

ek e

nhar

mon

ic o

r sep

timal

qua

rter-t

one

19-e

t jus

t dia

toni

c se

mito

ne, o

r maj

or h

alf-t

one

thre

e sy

nton

ic c

omm

as

trid

ecim

al g

rave

or s

mal

l chr

omat

ic s

emito

ne

18-e

t gra

ve o

r sm

all c

hrom

atic

sem

itone

, or e

qua

l thi

rd-to

ne

Pyth

agor

ean

dou

ble

dim

inish

ed th

ird

53-e

t gra

ve o

r sm

all c

hrom

atic

sem

itone

, or m

inor

hal

f-ton

e

thre

e an

d o

ne-q

uarte

r syn

toni

c co

mm

a

cycl

ic g

rave

or s

mal

l chr

omat

ic s

emito

ne, o

r min

or h

alf-t

one

(A)

XXXV

I

17-e

t gra

ve o

r sm

all c

hrom

atic

sem

itone

, or m

inor

hal

f-ton

e

grav

e or

sm

all j

ust c

hrom

atic

sem

itone

, or m

inor

hal

f-ton

e

trivi

gesim

al d

iato

nic

sem

itone

16-e

t gra

ve o

r sm

all c

hrom

atic

sem

itone

, or m

inor

hal

f-ton

e

thre

e an

d o

ne-h

alf s

ynto

nic

com

ma

mea

nton

e ch

rom

atic

sem

itone

, or m

inor

hal

f-ton

e (A

) V

II4

3 1ß

0.4

8081

222

32/3

1 243

)2

(

31/3

0

212

0.36

3636

8081

202

19

2

0.33

3333

8081, o

r 531

441/

5120

00

27/2

6

182

1342

1772

8/12

9140

163

353

)2

(

0.30

7692

8081

3 óö/2

õ÷ 17

2

25/2

4

24/2

3

162

0.28

5714

8081

2187

/204

8×0.

5714

298180

1.03

1544

1.03

2008

1.03

2258

1.03

2765

1.03

3333

1.03

3558

1.03

4752

1.03

5265

1.03

7155

1.03

7971

1.03

8462

1.03

9259

1.03

9318

1.04

0015

1.04

1199

1.04

1491

1.04

1616

1.04

1667

1.04

3478

1.04

4274

1.04

4438

1.04

4907

269.

878

270.

000

270.

065

270.

198

270.

346

270.

405

270.

718

270.

852

271.

346

271.

560

271.

688

271.

897

271.

912

272.

094

272.

404

272.

481

272.

513

272.

527

273.

001

273.

209

273.

252

273.

374

53.7

66

54.5

45

54.9

64

55.8

14

56.7

67

57.1

43

59.1

42

60.0

00

63.1

58

64.5

19

65.3

37

66.6

67

66.7

65

67.9

25

69.8

95

70.3

80

70.5

88

70.6

72

73.6

81

75.0

00

75.2

72

76.0

49

314 Appendices

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

ÖÖ

Ö×

ÖØ

ÖÙ

×Ð

×Ñ

×Ò

×Ó

×Ô

×Õ

×Ö

××

×Ø

×Ù

ØÐ

ØÑ

ØÒ

ØÓ

ØÔ

ØÕ

ØÖ

Ø×

31-e

t aug

men

ted

oct

ave,

or s

ubm

inor

sec

ond

67th

har

mon

ic

15-e

t gra

ve o

r sm

all c

hrom

atic

sem

itone

, or m

inor

hal

f-ton

e

nona

vige

simal

gra

ve o

r sm

all c

hrom

atic

sem

itone

thre

e an

d th

ree-

qua

rter s

ynto

nic

com

ma

43-e

t chr

omat

ic s

emito

ne, o

r min

or h

alf-t

one

sep

timal

dia

toni

c se

mito

ne

14-e

t Pyt

hago

rean

lim

ma

Pyth

agor

ean

limm

a, o

r dia

toni

c se

mito

ne (D

) V

53-e

t Pyt

hago

rean

lim

ma

acut

e or

larg

e Py

thag

orea

n lim

ma

13-e

t Pyt

hago

rean

lim

ma

nona

dec

imal

dia

toni

c se

mito

ne

cycl

ic P

ytha

gore

an li

mm

a (A

) XL

VIII

sep

tend

ecim

al d

iato

nic

sem

itone

equa

l sem

itone

23-e

t jus

t dia

toni

c se

mito

ne, o

r maj

or h

alf-t

one

sept

end

ecim

al c

hrom

atic

sem

itone

(17t

h ha

rmon

ic)

11-e

t jus

t dia

toni

c se

mito

ne, o

r maj

or h

alf-t

one

nona

deci

mal

chr

omat

ic s

emito

ne

43-e

t min

or s

econ

d

just

dia

toni

c se

mito

ne, o

r maj

or h

alf-t

one

231

)2

(

67/6

4 15

2

243/

232

0.26

6667

8081

343

)2

(

21/2

0

142

256/

243 4

53)

2(

135/

128

132

19/1

8

3 ôø/2

÷ö

18/1

7

122

, or a

ppro

xim

atel

y 10

24/9

67

223

)2

(

17/1

6

112

81/7

6 443

)2

(

16/1

5

1.04

5734

1.04

6875

1.04

7294

1.04

7414

1.04

7687

1.04

9547

1.05

0000

1.05

0757

1.05

3498

1.05

3705

1.05

4688

1.05

4766

1.05

5556

1.05

5700

1.05

8824

1.05

9463

1.06

2127

1.06

2500

1.06

5041

1.06

5789

1.06

6603

1.06

6667

273.

591

273.

889

273.

999

274.

030

274.

102

274.

588

274.

707

274.

905

275.

622

275.

676

275.

933

275.

954

276.

160

276.

198

277.

015

277.

183

277.

880

277.

977

278.

642

278.

838

279.

051

279.

067

77.4

19

79.0

70

80.0

00

80.1

98

80.6

49

83.7

21

84.4

67

85.7

14

90.2

25

90.5

66

92.1

79

92.3

08

93.6

03

93.8

40

98.9

55

100.

000

104.

348

104.

955

109.

091

110.

307

111.

628

111.

731

Appendices 315

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

ØØ

ØÙ

ÙÐ

ÙÑ

ÙÒ

ÙÓ

ÙÔ

ÙÕ

ÙÖ

Ù×

ÙØ

ÙÙ

ÑÐÐ

ÑÐÑ

ÑÐÒ

ÑÐÓ

ÑÐÔ

ÑÐÕ

ÑÐÖ

ÑÐ×

ÑÐØ

ÑÐÙ

53-e

t jus

t dia

toni

c se

mito

ne, o

r maj

or h

alf-t

one

Pyth

agor

ean

apot

ome,

or c

hrom

atic

sem

itone

(A)

VII

21-e

t jus

t dia

toni

c se

mito

ne, o

r maj

or h

alf-t

one

31-e

t sup

erau

gmen

ted

oct

ave,

or m

inor

sec

ond

mea

nton

e m

inor

sec

ond

(D)

V41

1�

sep

timal

chr

omat

ic s

emito

ne

10-e

t jus

t dia

toni

c se

mito

ne, o

r maj

or h

alf-t

one

nona

vige

simal

gra

ve o

r sm

all n

eutra

l sec

ond

19-e

t gre

at li

mm

a, o

r lar

ge h

alf-t

one

trivi

gesim

al c

hrom

atic

sem

itone

(69

th h

arm

onic

)

grea

t lim

ma,

acu

te o

r lar

ge h

alf-t

one

9-et

gre

at li

mm

a, o

r lar

ge h

alf-t

one

53-e

t gre

at li

mm

a, a

cute

or l

arge

hal

f-ton

e

cycl

ic g

reat

lim

ma,

acu

te o

r lar

ge h

alf-t

one

(A)

XIX

trid

ecim

al g

rave

or s

mal

l neu

tral s

econ

d

43-e

t dou

ble

dim

inish

ed th

ird

17-e

t thr

ee-q

uarte

r-ton

e

thre

e-q

uarte

r-ton

e

untri

gesim

al c

hrom

atic

sem

itone

equa

l thr

ee-q

uarte

r-ton

e

und

ecim

al g

rave

or s

mal

l neu

tral s

econ

d

mea

nton

e d

oubl

e au

gmen

ted

oct

ave

(A)

XIV

213

ß

553

)2

(

3 ÷/2

ññ, o

r 218

7/20

48

221

)2

(

331

)2

(

256/

243×

0.8

8081

15/1

4 10

2

29/2

7 219

)2

(

69/6

4

27/2

5 9

2 6

53)

2(

3 ñù/2

óð

13/1

2 543

)2

(

217

)2

(

135/

124

279/

256

82

, or a

ppro

xim

atel

y 10

24/9

39

12/1

1

4782

969/

4194

304×

0.28

5714

8180

1.06

7577

1.06

7871

1.06

8242

1.06

9380

1.06

9984

1.07

1429

1.07

1773

1.07

4074

1.07

5691

1.07

8125

1.08

0000

1.08

0060

1.08

1630

1.08

2440

1.08

3333

1.08

3936

1.08

4964

1.08

8710

1.08

9844

1.09

0508

1.09

0909

1.09

1830

279.

305

279.

382

279.

479

279.

777

279.

935

280.

313

280.

403

281.

005

281.

428

282.

065

282.

556

282.

571

282.

897

283.

194

283.

428

283.

585

283.

854

284.

834

285.

131

285.

305

285.

410

285.

651

113.

208

113.

685

114.

286

116.

129

117.

108

119.

443

120.

000

123.

712

126.

316

130.

229

133.

238

133.

333

135.

849

137.

145

138.

573

139.

535

147.

143

141.

176

148.

946

150.

000

150.

637

152.

098

316 Appendices

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

ÑÑÐ

ÑÑÑ

ÑÑÒ

ÑÑÓ

ÑÑÔ

ÑÑÕ

ÑÑÖ

ÑÑ×

ÑÑØ

ÑÑÙ

ÑÒÐ

ÑÒÑ

ÑÒÒ

ÑÒÓ

ÑÒÔ

ÑÒÕ

ÑÒÖ

ÑÒ×

ÑÒØ

ÑÒÙ

ÑÓÐ

ÑÓÑ

31-e

t dou

ble

augm

ente

d o

ctav

e, o

r neu

tral s

econ

d

sep

timal

neu

tral s

econ

d (3

5th

harm

onic

)

23-e

t gra

ve o

r sm

all t

one

53-e

t gra

ve o

r sm

all t

one

15-e

t gra

ve o

r sm

all t

one

cycl

ic g

rave

or s

mal

l ton

e (A

) XX

XI

grav

e or

sm

all t

one

acut

e or

larg

e d

oubl

e au

gmen

ted

oct

ave

22-e

t gra

ve o

r sm

all t

one

und

ecim

al a

cute

or l

arge

neu

tral s

econ

d

43-e

t dou

ble

augm

ente

d o

ctav

e

nona

vige

simal

acu

te o

r lar

ge n

eutra

l sec

ond

7-et

gra

ve o

r sm

all t

one

trid

ecim

al a

cute

or l

arge

neu

tral s

econ

d

71st

har

mon

ic

20-e

t jus

t min

or to

ne

Pyth

agor

ean

dim

inish

ed th

ird (D

) X

53-e

t jus

t min

or to

ne

just

min

or to

ne

cycl

ic m

inor

tone

(A)

XLIII

acut

e or

larg

e d

oubl

e su

per

augm

ente

d o

ctav

e

13-e

t jus

t min

or to

ne

431

)2

(

35/3

2 323

)2

(

753

)2

(

215

)2

(

3 óñ/2

ôù

800/

729

1125

/102

4 3

22)

2(

11/1

0 643

)2

(

32/2

9

72

72/6

5

71/6

4 320

)2

(

6553

6/59

049

853

)2

(

10/9

3 ôó/2

öø

1822

5/16

384

213

)2

(

1.09

3560

1.09

3750

1.09

4624

1.09

5869

1.09

6825

1.09

7208

1.09

7394

1.09

8633

1.09

9131

1.10

0000

1.10

1550

1.10

3448

1.10

4090

1.10

7692

1.10

9375

1.10

9569

1.10

9858

1.11

0295

1.11

1111

1.11

2178

1.11

2366

1.11

2531

286.

103

286.

153

286.

382

286.

707

286.

957

287.

058

287.

106

287.

430

287.

561

287.

788

288.

194

288.

690

288.

858

289.

801

290.

241

290.

292

290.

367

290.

482

290.

695

290.

974

291.

023

291.

067

154.

839

155.

140

156.

522

158.

491

160.

000

160.

605

160.

897

162.

851

163.

636

165.

004

167.

442

170.

423

171.

429

177.

069

179.

697

180.

000

180.

450

181.

132

182.

404

184.

065

184.

357

184.

616

Appendices 317

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

ÑÓÒ

ÑÓÓ

ÑÓÔ

ÑÓÕ

ÑÓÖ

ÑÓ×

ÑÓØ

ÑÓÙ

ÑÔÐ

ÑÔÑ

ÑÔÒ

ÑÔÓ

ÑÔÔ

ÑÔÕ

ÑÔÖ

ÑÔ×

ÑÔØ

ÑÔÙ

ÑÕÐ

ÑÕÑ

ÑÕÒ

ÑÕÓ

19-e

t jus

t min

or to

ne

third

-com

ma

mea

nton

e m

ajor

tone

(A)

II3

2ß

two-

seve

nth-

com

ma

mea

nton

e m

ajor

tone

(A)

II7

4ß

mea

nton

e m

ajor

tone

(A)

II21

ß

31-e

t jus

t and

Pyt

hago

rean

maj

or to

ne

two-

nint

h-co

mm

a m

eant

one

maj

or to

ne (A

) II

94

ß

fifth

-com

ma

mea

nton

e m

ajor

tone

(A)

II5

2ß

43-e

t jus

t and

Pyt

hago

rean

maj

or to

ne

sixth

-com

ma

mea

nton

e m

ajor

tone

(A)

II6

2ß

equa

l ton

e

nona

dec

imal

sup

erm

ajor

sec

ond

65-e

t jus

t and

Pyt

hago

rean

maj

or to

ne

118-

et ju

st a

nd P

ytha

gore

an m

ajor

tone

53-e

t jus

t and

Pyt

hago

rean

maj

or to

ne

just

and

Pyt

hago

rean

maj

or to

ne (A

) II (

9th

harm

onic

)

41-e

t jus

t and

Pyt

hago

rean

maj

or to

ne

140-

et ju

st a

nd P

ytha

gore

an m

ajor

tone

99-e

t jus

t and

Pyt

hago

rean

maj

or to

ne

87-e

t jus

t and

Pyt

hago

rean

maj

or to

ne

55th

cyc

lic fi

fth (A

) LV

23-e

t jus

t and

Pyt

hago

rean

maj

or to

ne

17-e

t jus

t and

Pyt

hago

rean

maj

or to

ne

319

)2

(

9/8×

1.5

8180

9/8×

1.75

8180

9/8×

28180

5

31)

2(

9/8×

2.25

8180

9/8×

2.5

8180

743

)2

(

9/8×

38180

6

2, o

r ap

pro

xim

atel

y 55

/49

64/5

7 1165

)2

(

2011

8)

2(

9

53)

2(

9/8

741

)2

(

2414

0)

2(

17

99)

2(

15

87)

2(

3 õõ/2

ø÷ 423

)2

(

317

)2

(

1.11

5658

1.11

5722

1.11

7042

1.11

8034

1.11

8287

1.11

8806

1.11

9424

1.11

9450

1.12

0351

1.12

2462

1.12

2807

1.12

4459

1.12

4662

1.12

4911

1.12

5000

1.12

5629

1.12

6173

1.12

6398

1.12

6942

1.12

7352

1.12

8114

1.13

0116

291.

885

291.

901

292.

247

292.

506

292.

572

292.

708

292.

870

292.

877

293.

113

293.

665

293.

755

294.

187

294.

240

294.

306

294.

329

294.

493

294.

636

294.

695

294.

837

294.

944

295.

143

295.

667

189.

474

189.

572

191.

621

193.

157

193.

548

194.

352

195.

307

195.

349

196.

741

200.

000

200.

532

203.

077

203.

390

203.

774

203.

910

204.

878

205.

714

206.

061

206.

897

207.

525

208.

696

211.

765

318 Appendices

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

ÑÕÔ

ÑÕÕ

ÑÕÖ

ÑÕ×

ÑÕØ

ÑÕÙ

ÑÖÐ

ÑÖÑ

ÑÖÒ

ÑÖÓ

ÑÖÔ

ÑÖÕ

ÑÖÖ

ÑÖ×

ÑÖØ

ÑÖÙ

Ñ×Ð

Ñ×Ñ

Ñ×Ò

Ñ×Ó

Ñ×Ô

Ñ×Õ

sep

tend

ecim

al s

uper

maj

or s

econ

d

11-e

t acu

te o

r lar

ge to

ne

43-e

t dim

inish

ed th

ird

subd

imin

ished

third

, or a

cute

or l

arge

tone

16-e

t acu

te o

r lar

ge to

ne

acut

e or

larg

e to

ne

53-e

t acu

te o

r lar

ge to

ne

Pyth

agor

ean

dou

ble

augm

ente

d o

ctav

e, o

r cyc

lic a

cute

or l

arge

tone

(A)

XIV

73rd

har

mon

ic

21-e

t acu

te o

r lar

ge to

ne

sep

timal

sup

erm

ajor

sec

ond

31-e

t sup

erm

ajor

sec

ond

, or d

imin

ished

third

mea

nton

e d

imin

ished

third

(D)

X21

2�

untri

gesim

al s

uper

maj

or s

econ

d

5-et

sup

erm

ajor

sec

ond

trivi

gesim

al s

uper

maj

or s

econ

d

und

ecim

al g

rave

or s

mal

l aug

men

ted

sec

ond

sup

erm

ajor

sec

ond

dim

inish

ed th

ird

trid

ecim

al g

rave

or s

mal

l aug

men

ted

sec

ond

53-e

t sup

erm

ajor

sec

ond

five

equa

l qua

rter-t

ones

17/1

5 211

)2

(

843

)2

(

256/

225 3

16)

2(

729/

640

1053

)2

(

3 ñô/2

òò, o

r 478

2969

/419

4304

73/6

4 421

)2

(

8/7

631

)2

(

6553

6/59

049×

0.4

8081

31/2

7 5

2

23/2

0

405/

352

5904

9/51

300

144/

125

15/1

3 1153

)2

(

524

)2

(, o

r ap

pro

xim

atel

y 52

/45

1.13

3333

1.13

4313

1.13

7642

1.13

7778

1.13

8789

1.13

9063

1.13

9720

1.14

0349

1.14

0625

1.14

1140

1.14

2857

1.14

3573

1.14

4867

1.14

8148

1.14

8698

1.15

0000

1.15

0568

1.15

1053

1.15

2000

1.15

3846

1.15

4723

1.15

5353

296.

509

296.

765

297.

636

297.

672

297.

936

298.

008

298.

180

298.

344

298.

417

298.

551

299.

001

299.

188

299.

526

300.

385

300.

529

300.

869

301.

018

301.

145

301.

393

301.

876

302.

105

302.

270

216.

687

218.

182

223.

256

223.

463

225.

000

225.

416

226.

415

227.

370

227.

789

228.

571

231.

174

232.

258

234.

216

239.

171

240.

000

241.

961

242.

816

243.

545

244.

969

247.

741

249.

057

250.

000

Appendices 319

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

Ñ×Ö

Ñ××

Ñ×Ø

Ñ×Ù

ÑØÐ

ÑØÑ

ÑØÒ

ÑØÓ

ÑØÔ

ÑØÕ

ÑØÖ

ÑØ×

ÑØØ

ÑØÙ

ÑÙÐ

ÑÙÑ

ÑÙÒ

ÑÙÓ

ÑÙÔ

ÑÙÕ

ÑÙÖ

ÑÙ×

cycl

ic s

uper

maj

or s

econ

d (A

) XX

VI

43-e

t trip

le a

ugm

ente

d o

ctav

e

37th

har

mon

ic

19-e

t fiv

e q

uarte

r-ton

es

14-e

t fiv

e q

uarte

r-ton

es

untri

gesim

al s

ubm

inor

third

five

qua

rter-t

ones

23-e

t fiv

e q

uarte

r-ton

es

nona

vige

simal

gra

ve o

r sm

all a

ugm

ente

d s

econ

d

9-et

five

qua

rter-t

ones

sep

timal

sub

min

or th

ird

mea

nton

e au

gmen

ted

sec

ond

(A)

IX41

2ß

31-e

t aug

men

ted

sec

ond

, or s

ubm

inor

third

53-e

t aug

men

ted

seco

nd

22-e

t aug

men

ted

seco

nd

cycl

ic a

ugm

ente

d se

cond

(A)

XXXV

III

augm

ente

d se

cond

(75t

h ha

rmon

ic)

13-e

t aug

men

ted

seco

nd

trivi

gesim

al s

ubm

inor

third

43-e

t aug

men

ted

seco

nd

sep

tend

ecim

al s

ubm

inor

third

17-e

t aug

men

ted

seco

nd

3 òö/2

ôñ 943

)2

(

37/3

2 419

)2

(

314

)2

(

36/3

1

93/8

0 523

)2

(

135/

116 2

9)

2(

7/6

1968

3/16

384×

0.44

4444

8180

731

)2

(

1253

)2

(

522

)2

(

3 óø/2

öð

75/6

4 313

)2

(

27/2

3 1043

)2

(

20/1

7 417

)2

(

1.15

5907

1.15

6129

1.15

6250

1.15

7110

1.16

0129

1.16

1290

1.16

2500

1.16

2629

1.16

3793

1.16

6529

1.16

6667

1.16

8241

1.16

9431

1.16

9924

1.17

0620

1.17

1677

1.17

1875

1.17

3460

1.17

3913

1.17

4916

1.17

6471

1.17

7147

302.

415

302.

473

302.

505

302.

730

303.

520

303.

823

304.

140

304.

174

304.

478

305.

194

305.

230

305.

642

305.

953

306.

082

306.

264

306.

541

306.

592

307.

007

307.

126

307.

388

307.

795

307.

972

250.

830

251.

163

251.

344

252.

632

257.

143

258.

874

260.

677

260.

870

262.

602

266.

667

266.

871

269.

206

270.

968

271.

698

272.

727

274.

290

274.

582

276.

923

277.

591

279.

070

281.

358

282.

353

320 Appendices

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

ÑÙØ

ÑÙÙ

ÒÐÐ

ÒÐÑ

ÒÐÒ

ÒÐÓ

ÒÐÔ

ÒÐÕ

ÒÐÖ

ÒÐ×

ÒÐØ

ÒÐÙ

ÒÑÐ

ÒÑÑ

ÒÑÒ

ÒÑÓ

ÒÑÔ

ÒÑÕ

ÒÑÖ

ÒÑ×

ÒÑØ

ÒÑÙ

21-e

t jus

t min

or th

ird

nona

dec

imal

sup

erau

gmen

ted

sec

ond

Pyth

agor

ean

min

or th

ird, o

r trih

emito

ne (D

) III

53-e

t Pyt

hago

rean

min

or th

ird, o

r trih

emito

ne

acut

e or

larg

e au

gmen

ted

sec

ond

nona

dec

imal

sub

min

or, o

r ove

rtone

min

or th

ird (1

9th

harm

onic

)

cycl

ic P

ytha

gore

an m

inor

third

, or t

rihem

itone

(A)

L

equa

l min

or th

ird

sixth

-com

ma

mea

nton

e m

inor

third

(D)

III6

3�

43-e

t jus

t min

or th

ird

fifth

-com

ma

mea

nton

e m

inor

third

(D)

III5

3�

two-

nint

h-co

mm

a m

eant

one

min

or th

ird (D

) III

96

�

sep

tend

ecim

al s

uper

augm

ente

d s

econ

d

31-e

t sup

erau

gmen

ted

sec

ond

, or j

ust m

inor

third

mea

nton

e m

inor

third

(D)

III4

33�

two-

seve

nth-

com

ma

mea

nton

e m

inor

third

(D)

III7

6�

trivi

gesim

al s

uper

augm

ente

d s

econ

d

23-e

t jus

t min

or th

ird

65-e

t jus

t min

or th

ird

99-e

t jus

t min

or th

ird

118-

et ju

st m

inor

third

third

-com

ma

mea

nton

e an

d ju

st m

inor

third

(D)

III1�

521

)2

(

45/3

8

32/2

7 1353

)2

(

1215

/102

4

19/1

6

3 õð/2

÷ù 4

2, o

r ap

pro

xim

atel

y 44

/37

32/2

7×2

8081

1143

)2

(

32/2

7×1.

6666

678081

32/2

7×1.

58081

153/

128 8

31)

2(

32/2

7×1.

3333

338081

32/2

7×1.

1666

678081

115/

96

623

)2

(

1765

)2

(

2699

)2

(

3111

8)

2(

32/2

7×81

/80,

or 6

/5

1.17

9434

1.18

4211

1.18

5185

1.18

5325

1.18

6523

1.18

7500

1.18

7663

1.18

9207

1.19

2570

1.19

4009

1.19

4052

1.19

5041

1.19

5313

1.19

5873

1.19

6279

1.19

7872

1.19

7917

1.19

8201

1.19

8756

1.19

9661

1.19

9732

1.20

0000

308.

570

309.

820

310.

075

310.

111

310.

425

310.

680

310.

723

311.

127

312.

007

312.

383

312.

395

312.

653

312.

724

312.

871

312.

977

313.

395

313.

406

313.

480

313.

625

313.

862

313.

880

313.

951

285.

714

292.

711

294.

135

294.

340

296.

089

297.

513

297.

750

300.

000

304.

888

306.

977

307.

039

308.

473

308.

865

309.

677

310.

265

312.

569

312.

633

313.

043

313.

846

315.

152

315.

254

315.

641

Appendices 321

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

ÒÒÐ

ÒÒÑ

ÒÒÒ

ÒÒÓ

ÒÒÔ

ÒÒÕ

ÒÒÖ

ÒÒ×

ÒÒØ

ÒÒÙ

ÒÓÐ

ÒÓÑ

ÒÓÒ

ÒÓÓ

ÒÓÔ

ÒÓÕ

ÒÓÖ

ÒÓ×

ÒÓØ

ÒÓÙ

ÒÔÐ

ÒÔÑ

19-e

t jus

t min

or th

ird

53-e

t jus

t min

or th

ird

140-

et ju

st m

inor

third

87-e

t jus

t min

or th

ird

Pyth

agor

ean

augm

ente

d s

econ

d, o

r cyc

lic m

inor

third

(A)

IX

15-e

t jus

t min

or th

ird

und

ecim

al n

eutra

l thi

rd (

77th

har

mon

ic)

41-e

t jus

t min

or th

ird

sep

timal

sup

erau

gmen

ted

sec

ond

11-e

t sev

en q

uarte

r-ton

es

nona

vige

simal

gra

ve o

r sm

all n

eutra

l thi

rd

seve

n q

uarte

r-ton

es

untri

gesim

al s

uper

augm

ente

d s

econ

d

und

ecim

al g

rave

or s

mal

l neu

tral t

hird

18-e

t sev

en q

uarte

r-ton

es

43-e

t dou

ble

dim

inish

ed fo

urth

grav

e or

sm

all n

eutra

l thi

rd

53-e

t neu

tral t

hird

cycl

ic n

eutra

l thi

rd (A

) XX

I

trid

ecim

al g

rave

or s

mal

l neu

tral t

hird

(39

th h

arm

onic

)

7-et

neu

tral t

hird

acut

e or

larg

e ne

utra

l thi

rd

519

)2

(

1453

)2

(

3714

0)

2(

23

87)

2(

3 ù/2

ñô, o

r 196

83/1

6384

4

15)

2(

77/6

4 1141

)2

(

135/

112 3

11)

2(

29/2

4

75/6

2

155/

128

40/3

3 518

)2

(

1243

)2

(

243/

200

1553

)2

(

3 òñ/2

óó

39/3

2 27

)2

(

8000

/656

1

1.20

0103

1.20

0929

1.20

1041

1.20

1110

1.20

1355

1.20

3025

1.20

3125

1.20

4382

1.20

5357

1.20

8089

1.20

8333

1.20

9677

1.21

0938

1.21

2121

1.21

2326

1.21

3412

1.21

5000

1.21

6738

1.21

7745

1.21

8750

1.21

9014

1.21

9326

313.

978

314.

194

314.

223

314.

241

314.

305

314.

742

314.

768

315.

097

315.

352

316.

067

316.

131

316.

483

316.

812

317.

122

317.

175

317.

460

317.

875

318.

330

318.

593

318.

856

318.

925

319.

007

315.

789

316.

981

317.

143

317.

241

317.

595

320.

000

320.

144

321.

951

323.

353

327.

273

327.

622

329.

547

331.

349

333.

041

333.

333

334.

884

337.

148

339.

623

341.

055

342.

483

342.

857

343.

301

322 Appendices

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

ÒÔÒ

ÒÔÓ

ÒÔÔ

ÒÔÕ

ÒÔÖ

ÒÔ×

ÒÔØ

ÒÔÙ

ÒÕÐ

ÒÕÑ

ÒÕÒ

ÒÕÓ

ÒÕÔ

ÒÕÕ

ÒÕÖ

ÒÕ×

ÒÕØ

ÒÕÙ

ÒÖÐ

ÒÖÑ

ÒÖÒ

ÒÖÓ

dou

ble

augm

ente

d se

cond

und

ecim

al a

cute

or l

arge

neu

tral t

hird

31-e

t dou

ble

augm

ente

d s

econ

d, o

r neu

tral t

hird

seve

n eq

ual q

uarte

r-ton

es

17-e

t neu

tral t

hird

trid

ecim

al a

cute

or l

arge

neu

tral t

hird

10-e

t gra

ve o

r sm

all m

ajor

third

53-e

t gra

ve o

r sm

all m

ajor

third

43-e

t dou

ble

augm

ente

d s

econ

d

cycl

ic g

rave

or s

mal

l maj

or th

ird (A

) XX

XIII

79th

har

mon

ic

grav

e or

sm

all m

ajor

third

23-e

t gra

ve o

r sm

all m

ajor

third

13-e

t gra

ve o

r sm

all m

ajor

third

nona

vige

simal

acu

te o

r lar

ge n

eutra

l thi

rd

16-e

t gra

ve o

r sm

all m

ajor

third

19-e

t jus

t maj

or th

ird

third

-com

ma

mea

nton

e m

ajor

third

(A)

IV31

1ß

41-e

t jus

t maj

or th

ird

22-e

t jus

t maj

or th

ird

two-

seve

nth-

com

ma

mea

nton

e m

ajor

third

(A)

IV71

1ß

Pyth

agor

ean

dim

inish

ed fo

urth

(D)

VIII

625/

512

11/9

931

)2

(

724

)2

(, o

r ap

pro

xim

atel

y 60

/49

517

)2

(

16/1

3 310

)2

(

1653

)2

(

1343

)2

(

3 óó/2

õò

79/6

4

100/

81

723

)2

(

413

)2

(

36/2

9 516

)2

(

619

)2

(

81/6

4×0.

758180

13

41)

2(

7

22)

2(

81/6

4×0.

875

8180

8192

/656

1

1.22

0703

1.22

2222

1.22

2914

1.22

4054

1.22

6135

1.23

0769

1.23

1144

1.23

2756

1.23

3131

1.23

4359

1.23

4375

1.23

4568

1.23

4860

1.23

7726

1.24

1379

1.24

1858

1.24

4693

1.24

4835

1.24

5801

1.24

6758

1.24

7784

1.24

8590

319.

367

319.

765

319.

945

320.

244

320.

788

322.

001

322.

099

322.

520

322.

618

322.

940

322.

944

322.

995

323.

071

323.

821

324.

777

324.

902

325.

643

325.

681

325.

933

326.

184

326.

452

326.

663

345.

255

347.

408

348.

387

350.

000

352.

941

359.

472

360.

000

362.

264

362.

791

364.

515

364.

537

364.

807

365.

217

369.

231

374.

333

375.

000

378.

947

379.

145

380.

488

381.

818

383.

241

384.

360

Appendices 323

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

ÒÖÔ

ÒÖÕ

ÒÖÖ

ÒÖ×

ÒÖØ

ÒÖÙ

Ò×Ð

Ò×Ñ

Ò×Ò

Ò×Ó

Ò×Ô

Ò×Õ

Ò×Ö

Ò××

Ò×Ø

Ò×Ù

ÒØÐ

ÒØÑ

ÒØÒ

ÒØÓ

ÒØÔ

ÒØÕ

53-e

t jus

t maj

or th

ird

140-

et ju

st m

ajor

third

87-e

t jus

t maj

or th

ird

mea

nton

e an

d ju

st m

ajor

third

(A)

IV1ß

(5t

h ha

rmon

ic)

118-

et ju

st m

ajor

third

31-e

t jus

t maj

or th

ird

65-e

t jus

t maj

or th

ird

99-e

t jus

t maj

or th

ird

cycl

ic m

ajor

third

(A)

XLV

two-

nint

h-co

mm

a m

eant

one

maj

or th

ird (A

) IV

98

�

fifth

-com

ma

mea

nton

e m

ajor

third

(A)

IV5

4ß

43-e

t jus

t maj

or th

ird

sixth

-com

ma

mea

nton

e m

ajor

third

(A)

IV6

4ß

sep

tend

ecim

al s

uper

maj

or th

ird

equa

l maj

or th

ird

nona

dec

imal

sup

erm

ajor

third

grav

e or

sm

all d

imin

ished

four

th

53-e

t Pyt

hago

rean

maj

or th

ird, o

r dito

ne

Pyth

agor

ean

maj

or th

ird, o

r dito

ne (A

) IV

(81s

t har

mon

ic)

57th

cyc

lic fi

fth (A

) LV

II

23-e

t Pyt

hago

rean

maj

or th

ird, o

r dito

ne

und

ecim

al d

imin

ished

four

th

1753

)2

(

4514

0)

2(

28

87)

2(

81/6

4×80

/81,

or 5

/4

3811

8)

2(

10

31)

2(

21

65)

2(

32

99)

2(

3 ôõ/2

֖

81/6

4×1.

125

8180

81/6

4×1.

258180

14

43)

2(

81/6

4×1.

58180

34/2

7

32

, or a

pp

roxi

mat

ely

63/5

0

24/1

9

512/

405

1853

)2

(

81/6

4

3 õ÷/2

ùð 823

)2

(

14/1

1

1.24

8984

1.24

9567

1.24

9923

1.25

0000

1.25

0092

1.25

0566

1.25

0996

1.25

1131

1.25

1200

1.25

1727

1.25

3109

1.25

3169

1.25

5187

1.25

9259

1.25

9921

1.26

3158

1.26

4198

1.26

5426

1.26

5625

1.26

8271

1.27

2642

1.27

2727

326.

766

326.

919

327.

012

327.

032

327.

056

327.

180

327.

292

327.

328

327.

346

327.

484

327.

845

327.

861

328.

389

329.

454

329.

628

330.

474

330.

746

331.

068

331.

120

331.

812

332.

956

332.

978

384.

906

385.

714

386.

207

386.

314

386.

441

387.

097

387.

692

387.

879

387.

975

388.

703

390.

615

390.

698

393.

482

399.

090

400.

000

404.

442

405.

866

407.

547

407.

820

411.

435

417.

391

417.

508

324 Appendices

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

ÒØÖ

ÒØ×

ÒØØ

ÒØÙ

ÒÙÐ

ÒÙÑ

ÒÙÒ

ÒÙÓ

ÒÙÔ

ÒÙÕ

ÒÙÖ

ÒÙ×

ÒÙØ

ÒÙÙ

ÓÐÐ

ÓÐÑ

ÓÐÒ

ÓÐÓ

ÓÐÔ

ÓÐÕ

ÓÐÖ

ÓÐ×

43-e

t dim

inish

ed fo

urth

20-e

t acu

te o

r lar

ge m

ajor

third

17-e

t acu

te o

r lar

ge m

ajor

third

trivi

gesim

al s

uper

maj

or th

ird

31-e

t sup

erm

ajor

third

, or d

imin

ished

four

th

mea

nton

e an

d d

imin

ished

four

th, o

r acu

te o

r lar

ge m

ajor

third

(D)

VIII

2�

14-e

t acu

te o

r lar

ge m

ajor

third

41st

har

mon

ic

53-e

t acu

te o

r lar

ge m

ajor

third

cycl

ic a

cute

or l

arge

maj

or th

ird (A

) XV

I

sep

timal

sup

erm

ajor

third

11-e

t nin

e q

uarte

r-ton

es

untri

gesim

al s

ubfo

urth

19-e

t nin

e q

uarte

r-ton

es

nine

qua

rter-t

ones

, or u

ntrig

esim

al s

uper

maj

or th

ird

43-e

t trip

le d

imin

ished

fifth

nine

eq

ual q

uarte

r-ton

es

83rd

har

mon

ic

53-e

t sub

four

th

cycl

ic s

ubfo

urth

(A)

XXV

III

augm

ente

d th

ird, o

r sub

four

th

21-e

t sub

four

th

1543

)2

(

720

)2

(

617

)2

(

23/1

8 1131

)2

(

8192

/656

1×0.

58081

, or 3

2/25

5

14)

2(

41/3

2 1953

)2

(

3 ñö/2

òõ

9/7

411

)2

(

40/3

1 719

)2

(

31/2

4 1643

)2

(

38

)2

(, o

r ap

pro

xim

atel

y 83

/64

83/6

4 2053

)2

(

3 òø/2

ôô

125/

96

821

)2

(

1.27

3534

1.27

4561

1.27

7162

1.27

7778

1.27

8843

1.28

0000

1.28

0887

1.28

1250

1.28

2084

1.28

2892

1.28

5714

1.28

6665

1.29

0323

1.29

0939

1.29

1667

1.29

4229

1.29

6840

1.29

6875

1.29

8961

1.30

0395

1.30

2083

1.30

2201

333.

189

333.

458

334.

138

334.

299

334.

578

334.

881

335.

113

335.

208

335.

426

335.

637

336.

376

336.

624

337.

581

337.

743

337.

933

338.

603

339.

286

339.

296

339.

841

340.

217

340.

658

340.

689

418.

605

420.

000

423.

529

424.

364

425.

806

427.

373

428.

571

429.

062

430.

189

431.

280

435.

084

436.

364

441.

278

442.

105

443.

081

446.

512

450.

000

450.

047

452.

830

454.

740

456.

986

457.

143

Appendices 325

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

ÓÐØ

ÓÐÙ

ÓÑÐ

ÓÑÑ

ÓÑÒ

ÓÑÓ

ÓÑÔ

ÓÑÕ

ÓÑÖ

ÓÑ×

ÓÑØ

ÓÑÙ

ÓÒÐ

ÓÒÑ

ÓÒÒ

ÓÒÓ

ÓÒÔ

ÓÒÕ

ÓÒÖ

ÓÒ×

ÓÒØ

ÓÒÙ

trivi

gesim

al s

ubfo

urth

13-e

t sub

four

th

mea

nton

e au

gmen

ted

third

(A) X

I4

32

ß

31-e

t aug

men

ted

third

, or s

ubfo

urth

18-e

t gra

ve o

r sm

all f

ourth

23-e

t gra

ve o

r sm

all f

ourth

sep

timal

sub

four

th (

21st

har

mon

ic)

43-e

t aug

men

ted

third

53-e

t gra

ve o

r sm

all f

ourth

grav

e or

sm

all f

ourth

cycl

ic g

rave

or s

mal

l fou

rth (A

) XL

acut

e or

larg

e au

gmen

ted

third

nona

dec

imal

sub

four

th

5-et

gra

ve o

r sm

all f

ourth

sep

tend

ecim

al s

ubfo

urth

22-e

t jus

t and

Pyt

hago

rean

per

fect

four

th

sep

tend

ecim

al s

uper

augm

ente

d th

ird (

85th

har

mon

ic)

17-e

t jus

t and

Pyt

hago

rean

per

fect

four

th

87-e

t jus

t and

Pyt

hago

rean

per

fect

four

th

99-e

t jus

t and

Pyt

hago

rean

per

fect

four

th

140-

et ju

st a

nd P

ytha

gore

an p

erfe

ct fo

urth

41-e

t jus

t and

Pyt

hago

rean

per

fect

four

th

30/2

3 513

)2

(

1771

47/1

3107

2×0.

3636

368180

12

31)

2(

7

18)

2(

9

23)

2(

21/1

6 1743

)2

(

2153

)2

(

320/

243

3 ôð/2

öó

675/

512

95/7

2 25

)2

(

45/3

4 922

)2

(

85/6

4 717

)2

(

3687

)2

(

4199

)2

(

5814

0)

2(

17

41)

2(

1.30

4348

1.30

5512

1.30

6133

1.30

7759

1.30

9385

1.31

1579

1.31

2500

1.31

5261

1.31

6061

1.31

6872

1.31

8137

1.31

8359

1.31

9444

1.31

9508

1.32

3529

1.32

7849

1.32

8125

1.33

0312

1.33

2184

1.33

2505

1.33

2639

1.33

2961

341.

251

341.

555

341.

718

342.

143

342.

568

343.

143

343.

384

344.

106

344.

315

344.

527

344.

858

344.

917

345.

200

345.

217

346.

269

347.

399

347.

471

348.

044

348.

533

348.

617

348.

652

348.

737

459.

994

461.

538

462.

363

464.

516

466.

667

469.

565

470.

781

474.

419

475.

472

476.

539

478.

200

478.

492

479.

917

480.

000

485.

268

490.

909

491.

269

494.

118

496.

552

496.

970

497.

143

497.

561

326 Appendices

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

ÓÓÐ

ÓÓÑ

ÓÓÒ

ÓÓÓ

ÓÓÔ

ÓÓÕ

ÓÓÖ

ÓÓ×

ÓÓØ

ÓÓÙ

ÓÔÐ

ÓÔÑ

ÓÔÒ

ÓÔÓ

ÓÔÔ

ÓÔÕ

ÓÔÖ

ÓÔ×

ÓÔØ

ÓÔÙ

ÓÕÐ

ÓÕÑ

just

and

Pyt

hago

rean

per

fect

four

th (D

) I

53-e

t jus

t and

Pyt

hago

rean

per

fect

four

th

118-

et ju

st a

nd P

ytha

gore

an p

erfe

ct fo

urth

65-e

t jus

t and

Pyt

hago

rean

per

fect

four

th

equa

l per

fect

four

th

sixth

-com

ma

mea

nton

e p

erfe

ct fo

urth

(D)

I61

�

cycl

ic p

erfe

ct fo

urth

(A)

LII

43-e

t jus

t and

Pyt

hago

rean

per

fect

four

th

fifth

-com

ma

mea

nton

e p

erfe

ct fo

urth

(D)

I51

�

two-

nint

h-co

mm

a m

eant

one

per

fect

four

th (D

) I

92

�

31-e

t jus

t and

Pyt

hago

rean

per

fect

four

th

mea

nton

e p

erfe

ct fo

urth

(D)

I 41

�

two-

seve

nth-

com

ma

mea

nton

e p

erfe

ct fo

urth

(D)

I7

2�

third

-com

ma

mea

nton

e p

erfe

ct fo

urth

(D)

I31

�

19-e

t jus

t and

Pyt

hago

rean

per

fect

four

th

43rd

har

mon

ic

7-et

acu

te o

r lar

ge fo

urth

subd

imin

ished

fifth

acut

e or

larg

e fo

urth

53-e

t acu

te o

r lar

ge fo

urth

Pyth

agor

ean

augm

ente

d th

ird, o

r cyc

lic a

cute

or l

arge

four

th (A

) XI

23-e

t acu

te o

r lar

ge fo

urth

4/3

2253

)2

(

4911

8)

2(

27

65)

2(

5

12)

2(

, or a

ppro

xim

atel

y 10

24/7

67

4/3×

68081

3 õò/2

øò 1843

)2

(

4/3×

58081

4/3×

4.5

8081

1331

)2

(

4/3×

48081

4/3×

3.5

8081

4/3×

38081

8

19)

2(

43/3

2 37

)2

(

8192

/607

5

27/2

0 2353

)2

(

3 ññ/2

ñ÷, o

r 177

147/

1310

72

1023

)2

(

1.33

3333

1.33

3386

1.33

3534

1.33

3654

1.33

4840

1.33

6097

1.33

6120

1.33

6634

1.33

6650

1.33

7019

1.33

7329

1.33

7481

1.33

8074

1.33

8866

1.33

8904

1.34

3750

1.34

5900

1.34

8477

1.35

0000

1.35

0939

1.35

1524

1.35

1707

348.

834

348.

848

348.

886

348.

918

349.

228

349.

557

349.

563

349.

698

349.

702

349.

798

349.

880

349.

919

350.

074

350.

282

350.

292

351.

559

352.

122

352.

796

353.

195

353.

440

353.

593

353.

641

498.

045

498.

113

498.

305

498.

462

500.

000

501.

629

501.

660

502.

326

502.

346

502.

824

503.

226

503.

422

504.

190

505.

214

505.

263

511.

518

514.

286

517.

598

519.

551

520.

755

521.

505

521.

739

Appendices 327

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

ÓÕÒ

ÓÕÓ

ÓÕÔ

ÓÕÕ

ÓÕÖ

ÓÕ×

ÓÕØ

ÓÕÙ

ÓÖÐ

ÓÖÑ

ÓÖÒ

ÓÖÓ

ÓÖÔ

ÓÖÕ

ÓÖÖ

ÓÖ×

ÓÖØ

ÓÖÙ

Ó×Ð

Ó×Ñ

Ó×Ò

Ó×Ó

trid

ecim

al s

uper

four

th

16-e

t acu

te o

r lar

ge fo

urth

43-e

t dou

ble

dim

inish

ed fi

fth

nona

vige

simal

sup

erfo

urth

(87

th h

arm

onic

)

9-et

sup

erfo

urth

und

ecim

al tr

itone

, or a

ugm

ente

d fo

urth

grav

e or

sm

all a

ugm

ente

d fo

urth

, or s

uper

four

th

20-e

t sup

erfo

urth

31-e

t sup

erfo

urth

, or d

imin

ished

fifth

53-e

t sup

erfo

urth

cycl

ic s

uper

four

th (A

) XX

III

11-e

t sup

erfo

urth

elev

en e

qua

l qua

rter-t

ones

und

ecim

al s

uper

four

th (

11th

har

mon

ic)

untri

gesim

al s

ubd

imin

ished

fifth

13-e

t ele

ven

qua

rter-t

ones

elev

en q

uarte

r-ton

es

nona

vige

simal

trito

ne, o

r aug

men

ted

four

th

43-e

t dou

ble

augm

ente

d th

ird

15-e

t ele

ven

qua

rter-t

ones

trid

ecim

al tr

itone

, or a

ugm

ente

d fo

urth

17-e

t gra

ve o

r sm

all a

ugm

ente

d fo

urth

65/4

8 716

)2

(

1943

)2

(

87/6

4 49

)2

(

15/1

1

512/

375 9

20)

2(

14

31)

2(

24

53)

2(

3 òó/2

óö 511

)2

(

1124

)2

(, o

r app

roxi

mat

ely

1024

/745

11/8

128/

93

613

)2

(

62/4

5

40/2

9 2043

)2

(

715

)2

(

18/1

3 817

)2

(

1.35

4167

1.35

4256

1.35

8355

1.35

9375

1.36

0790

1.36

3636

1.36

5333

1.36

6040

1.36

7568

1.36

8723

1.36

9964

1.37

0351

1.37

3954

1.37

5000

1.37

6344

1.37

7009

1.37

7778

1.37

9310

1.38

0429

1.38

1913

1.38

4615

1.38

5674

354.

285

354.

308

355.

380

355.

647

356.

017

356.

762

357.

206

357.

391

357.

791

358.

093

358.

417

358.

519

359.

461

359.

735

360.

087

360.

261

360.

462

360.

863

361.

155

361.

544

362.

251

362.

528

524.

886

525.

000

530.

233

531.

532

533.

333

536.

951

539.

104

540.

000

541.

935

543.

396

544.

965

545.

455

550.

000

551.

318

553.

009

553.

846

554.

812

556.

737

558.

140

560.

000

563.

382

564.

706

328 Appendices

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

Ó×Ô

Ó×Õ

Ó×Ö

Ó××

Ó×Ø

Ó×Ù

ÓØÐ

ÓØÑ

ÓØÒ

ÓØÓ

ÓØÔ

ÓØÕ

ÓØÖ

ÓØ×

ÓØØ

ÓØÙ

ÓÙÐ

ÓÙÑ

ÓÙÒ

ÓÙÓ

ÓÙÔ

ÓÙÕ

53-e

t gra

ve o

r sm

all a

ugm

ente

d fo

urth

19-e

t gra

ve o

r sm

all a

ugm

ente

d fo

urth

cycl

ic g

rave

or s

mal

l aug

men

ted

four

th (A

) XX

XV

grav

e or

sm

all a

ugm

ente

d fo

urth

89th

har

mon

ic

21-e

t gra

ve o

r sm

all a

ugm

ente

d fo

urth

trivi

gesim

al s

ubd

imin

ished

fifth

23-e

t gra

ve o

r sm

all a

ugm

ente

d fo

urth

mea

nton

e tri

tone

, or a

ugm

ente

d fo

urth

(A)

VI

211ß

31-e

t aug

men

ted

four

th, o

r sub

dim

inish

ed fi

fth

sep

timal

sub

dim

inish

ed fi

fth

43-e

t jus

t trit

one,

or a

ugm

ente

d fo

urth

Pyth

agor

ean

dim

inish

ed fi

fth (D

) V

I

53-e

t jus

t trit

one,

or a

ugm

ente

d fo

urth

just

trito

ne, o

r aug

men

ted

four

th (

45th

har

mon

ic)

nona

dec

imal

sub

dim

inish

ed fi

fth

cycl

ic tr

itone

, or a

ugm

ente

d fo

urth

(A)

XLV

II

sep

tend

ecim

al s

ubd

imin

ished

fifth

equa

l trit

one,

or a

ugm

ente

d fo

urth

sep

tend

ecim

al tr

itone

, or s

uper

augm

ente

d fo

urth

nona

dec

imal

trito

ne, o

r sup

erau

gmen

ted

four

th

trid

ecim

al d

imin

ished

fifth

(91

st h

arm

onic

)

2553

)2

(

919

)2

(

3 óõ/2

õõ

25/1

8

89/6

4 1021

)2

(

32/2

3 1123

)2

(

729/

512×

0.66

6667

8180

1531

)2

(

7/5

2143

)2

(

1024

/729

26

53)

2(

45/3

2

38/2

7

3 ô÷/2

÷ô

24/1

7

22

, or a

ppro

xim

atel

y 18

1/12

8

17/1

2

27/1

9

91/6

4

1.38

6741

1.38

8651

1.38

8654

1.38

8889

1.39

0625

1.39

1066

1.39

1304

1.39

3063

1.39

7542

1.39

8491

1.40

0000

1.40

2861

1.40

4664

1.40

4996

1.40

6250

1.40

7407

1.40

7600

1.41

1765

1.41

4214

1.41

6667

1.42

1053

1.42

1875

362.

807

363.

307

363.

307

363.

369

363.

823

363.

938

364.

001

364.

461

365.

633

365.

881

366.

276

367.

024

367.

496

367.

583

367.

911

368.

214

368.

264

369.

354

369.

994

370.

636

371.

784

371.

999

566.

038

568.

421

568.

425

568.

717

570.

880

571.

429

571.

726

573.

913

579.

471

580.

645

582.

512

586.

047

588.

270

588.

679

590.

224

591.

648

591.

885

597.

000

600.

000

603.

000

608.

352

609.

354

Appendices 329

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

ÓÙÖ

ÓÙ×

ÓÙØ

ÓÙÙ

ÔÐÐ

ÔÐÑ

ÔÐÒ

ÔÐÓ

ÔÐÔ

ÔÐÕ

ÔÐÖ

ÔÐ×

ÔÐØ

ÔÐÙ

ÔÑÐ

ÔÑÑ

ÔÑÒ

ÔÑÓ

ÔÑÔ

ÔÑÕ

ÔÑÖ

ÔÑ×

dim

inish

ed fi

fth, o

r acu

te o

r lar

ge tr

itone

, or a

ugm

ente

d fo

urth

53-e

t acu

te o

r lar

ge tr

itone

, or a

ugm

ente

d fo

urth

Pyth

agor

ean

trito

ne, o

r aug

men

ted

four

th (A

) VI

43-e

t dim

inish

ed fi

fth

59th

cyc

lic fi

fth (A

) LIX

sep

timal

trito

ne, o

r sup

erau

gmen

ted

four

th

31-e

t sup

erau

gmen

ted

four

th, o

r dim

inish

ed fi

fth

mea

nton

e d

imin

ished

fifth

(D) V

I21

1�

23-e

t acu

te o

r lar

ge d

imin

ished

fifth

trivi

gesim

al s

uper

augm

ente

d fo

urth

(23r

d ha

rmon

ic)

21-e

t acu

te o

r lar

ge d

imin

ished

fifth

acut

e or

larg

e d

imin

ished

fifth

19-e

t acu

te o

r lar

ge d

imin

ished

fifth

53-e

t acu

te o

r lar

ge d

imin

ished

fifth

cycl

ic a

cute

or l

arge

dim

inish

ed fi

fth (A

) XV

III

17-e

t acu

te o

r lar

ge d

imin

ished

fifth

trid

ecim

al s

ubd

imin

ished

fifth

15-e

t thi

rteen

qua

rter-t

ones

43-e

t dou

ble

dim

inish

ed s

ixth

nona

vige

simal

sub

dim

inish

ed fi

fth

thirt

een

qua

rter-t

ones

13-e

t thi

rteen

qua

rter-t

ones

64/4

5 2753

)2

(

3 ö/2

ù, or

729

/512

22

43)

2(

3 õù/2

ùó

10/7

1631

)2

(

1024

/729

×0.

6666

678081

12

23)

2(

23/1

6 1121

)2

(

36/2

5 1019

)2

(

2853

)2

(

3 ñø/2

òø 917

)2

(

13/9

815

)2

(

2343

)2

(

29/2

0

90/6

2 713

)2

(

1.42

2222

1.42

3492

1.42

3828

1.42

5658

1.42

6804

1.42

8571

1.43

0113

1.43

1084

1.43

5685

1.43

7500

1.43

7747

1.44

0000

1.44

0247

1.44

2231

1.44

3254

1.44

3341

1.44

4444

1.44

7269

1.44

8825

1.45

0000

1.45

1613

1.45

2423

372.

090

372.

422

372.

510

372.

989

373.

289

373.

751

374.

154

374.

408

375.

612

376.

087

376.

151

376.

741

376.

805

377.

324

377.

592

377.

615

377.

904

378.

643

379.

050

379.

357

379.

779

379.

991

609.

776

611.

321

611.

730

613.

953

615.

345

617.

488

619.

355

620.

529

626.

087

628.

274

628.

571

631.

283

631.

579

633.

962

635.

190

635.

294

636.

618

640.

000

641.

860

643.

263

645.

188

646.

154

330 Appendices

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

ÔÑØ

ÔÑÙ

ÔÒÐ

ÔÒÑ

ÔÒÒ

ÔÒÓ

ÔÒÔ

ÔÒÕ

ÔÒÖ

ÔÒ×

ÔÒØ

ÔÒÙ

ÔÓÐ

ÔÓÑ

ÔÓÒ

ÔÓÓ

ÔÓÔ

ÔÓÕ

ÔÓÖ

ÔÓ×

ÔÓØ

ÔÓÙ

untri

gesim

al s

uper

augm

ente

d fo

urth

(93

rd h

arm

onic

)

und

ecim

al s

ubfif

th

thirt

een

equa

l qua

rter-t

ones

11-e

t sub

fifth

mea

nton

e d

oubl

e au

gmen

ted

four

th (A

) XI

II41

3ß

53-e

t sub

fifth

31-e

t dou

ble

augm

ente

d fo

urth

, or s

ubfif

th

cycl

ic s

ubfif

th (A

) XX

X

20-e

t sub

fifth

acut

e or

larg

e d

oubl

e au

gmen

ted

four

th, o

r sub

fifth

und

ecim

al s

ubd

imin

ished

fifth

47th

har

mon

ic

9-et

sub

fifth

nona

vige

simal

sub

fifth

43-e

t dou

ble

augm

ente

d fo

urth

16-e

t gra

ve o

r sm

all f

ifth

trid

ecim

al s

ubfif

th

23-e

t gra

ve o

r sm

all f

ifth

Pyth

agor

ean

dim

inish

ed s

ixth

(D)

XI

53-e

t gra

ve o

r sm

all f

ifth

grav

e or

sm

all f

ifth

cycl

ic g

rave

or s

mal

l fift

h (A

) XL

II

93/6

4

16/1

1 13

24)

2(

, or a

ppro

xim

atel

y 74

5/51

2 6

11)

2(

1594

323/

1048

576×

0.30

7692

8180

2953

)2

(

1731

)2

(

3 óð/2

ô÷ 1120

)2

(

375/

256

22/1

5

47/3

2 59

)2

(

128/

87

2443

)2

(

916

)2

(

96/6

5 1323

)2

(

2621

44/1

7714

7 30

53)

2(

40/2

7

3 ôò/2

öö

1.45

3125

1.45

4545

1.45

5653

1.45

9480

1.46

0302

1.46

1216

1.46

2450

1.46

2944

1.46

4086

1.46

4844

1.46

6667

1.46

8750

1.46

9734

1.47

1264

1.47

2369

1.47

6826

1.47

6923

1.47

9610

1.47

9811

1.48

0452

1.48

1481

1.48

2904

380.

175

380.

546

380.

836

381.

837

382.

052

382.

292

382.

614

382.

744

383.

042

383.

241

383.

717

384.

263

384.

520

384.

920

385.

209

386.

375

386.

401

387.

104

387.

156

387.

324

387.

593

387.

966

646.

991

648.

682

650.

000

654.

545

655.

536

656.

604

658.

065

658.

650

660.

000

660.

896

663.

049

665.

507

666.

667

668.

468

669.

767

675.

000

675.

114

678.

261

678.

495

679.

245

680.

449

682.

110

Appendices 331

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

ÔÔÐ

ÔÔÑ

ÔÔÒ

ÔÔÓ

ÔÔÔ

ÔÔÕ

ÔÔÖ

ÔÔ×

ÔÔØ

ÔÔÙ

ÔÕÐ

ÔÕÑ

ÔÕÒ

ÔÕÓ

ÔÕÔ

ÔÕÕ

ÔÕÖ

ÔÕ×

ÔÕØ

ÔÕÙ

ÔÖÐ

ÔÖÑ

acut

e or

larg

e d

oubl

e su

per

augm

ente

d fo

urth

nona

dec

imal

sub

fifth

(95

th h

arm

onic

)

7-et

gra

ve o

r sm

all f

ifth

19-e

t jus

t and

Pyt

hago

rean

per

fect

fifth

third

-com

ma

mea

nton

e p

erfe

ct fi

fth (A

) I31

ß

two-

seve

nth-

com

ma

mea

nton

e p

erfe

ct fi

fth (A

) I7

2ß

mea

nton

e p

erfe

ct fi

fth (A

) I41

ß

31-e

t jus

t and

Pyt

hago

rean

per

fect

fifth

two-

nint

h-co

mm

a m

eant

one

per

fect

fifth

(A)

I9

2ß

fifth

-com

ma

mea

nton

e p

erfe

ct fi

fth (A

) I51

ß

43-e

t jus

t and

Pyt

hago

rean

per

fect

fifth

sixth

-com

ma

mea

nton

e p

erfe

ct fi

fth (A

) I61

ß

equa

l per

fect

fifth

65-e

t jus

t and

Pyt

hago

rean

per

fect

fifth

118-

et ju

st a

nd P

ytha

gore

an p

erfe

ct fi

fth

53-e

t jus

t and

Pyt

hago

rean

per

fect

fifth

just

and

Pyt

hago

rean

per

fect

fifth

(A) I

(3rd

har

mon

ic)

41-e

t jus

t and

Pyt

hago

rean

per

fect

fifth

140-

et ju

st a

nd P

ytha

gore

an p

erfe

ct fi

fth

99-e

t jus

t and

Pyt

hago

rean

per

fect

fifth

87-e

t jus

t and

Pyt

hago

rean

per

fect

fifth

54th

cyc

lic fi

fth (A

) LIV

6075

/409

6

95/6

4 47

)2

(

1119

)2

(

3/2×

38180

3/2×

3.5

8180

3/2×

48180

18

31)

2(

3/2×

4.5

8180

3/2×

58180

25

43)

2(

3/2×

68180

7

12)

2(

, or a

ppro

xim

atel

y 76

7/51

2 38

65)

2(

69

118

)2

(

3153

)2

(

3/2 24

41)

2(

82

140

)2

(

5899

)2

(

5187

)2

(

3 õô/2

øõ

1.48

3154

1.48

4375

1.48

5994

1.49

3759

1.49

3802

1.49

4686

1.49

5349

1.49

5518

1.49

5865

1.49

6278

1.49

6296

1.49

6898

1.49

8307

1.49

9639

1.49

9775

1.49

9941

1.50

0000

1.50

0419

1.50

0782

1.50

0932

1.50

1294

1.50

3135

388.

031

388.

350

388.

774

390.

806

390.

817

391.

048

391.

221

391.

266

391.

356

393.

415

391.

469

391.

627

391.

995

392.

344

392.

379

392.

423

392.

438

392.

548

392.

643

392.

682

392.

777

393.

259

682.

402

683.

827

685.

714

694.

737

694.

786

695.

810

696.

578

696.

774

697.

176

697.

654

697.

674

698.

371

700.

000

701.

538

701.

695

701.

887

701.

955

702.

439

702.

857

703.

030

703.

448

705.

570

332 Appendices

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

ÔÖÒ

ÔÖÓ

ÔÖÔ

ÔÖÕ

ÔÖÖ

ÔÖ×

ÔÖØ

ÔÖÙ

Ô×Ð

Ô×Ñ

Ô×Ò

Ô×Ó

Ô×Ô

Ô×Õ

Ô×Ö

Ô××

Ô×Ø

Ô×Ù

ÔØÐ

ÔØÑ

ÔØÒ

ÔØÓ

17-e

t jus

t and

Pyt

hago

rean

per

fect

fifth

22-e

t jus

t and

Pyt

hago

rean

per

fect

fifth

sep

tend

ecim

al s

uper

fifth

und

ecim

al g

rave

or s

mal

l aug

men

ted

fifth

97th

har

mon

ic

5-et

acu

te o

r lar

ge fi

fth

nona

dec

imal

sup

erfif

th

subd

imin

ished

six

th

acut

e or

larg

e fif

th

53-e

t acu

te o

r lar

ge fi

fth

Pyth

agor

ean

dou

ble

augm

ente

d fo

urth

, or c

yclic

acu

te o

r lar

ge fi

fth (A

) XI

II

43-e

t dim

inish

ed s

ixth

sep

timal

sup

erfif

th

23-e

t acu

te o

r lar

ge fi

fth

18-e

t sup

erfif

th

31-e

t sup

erfif

th, o

r dim

inish

ed s

ixth

mea

nton

e d

imin

ished

six

th a

nd w

olf f

ifth

(D) X

I4

32�

sep

timal

dim

inish

ed s

ixth

(49

th h

arm

onic

)

13-e

t sup

erfif

th

trivi

gesim

al s

uper

fifth

21-e

t sup

erfif

th

dim

inish

ed s

ixth

, or s

uper

fifth

1017

)2

(

1322

)2

(

68/4

5

50/3

3

97/6

4 35

)2

(

144/

95

1024

/675

243/

160

3253

)2

(

3 ñó/2

òð, o

r 159

4323

/104

8576

26

43)

2(

32/2

1 1423

)2

(

1118

)2

(

1931

)2

(

2621

44/1

7714

7×0.

3636

368081

49/3

2 813

)2

(

23/1

5 1321

)2

(

192/

125

1.50

3407

1.50

6196

1.51

1111

1.51

5152

1.51

5625

1.51

5717

1.51

5789

1.51

7037

1.51

8750

1.51

9686

1.52

0465

1.52

0611

1.52

3810

1.52

4880

1.52

7435

1.52

9334

1.53

1237

1.53

1250

1.53

1966

1.53

3333

1.53

5861

1.53

6000

393.

330

394.

059

395.

345

396.

402

396.

526

396.

550

396.

569

396.

896

397.

344

397.

589

397.

792

397.

831

398.

104

398.

948

399.

616

400.

113

400.

611

400.

614

400.

802

401.

159

401.

820

401.

857

705.

882

709.

091

714.

732

719.

354

719.

895

720.

000

720.

083

721.

508

723.

014

724.

528

725.

415

725.

581

729.

219

730.

435

733.

333

735.

484

737.

637

737.

652

738.

462

740.

006

742.

857

743.

014

Appendices 333

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

ÔØÔ

ÔØÕ

ÔØÖ

ÔØ×

ÔØØ

ÔØÙ

ÔÙÐ

ÔÙÑ

ÔÙÒ

ÔÙÓ

ÔÙÔ

ÔÙÕ

ÔÙÖ

ÔÙ×

ÔÙØ

ÔÙÙ

ÕÐÐ

ÕÐÑ

ÕÐÒ

ÕÐÓ

ÕÐÔ

ÕÐÕ

trid

ecim

al g

rave

or s

mal

l aug

men

ted

fifth

53-e

t sup

erfif

th

cycl

ic s

uper

fifth

(A)

XXV

fifte

en e

qua

l qua

rter-t

ones

43-e

t trip

le a

ugm

ente

d fo

urth

und

ecim

al s

uper

fifth

(99t

h ha

rmon

ic)

untri

gesim

al s

ubm

inor

six

th

19-e

t Pyt

hago

rean

min

or s

ixth

fifte

en q

uarte

r-ton

es, o

r unt

riges

imal

sup

erfif

th

nona

vige

simal

gra

ve o

r sm

all a

ugm

ente

d s

ixth

11-e

t fift

een

qua

rter-t

ones

sep

timal

sub

min

or s

ixth

53-e

t aug

men

ted

fifth

14-e

t aug

men

ted

fifth

cycl

ic a

ugm

ente

d fi

fth (A

) XX

XVII

mea

nton

e au

gmen

ted

and

aug

men

ted

fifth

(A) V

III2

ß (

25th

har

mon

ic)

31-e

t aug

men

ted

fifth

, or s

ubm

inor

six

th

trivi

gesim

al s

ubm

inor

six

th

17-e

t aug

men

ted

fifth

20-e

t aug

men

ted

fifth

43-e

t aug

men

ted

fifth

und

ecim

al a

ugm

ente

d fi

fth

20/1

3 3353

)2

(

3 òõ/2

óù 5

8)

2(

, or a

ppro

xim

atel

y 12

8/83

27

43)

2(

99/6

4

48/3

1 1219

)2

(

31/2

0

45/2

9 711

)2

(

14/9

3453

)2

(

914

)2

(

3 ó÷/2

õø

6561

/409

6×0.

58180

, or 2

5/16

20

31)

2(

36/2

3 1117

)2

(

1320

)2

(

2843

)2

(

11/7

1.53

8462

1.53

9692

1.54

1209

1.54

2211

1.54

5321

1.54

6875

1.54

8387

1.54

9260

1.55

0000

1.55

1724

1.55

4406

1.55

5556

1.55

9960

1.56

1418

1.56

2236

1.56

2500

1.56

3914

1.56

5217

1.56

5972

1.56

9168

1.57

0433

1.57

1429

402.

501

402.

823

403.

220

403.

482

404.

296

404.

702

405.

098

405.

326

405.

520

405.

971

406.

672

406.

973

408.

126

408.

507

408.

721

408.

790

409.

160

409.

501

409.

698

410.

535

410.

866

411.

126

745.

786

747.

170

748.

875

750.

000

753.

488

755.

228

756.

919

757.

895

758.

722

760.

647

763.

636

764.

916

769.

811

771.

429

772.

335

772.

627

774.

194

775.

636

776.

471

780.

000

781.

395

782.

492

334 Appendices

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

ÕÐÖ

ÕÐ×

ÕÐØ

ÕÐÙ

ÕÑÐ

ÕÑÑ

ÕÑÒ

ÕÑÓ

ÕÑÔ

ÕÑÕ

ÕÑÖ

ÕÑ×

ÕÑØ

ÕÑÙ

ÕÒÐ

ÕÒÑ

ÕÒÒ

ÕÒÓ

ÕÒÔ

ÕÒÕ

ÕÒÖ

ÕÒ×

23-e

t Pyt

hago

rean

min

or s

ixth

sep

tend

ecim

al s

uper

augm

ente

d fi

fth

101s

t har

mon

ic

nona

dec

imal

sup

erau

gmen

ted

fifth

Pyth

agor

ean

min

or s

ixth

(D)

IV

53-e

t Pyt

hago

rean

min

or s

ixth

acut

e or

larg

e au

gmen

ted

fifth

nona

dec

imal

sub

min

or s

ixth

cycl

ic m

inor

six

th (A

) XL

IX

equa

l min

or s

ixth

sep

tend

ecim

al s

ubm

inor

six

th

sep

tend

ecim

al s

uper

augm

ente

d fi

fth (5

1st h

arm

onic

)

43-e

t jus

t min

or s

ixth

trivi

gesim

al s

uper

augm

ente

d fi

fth

31-e

t sup

erau

gmen

ted

fifth

, or j

ust m

inor

six

th

mea

nton

e an

d ju

st m

inor

six

th (D

) IV

1�

53-e

t jus

t min

or s

ixth

Pyth

agor

ean

augm

ente

d fi

fth (A

) VIII

22-e

t jus

t min

or s

ixth

19-e

t sev

ente

en q

uarte

r-ton

es

sep

timal

sup

erau

gmen

ted

six

th

103r

d ha

rmon

ic

1523

)2

(

85/5

4

101/

64

30/1

9

128/

81

3553

)2

(

405/

256

19/1

2

3 ôù/2

÷÷ 2

3)

2(

, or a

ppro

xim

atel

y 10

0/63

27/1

7

51/3

2 2943

)2

(

115/

72

2131

)2

(

128/

81×

81/8

0, o

r 8/5

36

53)

2(

3 ø/2

ñò, o

r 656

1/40

96

1522

)2

(

1319

)2

(

45/2

8

103/

64

1.57

1534

1.57

4074

1.57

8125

1.57

8947

1.58

0247

1.58

0496

1.58

2031

1.58

3333

1.58

3550

1.58

7401

1.58

8235

1.59

3750

1.59

5953

1.59

7222

1.59

9276

1.60

0000

1.60

1302

1.60

1807

1.60

4160

1.60

6822

1.60

7143

1.60

9375

411.

154

411.

818

412.

878

413.

093

413.

433

413.

498

413.

900

414.

240

414.

297

415.

305

415.

523

416.

966

417.

542

417.

874

418.

412

418.

601

418.

942

419.

074

419.

689

420.

386

420.

470

421.

054

782.

609

785.

404

789.

854

790.

756

792.

180

792.

453

794.

134

795.

558

795.

795

800.

000

800.

910

806.

910

809.

302

810.

678

812.

903

813.

686

815.

094

815.

640

818.

182

821.

053

821.

398

823.

801

Appendices 335

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

ÕÒØ

ÕÒÙ

ÕÓÐ

ÕÓÑ

ÕÓÒ

ÕÓÓ

ÕÓÔ

ÕÓÕ

ÕÓÖ

ÕÓ×

ÕÓØ

ÕÓÙ

ÕÔÐ

ÕÔÑ

ÕÔÒ

ÕÔÓ

ÕÔÔ

ÕÔÕ

ÕÔÖ

ÕÔ×

ÕÔØ

ÕÔÙ

16-e

t sev

ente

en q

uarte

r-ton

es

nona

vige

simal

gra

ve o

r sm

all n

eutra

l six

th

seve

ntee

n q

uarte

r-ton

es

untri

gesim

al s

uper

augm

ente

d fi

fth

13-e

t sev

ente

en q

uarte

r-ton

es

23-e

t neu

tral s

ixth

neut

ral s

ixth

43-e

t dou

ble

dim

inish

ed s

even

th

53-e

t neu

tral s

ixth

cycl

ic n

eutra

l six

th (A

) XX

10-e

t neu

tral s

ixth

trid

ecim

al g

rave

or s

mal

l neu

tral,

or o

verto

ne s

ixth

(13

th h

arm

onic

)

dou

ble

augm

ente

d fi

fth

17-e

t neu

tral s

ixth

mea

nton

e d

oubl

e au

gmen

ted

fifth

(A) X

V4

33

ß

seve

ntee

n eq

ual q

uarte

r-ton

es

31-e

t dou

ble

augm

ente

d fi

fth, o

r neu

tral s

ixth

und

ecim

al g

rave

or s

mal

l neu

tral s

ixth

sept

imal

neu

tral s

ixth

(10

5th

harm

onic

)

7-et

gra

ve o

r sm

all m

ajor

six

th

trid

ecim

al a

cute

or l

arge

neu

tral s

ixth

53-e

t gra

ve o

r sm

all m

ajor

six

th

1116

)2

(

29/1

8

50/3

1

155/

96

913

)2

(

1623

)2

(

81/5

0 3043

)2

(

3753

)2

(

3 òð/2

óñ 710

)2

(

13/8

625/

384

1217

)2

(

1434

8907

/838

8608

×0.

2666

678180

17

24)

2(

, or a

pp

roxi

mat

ely

49/3

0 22

31)

2(

18/1

1

105/

64

57

)2

(

64/3

9 3853

)2

(

1.61

0490

1.61

1111

1.61

2903

1.61

4583

1.61

5866

1.61

9616

1.62

0000

1.62

1888

1.62

2382

1.62

3661

1.62

4505

1.62

5000

1.62

7604

1.63

1142

1.63

2667

1.63

3915

1.63

5438

1.63

6364

1.64

0625

1.64

0671

1.64

1026

1.64

3739

421.

345

421.

508

421.

977

422.

416

422.

752

423.

733

423.

833

424.

327

424.

457

424.

791

425.

012

425.

142

425.

823

426.

748

427.

147

427.

474

427.

872

428.

115

429.

229

429.

241

429.

334

430.

044

825.

000

825.

667

827.

592

829.

394

830.

769

834.

783

835.

193

837.

209

837.

736

839.

100

840.

000

840.

528

843.

300

847.

059

848.

676

850.

000

851.

613

852.

592

857.

095

857.

143

857.

517

860.

377

336 Appendices

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

ÕÕÐ

ÕÕÑ

ÕÕÒ

ÕÕÓ

ÕÕÔ

ÕÕÕ

ÕÕÖ

ÕÕ×

ÕÕØ

ÕÕÙ

ÕÖÐ

ÕÖÑ

ÕÖÒ

ÕÖÓ

ÕÖÔ

ÕÖÕ

ÕÖÖ

ÕÖ×

ÕÖØ

ÕÖÙ

Õ×Ð

Õ×Ñ

cycl

ic g

rave

or s

mal

l maj

or s

ixth

(A)

XXXI

I

grav

e or

sm

all m

ajor

six

th

43-e

t dou

ble

augm

ente

d fi

fth

18-e

t gra

ve o

r sm

all m

ajor

six

th

und

ecim

al a

cute

or l

arge

neu

tral s

ixth

nona

vige

simal

acu

te o

r lar

ge n

eutra

l six

th

11-e

t gra

ve o

r sm

all m

ajor

six

th

53rd

har

mon

ic

15-e

t jus

t maj

or s

ixth

Pyth

agor

ean

dim

inish

ed s

even

th (D

) IX

53-

et ju

st m

ajor

six

th

19-e

t jus

t maj

or s

ixth

just

maj

or s

ixth

cycl

ic m

ajor

six

th (A

) XL

IV

23-e

t jus

t maj

or s

ixth

mea

nton

e m

ajor

six

th (A

) III

43

ß

107t

h ha

rmon

ic

31-e

t jus

t maj

or s

ixth

43-e

t jus

t maj

or s

ixth

equa

l maj

or s

ixth

nona

dec

imal

sup

erm

ajor

six

th

grav

e or

sm

all d

imin

ished

sev

enth

3 óò/2

õð

400/

243 31

43)

2(

13

18)

2(

33/2

0

48/2

9 811

)2

(

53/3

2 1115

)2

(

3276

8/19

683

3953

)2

(

1419

)2

(

5/3

3 ôô/2

öù 1723

)2

(

27/1

6×1.

3333

338180

107/

64

2331

)2

(

3243

)2

(

34

)2

(, o

r ap

pro

xim

atel

y 37

/22

32/1

9

2048

/121

5

1.64

5813

1.64

6091

1.64

8244

1.64

9721

1.65

0000

1.65

5172

1.65

5507

1.65

6250

1.66

2476

1.66

4787

1.66

5377

1.66

6524

1.66

6667

1.66

8267

1.66

9169

1.67

1851

1.67

1875

1.67

2418

1.67

5029

1.68

1793

1.68

4211

1.68

5597

430.

587

430.

659

431.

223

431.

609

431.

682

433.

035

433.

123

433.

317

434.

946

435.

551

435.

705

436.

005

436.

043

436.

461

436.

697

437.

399

437.

405

437.

547

438.

230

440.

000

440.

633

440.

995

862.

560

862.

852

865.

116

866.

667

866.

959

872.

378

872.

727

873.

505

880.

000

882.

405

883.

019

884.

211

884.

359

886.

020

886.

957

889.

735

889.

760

890.

323

893.

023

900.

000

902.

487

903.

911

Appendices 337

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

Õ×Ò

Õ×Ó

Õ×Ô

Õ×Õ

Õ×Ö

Õ××

Õ×Ø

Õ×Ù

ÕØÐ

ÕØÑ

ÕØÒ

ÕØÓ

ÕØÔ

ÕØÕ

ÕØÖ

ÕØ×

ÕØØ

ÕØÙ

ÕÙÐ

ÕÙÑ

ÕÙÒ

ÕÙÓ

53-e

t Pyt

hago

rean

maj

or s

ixth

Pyth

agor

ean

maj

or s

ixth

(A) I

II (27

th h

arm

onic

)

56th

cyc

lic fi

fth (A

) LV

I

21-e

t Pyt

hago

rean

maj

or s

ixth

17-e

t acu

te o

r lar

ge m

ajor

six

th

sep

tend

ecim

al s

uper

maj

or s

ixth

43-e

t dim

inish

ed s

even

th

109t

h ha

rmon

ic

trivi

gesim

al s

uper

maj

or s

ixth

13-e

t acu

te o

r lar

ge m

ajor

six

th

und

ecim

al g

rave

or s

mal

l aug

men

ted

six

th

dim

inish

ed s

even

th, o

r acu

te o

r lar

ge m

ajor

six

th

22-e

t acu

te o

r lar

ge m

ajor

six

th

53-e

t acu

te o

r lar

ge m

ajor

six

th

31-e

t sup

erm

ajor

six

th, o

r dim

inish

ed s

even

th

Pyth

agor

ean

dou

ble

augm

ente

d fi

fth, o

r cyc

lic a

cute

or l

arge

maj

or s

ixth

(A) X

V

mea

nton

e d

imin

ished

sev

enth

(D)

IX41

2�

sep

timal

sup

erm

ajor

six

th

9-et

nin

etee

n q

uarte

r-ton

es

und

ecim

al s

uper

maj

or s

ixth

(55t

h ha

rmon

ic)

23-e

t nin

etee

n q

uarte

r-ton

es

nine

teen

qua

rter-t

ones

, or u

ntrig

esim

al s

uper

maj

or s

ixth

4053

)2

(

3 ó/2

ô, or

27/

16

3 õö/2

øø 1621

)2

(

1317

)2

(

17/1

0 3343

)2

(

109/

64

46/2

7 1013

)2

(

75/4

4

128/

75

1722

)2

(

4153

)2

(

2431

)2

(

3 ñõ/2

òó, o

r 143

4890

7/83

8860

8

3276

8/19

683×

0.44

4444

8081

12/7

79

)2

(

55/3

2 1823

)2

(

31/1

8

1.68

7301

1.68

7500

1.69

1027

1.69

5728

1.69

9024

1.70

0000

1.70

2249

1.70

3125

1.70

3704

1.70

4361

1.70

4545

1.70

6667

1.70

8496

1.70

9512

1.71

0234

1.71

0523

1.71

1975

1.71

4286

1.71

4488

1.71

8750

1.72

0239

1.72

2222

441.

441

441.

493

442.

416

443.

646

444.

508

444.

763

445.

352

445.

581

445.

732

445.

904

445.

953

446.

508

446.

986

447.

252

447.

441

447.

517

447.

896

448.

501

448.

554

449.

669

450.

058

450.

577

905.

660

905.

865

909.

480

914.

286

917.

647

918.

642

920.

930

921.

821

922.

409

923.

077

923.

264

925.

418

927.

273

928.

302

929.

032

929.

325

930.

794

933.

129

933.

333

937.

632

939.

130

941.

126

338 Appendices

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

ÕÙÔ

ÕÙÕ

ÕÙÖ

ÕÙ×

ÕÙØ

ÕÙÙ

ÖÐÐ

ÖÐÑ

ÖÐÒ

ÖÐÓ

ÖÐÔ

ÖÐÕ

ÖÐÖ

ÖÐ×

ÖÐØ

ÖÐÙ

ÖÑÐ

ÖÑÑ

ÖÑÒ

ÖÑÓ

ÖÑÔ

ÖÑÕ

14-e

t aug

men

ted

six

th

nona

vige

simal

gra

ve o

r sm

all a

ugm

ente

d s

ixth

19-e

t nin

etee

n q

uarte

r-ton

es

43-e

t trip

le d

imin

ished

oct

ave

trid

ecim

al g

rave

or s

mal

l aug

men

ted

six

th

nine

teen

eq

ual q

uarte

r-ton

es

53-e

t aug

men

ted

six

th

cycl

ic a

ugm

ente

d six

th (A

) XX

VII

111t

h ha

rmon

ic

augm

ente

d s

ixth

trivi

gesim

al s

ubm

inor

sev

enth

5-et

aug

men

ted

six

th

untri

gesim

al s

ubm

inor

sev

enth

mea

nton

e au

gmen

ted

six

th (A

) X21

2ß

31-e

t aug

men

ted

six

th, o

r sub

min

or s

even

th

sep

timal

sub

min

or s

even

th (

7th

harm

onic

)

21-e

t gra

ve o

r sm

all m

inor

sev

enth

nona

dec

imal

sup

erau

gmen

ted

six

th

53-e

t gra

ve o

r sm

all m

inor

sev

enth

16-e

t gra

ve o

r sm

all m

inor

sev

enth

cycl

ic g

rave

or s

mal

l min

or s

even

th (A

) XX

XIX

acut

e or

larg

e au

gmen

ted

six

th, o

r gra

ve o

r sm

all m

inor

sev

enth

1114

)2

(

50/2

9 1519

)2

(

3443

)2

(

45/2

6 19

24)

2(

, or a

pp

roxi

mat

ely

45/2

6 42

53)

2(

3 ò÷/2

ôò

111/

64

125/

72

40/2

3 45

)2

(

54/3

1

5904

9/32

768×

0.4

8180

2531

)2

(

7/4 17

21)

2(

100/

57

4353

)2

(

1316

)2

(

3 óù/2

öñ

225/

128

1.72

3946

1.72

4138

1.72

8444

1.72

9911

1.73

0769

1.73

1073

1.73

2017

1.73

3860

1.73

4375

1.73

6111

1.73

9130

1.74

1101

1.74

1935

1.74

6928

1.74

8905

1.75

0000

1.75

2633

1.75

4386

1.75

4817

1.75

6252

1.75

7516

1.75

7813

451.

028

451.

079

452.

205

452.

589

452.

813

452.

893

453.

140

453.

622

453.

757

454.

211

455.

001

455.

517

455.

735

457.

041

457.

558

457.

845

458.

534

458.

992

459.

105

459.

480

459.

811

459.

889

942.

857

943.

050

947.

368

948.

837

949.

696

950.

000

950.

943

952.

785

953.

299

955.

031

958.

039

960.

000

960.

829

965.

784

967.

742

968.

826

971.

429

973.

159

973.

585

975.

000

976.

245

976.

537

Appendices 339

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

ÖÑÖ

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ÖÒÑ

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ÖÓÔ

ÖÓÕ

ÖÓÖ

ÖÓ×

43-e

t aug

men

ted

six

th

11-e

t gra

ve o

r sm

all m

inor

sev

enth

sep

tend

ecim

al s

ubm

inor

sev

enth

113t

h ha

rmon

ic

17-e

t Pyt

hago

rean

min

or s

even

th

sep

tend

ecim

al s

uper

augm

ente

d s

ixth

23-e

t Pyt

hago

rean

min

or s

even

th

Pyth

agor

ean

min

or s

even

th (D

) II

53-e

t Pyt

hago

rean

min

or s

even

th

sup

erau

gmen

ted

six

th

nona

dec

imal

sub

min

or s

even

th (

57th

har

mon

ic)

cycl

ic m

inor

sev

enth

(A)

LI

equa

l min

or s

even

th

sep

timal

sup

erau

gmen

ted

six

th

43-e

t Pyt

hago

rean

min

or s

even

th

31-e

t sup

erau

gmen

ted

six

th, o

r min

or s

even

th

mea

nton

e m

inor

sev

enth

(D)

21�

19-e

t acu

te o

r lar

ge m

inor

sev

enth

untri

gesim

al s

uper

augm

ente

d s

ixth

trivi

gesim

al s

uper

augm

ente

d six

th (

115t

h ha

rmon

ic)

13-e

t Pyt

hago

rean

min

or s

even

th

acut

e or

larg

e m

inor

sev

enth

3543

)2

(

911

)2

(

30/1

7

113/

64

1417

)2

(

85/4

8 1923

)2

(

16/9

4453

)2

(

3645

/204

8

57/3

2

3 õñ/2

øð 5

6)

2(

, or a

pp

roxi

mat

ely

98/5

5

25/1

4 3643

)2

(

2631

)2

(

16/9

×2

8081

1619

)2

(

775/

432

115/

64

1113

)2

(

9/5

1.75

8022

1.76

3183

1.76

4706

1.76

5625

1.76

9730

1.77

0833

1.77

2870

1.77

7778

1.77

7918

1.77

9785

1.78

1250

1.78

1494

1.78

1797

1.78

5714

1.78

6591

1.78

8450

1.78

8854

1.79

2664

1.79

3981

1.79

6875

1.79

7702

1.80

0000

459.

944

461.

294

461.

692

461.

933

463.

007

463.

295

463.

828

465.

112

465.

149

465.

637

466.

021

466.

084

466.

164

467.

885

467.

418

467.

904

468.

010

469.

007

469.

351

470.

108

470.

325

470.

926

976.

744

981.

818

983.

313

984.

215

988.

235

989.

314

991.

304

996.

090

996.

226

998.

044

999.

468

999.

705

1000

.000

1003

.802

1004

.651

1006

.452

1006

.843

1010

.526

1011

.798

1014

.588

1015

.385

1017

.596

340 Appendices

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

ÖÓØ

ÖÓÙ

ÖÔÐ

ÖÔÑ

ÖÔÒ

ÖÔÓ

ÖÔÔ

ÖÔÕ

ÖÔÖ

ÖÔ×

ÖÔØ

ÖÔÙ

ÖÕÐ

ÖÕÑ

ÖÕÒ

ÖÕÓ

ÖÕÔ

ÖÕÕ

ÖÕÖ

ÖÕ×

ÖÕØ

ÖÕÙ

53-e

t acu

te o

r lar

ge m

inor

sev

enth

Pyth

agor

ean

augm

ente

d s

ixth

, or c

yclic

acu

te o

r lar

ge m

inor

sev

enth

(A) X

20-e

t acu

te o

r lar

ge m

inor

sev

enth

trid

ecim

al g

rave

or s

mal

l neu

tral s

even

th

7-et

twen

ty-o

ne q

uarte

r-ton

es

nona

vige

simal

gra

ve o

r sm

all n

eutra

l sev

enth

(29

th h

arm

onic

)

43-e

t dou

ble

dim

inish

ed o

ctav

e

und

ecim

al g

rave

or s

mal

l neu

tral s

even

th

22-e

t tw

enty

-one

qua

rter-t

ones

neut

ral s

even

th

15-e

t neu

tral s

even

th

53-e

t neu

tral s

even

th

cycl

ic n

eutra

l sev

enth

(A)

XXII

23-e

t neu

tral s

even

th

trid

ecim

al n

eutra

l sev

enth

(117

th h

arm

onic

)

31-e

t dou

ble

augm

ente

d s

ixth

, or n

eutra

l sev

enth

neut

ral s

even

th

acut

e or

larg

e d

oubl

e au

gmen

ted

six

th

und

ecim

al a

cute

or l

arge

neu

tral s

even

th

twen

ty-o

ne e

qua

l qua

rter-t

ones

17-e

t gra

ve o

r sm

all m

ajor

sev

enth

59th

har

mon

ic

4553

)2

(

3 ñð/2

ñõ, o

r 590

49/3

2768

17

20)

2(

65/3

6 67

)2

(

29/1

6 3743

)2

(

20/1

1 1922

)2

(

729/

400

1315

)2

(

4653

)2

(

3 òò/2

óô 2023

)2

(

117/

64

2731

)2

(

4000

/218

7

1875

/102

4

11/6

7

8)

2(

, or a

ppro

xim

atel

y 93

9/51

2 15

17)

2(

59/3

2

1.80

1323

1.80

2032

1.80

2501

1.80

5556

1.81

1447

1.81

2500

1.81

5624

1.81

8181

1.81

9619

1.82

2500

1.82

3445

1.82

5036

1.82

6618

1.82

7112

1.82

8125

1.82

8889

1.82

8989

1.83

1055

1.83

3333

1.83

4008

1.84

3379

1.84

3750

471.

272

471.

458

471.

580

472.

379

473.

921

474.

196

475.

014

475.

683

476.

059

476.

813

477.

060

477.

476

477.

890

478.

019

478.

284

478.

484

478.

510

479.

051

479.

647

479.

823

482.

275

482.

372

1018

.868

1019

.550

1020

.000

1022

.931

1028

.571

1029

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1032

.558

1034

.996

1036

.364

1039

.103

1040

.000

1041

.509

1043

.010

1043

.478

1044

.438

1045

.161

1045

.256

1047

.210

1049

.363

1050

.000

1058

.824

1059

.172

Appendices 341

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

ÖÖÐ

ÖÖÑ

ÖÖÒ

ÖÖÓ

ÖÖÔ

ÖÖÕ

ÖÖÖ

ÖÖ×

ÖÖØ

ÖÖÙ

Ö×Ð

Ö×Ñ

Ö×Ò

Ö×Ó

Ö×Ô

Ö×Õ

Ö×Ö

Ö××

Ö×Ø

Ö×Ù

ÖØÐ

ÖØÑ

43-e

t dou

ble

augm

ente

d s

ixth

trid

ecim

al a

cute

or l

arge

neu

tral s

even

th

53-e

t gra

ve o

r sm

all m

ajor

sev

enth

cycl

ic g

rave

or s

mal

l maj

or s

even

th (A

) XX

XIV

9-et

gra

ve o

r sm

all m

ajor

sev

enth

grav

e or

sm

all m

ajor

sev

enth

19-e

t gra

ve o

r sm

all m

ajor

sev

enth

sept

ende

cim

al n

eutra

l sev

enth

(11

9th

harm

onic

)

nona

vige

simal

acu

te o

r lar

ge n

eutra

l sev

enth

10-e

t jus

t dia

toni

c m

ajor

sev

enth

mea

nton

e m

ajor

sev

enth

(A) V

411ß

31-e

t jus

t dia

toni

c m

ajor

sev

enth

21-e

t jus

t dia

toni

c m

ajor

sev

enth

Pyth

agor

ean

dim

inish

ed o

ctav

e (D

) VII

53-e

t jus

t dia

toni

c m

ajor

sev

enth

just

dia

toni

c m

ajor

sev

enth

(15

th h

arm

onic

)

43-e

t jus

t dia

toni

c m

ajor

sev

enth

cycl

ic d

iato

nic

maj

or s

even

th (A

) XLV

I

11-e

t jus

t dia

toni

c m

ajor

sev

enth

23-e

t jus

t dia

toni

c m

ajor

sev

enth

equa

l maj

or s

even

th

sep

tend

ecim

al s

uper

maj

or s

even

th

3843

)2

(

24/1

3 4753

)2

(

3 óô/2

õó 89

)2

(

50/2

7 1719

)2

(

119/

64

54/2

9 910

)2

(

243/

128×

0.8

8180

2831

)2

(

1921

)2

(

4096

/218

7 48

53)

2(

15/8

3943

)2

(

3 ôö/2

÷ò 1011

)2

(

2123

)2

(

1112

)2

(, o

r app

roxi

mat

ely

967/

512

17/9

1.84

5128

1.84

6154

1.84

9061

1.85

1539

1.85

1749

1.85

1852

1.85

9271

1.85

9375

1.86

2069

1.86

6066

1.86

9186

1.87

0243

1.87

2235

1.87

2885

1.87

3402

1.87

5000

1.87

5112

1.87

6800

1.87

7862

1.88

3014

1.88

7749

1.88

8889

482.

733

483.

001

483.

762

484.

410

484.

465

484.

492

486.

433

486.

460

487.

165

488.

211

489.

027

489.

303

489.

825

489.

995

490.

130

490.

548

490.

577

491.

019

491.

297

492.

645

493.

883

494.

182

1060

.465

1061

.427

1064

.151

1066

.470

1066

.667

1066

.762

1073

.684

1073

.781

1076

.288

1080

.000

1082

.892

1083

.871

1085

.714

1086

.315

1086

.792

1088

.269

1088

.372

1089

.930

1090

.909

1095

.652

1100

.000

1101

.045

342 Appendices

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

ÖØÒ

ÖØÓ

ÖØÔ

ÖØÕ

ÖØÖ

ÖØ×

ÖØØ

ÖØÙ

ÖÙÐ

ÖÙÑ

ÖÙÒ

ÖÙÓ

ÖÙÔ

ÖÙÕ

ÖÙÖ

ÖÙ×

ÖÙØ

ÖÙÙ

×ÐÐ

×ÐÑ

×ÐÒ

×ÐÓ

grav

e or

sm

all n

eutra

l sev

enth

(12

1st h

arm

onic

)

nona

dec

imal

sup

erm

ajor

sev

enth

13-e

t Pyt

hago

rean

maj

or s

even

th

grav

e or

sm

all d

imin

ished

oct

ave

53-e

t Pyt

hago

rean

maj

or s

even

th

Pyth

agor

ean

maj

or s

even

th (A

) V

58th

cyc

lic fi

fth (A

) LV

III

14-e

t Pyt

hago

rean

maj

or s

even

th

sep

timal

sup

erm

ajor

sev

enth

43-e

t dim

inish

ed o

ctav

e

61st

har

mon

ic

15-e

t jus

t dia

toni

c m

ajor

sev

enth

31-e

t sup

erm

ajor

sev

enth

, or d

imin

ished

oct

ave

mea

nton

e d

imin

ished

oct

ave

(D) V

II4

3 1�

16-e

t acu

te o

r lar

ge m

ajor

sev

enth

trivi

gesim

al s

uper

maj

or s

even

th

dim

inish

ed o

ctav

e, o

r acu

te o

r lar

ge m

ajor

sev

enth

17-e

t acu

te o

r lar

ge m

ajor

sev

enth

123r

d ha

rmon

ic

53-e

t acu

te o

r lar

ge m

ajor

sev

enth

cycl

ic a

cute

or l

arge

maj

or s

even

th (A

) XV

II

18-e

t acu

te o

r lar

ge m

ajor

sev

enth

121/

64

36/1

9 1213

)2

(

256/

135

4953

)2

(

3 õ/2

÷, or

243

/128

3 õø/2

ùñ 1314

)2

(

40/2

1 4043

)2

(

61/3

2 1415

)2

(

2931

)2

(

4096

/218

7×0.

5714

298081

15

16)

2(

23/1

2

48/2

5 1617

)2

(

123/

64

5053

)2

(

3 ñ÷/2

òö 1718

)2

(

1.89

0625

1.89

4737

1.89

6155

1.89

6296

1.89

8064

1.89

8438

1.90

2406

1.90

3390

1.90

4762

1.90

5583

1.90

6250

1.90

9683

1.91

2532

1.91

4046

1.91

5207

1.91

6667

1.92

0000

1.92

0093

1.92

1875

1.92

3050

1.92

4338

1.92

4448

494.

636

495.

712

496.

083

496.

120

496.

582

496.

680

497.

718

497.

976

498.

334

498.

549

498.

724

499.

622

500.

367

500.

763

501.

067

501.

449

502.

321

502.

346

502.

812

503.

119

503.

456

503.

485

1102

.636

1106

.397

1107

.692

1107

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1109

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1109

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1113

.390

1114

.286

1115

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1116

.279

1116

.885

1120

.000

1122

.581

1123

.951

1125

.000

1126

.319

1129

.328

1129

.412

1131

.017

1132

.075

1133

.235

1133

.333

Appendices 343

DEG

REE

NUM

BER

INTE

RVA

L FA

CTO

R RA

TIO

(DEC

IMA

L)

FREQ

UEN

CY

(HER

TZ)

CEN

TS

×ÐÔ

×ÐÕ

×ÐÖ

×Ð×

×ÐØ

×ÐÙ

×ÑÐ

×ÑÑ

×ÑÒ

×ÑÓ

×ÑÔ

×ÑÕ

×ÑÖ

×Ñ×

×ÑØ

×ÑÙ

×ÒÐ

×ÒÑ

×ÒÒ

×ÒÓ

×ÒÔ ÐÑ

19-e

t tw

enty

-thre

e q

uarte

r-ton

es

20-e

t tw

enty

-thre

e q

uarte

r-ton

es

21-e

t tw

enty

-thre

e q

uarte

r-ton

es

twen

ty-th

ree

qua

rter-t

ones

43-e

t dou

ble

dim

inish

ed s

econ

d

untri

gesim

al s

uper

maj

or s

even

th (

31st

har

mon

ic)

22-e

t tw

enty

-thre

e q

uarte

r-ton

es

und

ecim

al s

ubd

imin

ished

oct

ave

23-e

t tw

enty

-thre

e q

uarte

r-ton

es

twen

ty-th

ree

equa

l qua

rter-t

ones

53-e

t sub

octa

ve

cycl

ic s

uboc

tave

(A)

XXIX

mea

nton

e au

gmen

ted

sev

enth

, or s

uboc

tave

(A)

XII

3ß

(12

5th

harm

onic

)

31-e

t aug

men

ted

sev

enth

, sub

octa

ve

43-e

t aug

men

ted

sev

enth

sep

timal

sub

dim

inish

ed o

ctav

e (6

3rd

har

mon

ic)

53-e

t gra

ve o

r sm

all o

ctav

e

grav

e or

sm

all o

ctav

e

cycl

ic g

rave

or s

mal

l oct

ave

(A)

XLI

acut

e or

larg

e m

eant

one

augm

ente

d s

even

th

127t

h ha

rmon

ic

octa

ve (

2nd

har

mon

ic)

1819

)2

(

1920

)2

(

2021

)2

(

60/3

1 4143

)2

(

31/1

6 2122

)2

(

64/3

3 2223

)2

(

2324

)2

(, o

r app

roxi

mat

ely

1024

/527

51

53)

2(

3òù/2

ôõ

5314

41/5

2428

8×0.

3333

338180

, or 1

25/6

4 30

31)

2(

42

43)

2(

63/3

2 5253

)2

(

160/

81

3ôñ/2

öô

2025

/102

4

127/

64

2/1

1.92

8352

1.93

1873

1.93

5064

1.93

5484

1.93

6549

1.93

7500

1.93

7969

1.93

9394

1.94

0626

1.94

3064

1.94

8365

1.95

0593

1.95

3125

1.95

5777

1.96

8019

1.96

8750

1.97

4014

1.97

5309

1.97

7205

1.97

7539

1.98

4375

2.00

0000

504.

506

505.

427

506.

262

506.

372

506.

651

506.

900

507.

022

507.

395

507.

717

508.

355

509.

742

510.

325

510.

987

511.

681

514.

884

515.

075

516.

452

516.

791

517.

287

517.

375

519.

163

523.

251

1136

.842

1140

.000

1142

.857

1143

.233

1144

.186

1145

.036

1145

.455

1146

.727

1147

.826

1150

.000

1154

.717

1156

.695

1158

.941

1161

.290

1172

.093

1172

.736

1177

.358

1178

.494

1180

.155

1180

.447

1186

.422

1200

.000

344 Appendices

Appendix B

Microtonal Notation Font

! " # $ % & '

32 33 34 35 36 37 38 39

( ) * + , - . /

40 41 42 43 44 45 46 47

0 1 2 3 4 5 6 7

48 49 50 51 52 53 54 55

8 9 : ; < = > ? 56 57 58 59 60 61 62 63

@ A B C D E F G

64 65 66 67 68 69 70 71

H I J K L M N O

72 73 74 75 76 77 78 79

P Q R S T U V W

80 81 82 83 84 85 86 87

X Y Z [ \ ] ^ _

88 89 90 91 92 93 94 95

` a b c d e f g

96 97 98 99 100 101 102 103

h i j k l m n o

104 105 106 107 108 109 110 111

p q r s t u v w

112 113 114 115 116 117 118 119

x y z { | } ~

120 121 122 123 124 125 126 127

‚ ƒ „ … † ‡

128 129 130 131 132 133 134 135

ˆ ‰ Š ‹ Œ

136 137 138 139 140 141 142 143

346 Appendices

‘ ’ “ ” • – —

144 145 146 147 148 149 150 151

˜ ™ š › œ Ÿ

152 153 154 155 156 157 158 159

¡ ¢ £ ¤ ¥ ¦ §

160 161 162 163 164 165 166 167

¨ © ª « ® ¯

168 169 170 171 172 173 174 175

° ± ² ³ ´ μ ¶ ·

176 177 178 179 180 181 182 183

¸ ¹ º » ¼ ½ ¾ ¿

184 185 186 187 188 189 190 191

À Á Â Ã Ä Å Æ Ç

192 193 194 195 196 197 198 199

Appendices 347

È É Ê Ë Ì Í Î Ï

200 201 202 203 204 205 206 207

Ð Ñ Ò Ó Ô Õ Ö ×

208 209 210 211 212 213 214 215

Ø Ù Ú Û Ü Ý

216 217 218 219 220 221 222 223

à á â ã ä å æ ç

224 225 226 227 228 229 230 231

è é ê ë ì í î ï

232 233 234 235 236 237 238 239

ð ñ ò ó ô õ

240 241 242 243 244 245 246 247

ø ù ú û ü ý þ ÿ

248 249 250 251 252 253 254 255

348 Appendices