i

c. 2012 Peter Alexander Thoegersen

ii

POLYTEMPIC POLYMICROTONAL MUSIC: “A Road Less Traveled”

BY

PETER ALEXANDER THOEGERSEN

THESIS

Submitted in partial fulfillment of the requirements

for the degree of Doctor of Musical Arts in Music

with a concentration in Music Composition

in the Graduate College of the

University of Illinois at Urbana-Champaign, 2012

Urbana, Illinois

Doctoral Committee:

Associate Professor Stephen A. Taylor (chair)

Professor Heinrich Taube

Assistant Professor Reynold Tharp

Associate Professor Chester Alwes

iii

Dedicated to my father, Egil, and Milko, my cat and best friend. Also, in memory of Astrid and Meash

iv

Acknowledgements

I would never have even been here were it not for the support of my father, Egil

Thoegersen, and his wife, Suzanne Olsen Thoegersen.

I would like to give thanks to certain individuals without whom my research

would have been much more difficult, in particular my drum teacher Freddie Gruber, for

his having advised me that I had the thinking of a composer way back when, while I was

trying to become the “greatest drummer in the world.”

I would like to thank Johnny Reinhard for his help in understanding the nature of

Ives’s Universe Symphony, and polymicrotonality in general.

I would like to thank the faculty at the School of Music from the University of

Illinois for allowing me back into the fold after a personal period of illness: Erik Lund,

Rick Taube, Scott Wyatt, Steve Taylor, and Reynold Tharp.

I thank my committee for their time and well needed criticism.

I would like to thank John Wagstaff and Chris Pawlicki for helping me in my

queries at the greatest music library in the country, the UIUC Music library.

I would like to thank Rod Butler, now deceased, who was my first composition

teacher at Cal State Dominguez Hills, and who believed in a late starter like me. Also, I

would like to thank Burns Taft, of Ventura College, for his recognition in my interest in

composition very early in my development.

Lastly, I would like to thank John M Kennedy, and Bill Kraft, for both of their

open mindedness regarding my compositional aesthetic. I would also like to thank Kyle

Gann and Brian Ferneyhough for their supportive email correspondences, which have

been very encouraging for me.

v

Abstract

This paper introduces polytempic polymicrotonality as a new musical aesthetic.

Microtonality is the basis for its inception, from which the discussion proposes music

with more than one microtonal tuning system. Examples from the literature are discussed

to give an historic framework showing that this tendency has been present throughout

human musical history. Polytempo is a tool for which polymicrotonal structures can

function in relief from its background. Polytempo acts as a frame, or ground structure,

that is multi-dimensional, akin to the advancement of perspective in Renaissance art.

Examples of music literature are displayed for musical precedence in this area, focusing

on Charles Ives’s Universe Symphony, unfinished since 1925, and realized recently by

Johnny Reinhard in 1996.

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TABLE OF CONTENTS

CHAPTER 1: Introduction ................................................................................................. 1

CHAPTER 2: PITCH: Historical Information Leading to the Possibility of

Polymicrotonality: Representative Figures and Works .....................................................11

CHAPTER 3: RHYTHM: Representative Twentieth Century Polytempic Works ...........98

CHAPTER 4: PITCH + RHYTHM: Literature and Compositions Suggesting

Relationships between Microtonal Pitch and Rhythm, in Terms of Tempo,

Leading to the Union of Polymicrotonality and Polytempo; Henry Cowell,

Ivan Wyschnegradsky, and Ben Johnston ...................................................................... 128

CHAPTER 5: The Progenitor Charles Ives, His Universe Symphony and its Legacy:

Polymicrotonal Polytempic Art Music and Its Practice .................................................. 167

CHAPTER 6: Conclusions ..............................................................................................204 203

BIBLIOGRAPHY ......................................................................................................... 212

1

CHAPTER 1: INTRODUCTION

“All traditions belong to me, if I claim them.”

Ben Johnston1

1.1. THE SIGNIFICANCE OF POLYMICROTONAL POLYTEMPIC MUSIC

Charles Ives’s Universe Symphony is the first, and up to now, possibly the only

polytempic polymicrotonal work. Written between 1915 -1926, the Universe Symphony

lay in disrepair for half a century until two individuals, Larry Austin in 1974, and Johnny

Reinhard in 1996, pieced together the scraps and notes of the score, [in their original

intention, as Ives proscribed in his notes], and, released two vastly different versions. The

version that best suits the scope of polymicrotonality is Reinhard’s version, from 1996,

whom released the official Universe Symphony score and recording through the

American Festival of Microtonal Music, approved by the Charles Ives Society.2

Within the Universe Symphony, there are three levels of tempi and four different

tunings. These impart a structurally-stratified, deeper, and more powerful listening

experience for the audience than even Ives’s Symphony No. 4, which is already a

wonderfully temporally and rhythmically stratified architecture. The differing tunings

yield microtones that go well beyond Ives’s Three Quartertone Pieces, and point to a

completely startling, evocative, and pioneering use of pitch color never before heard.

The significance and vast potential of polytempic-polymicrotonal music can be

seen by comparing this phenomenon with the similar polytextuality of the early motets of

the fourteenth century, the Italian trecento. Then polyphony was in its infancy, and

1 Johnston, 2006. Maximum Clarity. p. 119.

2 Reinhard. 2004. The Ives Universe: a symphonic odyssey. Self -Published via AFMM, NY. New York.

2

composers were willing to separate the simultaneous melodies by using differing texts,

which were often polytextual literature (e.g. French and Latin3) in order to create

independence and individuality in the melodic lines. The trecento polytextuality is a

perfect analogy for polymicrotonality, except gradations of pitch are used in the latter to

define the voice part in extreme subtlety; further independence is achieved by setting

each line in a different tempo. Just as it is possible that the trecento composer

experienced a greater freedom than many twentieth-century composers, it is this author’s

intent to help define a style that takes Ives’s Universe Symphony as a foundation, building

thereon music of an unprecedented freedom for the twenty-first century.

1.2. EXTANT MUSICAL WORKS APPROACHING POLYMICROTONALITY AND

POLYTEMPO AND THE HISTORIC SIGNIFICANCE OF MICROTONALITY

Polytempic structures, found in the works of Conlon Nancarrow, Karlheinz

Stockhausen [Gruppen] Charles Ives, among others, will also be explored culminating in

the actual simultaneous use of multiple tunings and multiple tempi in Ives’s Universe

Symphony.

3 The thirteenth century motet has often been regarded as the most difficult type of composition to

understand and appreciate in the whole history of western music before 20th

century, not only because of

the simultaneous performance of two or more different texts, sometimes in two different languages.

(Harmon, 1958)

3

1.3. THE GOALS AND CONTENTS OF THIS PAPER

This is not an analytical paper, but rather a type of Hegelian dialectic synthesis by

examining the historical evidence for polymicrotonality in Greece and potentially Europe

from Zarlino to Werckmeister, and building thereupon for a new music for the Twenty-

first century. There will be some examples illustrating tempi and tunings, but there is no

systematic attempt to construct an analytical philosophy, or even a theory of

polymicrotonality or polytempic strata. Composition is the converse of analysis, and this

paper hopes to find its way to elucidate this relatively unknown area of music. I leave it

to future theorists and musicologists to devise an analytical strategy with respect to this

area of music, if it does, in fact, become a genre in its own right. Here, I present a brief

history of microtonality and its theoretical implications, and an original composition: a

polytempic polymicrotonal string quartet, entitled Hypercube, in order to demonstrate the

some of the key ideas in this paper.

The following numbered points will provide descriptions of the materials and

information covered in this paper.

1.3.1. Historical Background of Polymicrotonality and Representative Works: Exclusion

of Certain Historical Figures

This inquiry will briefly cover the Greek Generas divisions of the tetrachord,

which yielded its own unique, microtonal tuning; the obvious question is how it managed

to influence European tuning into the modern era.

4

Just intonation in the early Renaissance through Zarlino will be explored, along

with its influence on tuning by introducing the just third. Among others Mersenne’s

experiments with split key organs having its influence, in turn, on Vicentino’s tunings

including 19 tones to the octave all the way up to Jean Etienne Marie’s polymicrotonal

work Le Tombeau de Carrillo will also be reviewed.

The curious reader may wonder why Harry Partch or other microtonal pioneers

are not included in this paper. The reason is twofold: one, he is neither polymicrotonal

nor polyphonic in his approach to music, and two, he is similarly not polytempic. Alois

Hába, similarly, will also not receive a more in depth consideration for the same reasons:

he does not fit the scope of this thesis due to their not falling into the class of

polymicrotonal or polytempo composers, and additionally, much literature is already

written about both of these great pioneers. This author’s lack of inclusion of both these

composers is in no way meant as deprecation, whatsoever. I do include

Wyschnegradsky, who did employ polymicrotonality along with a rhythmic system based

on the overtone series.

It is understandable that there will be dissension about Harry Partch not being

included in this paper, when in fact he does not fit into this author’s discussion with

respect to polymicrotonality and that recently, the notion of undertones as an actual

acoustic reality has been proven false: they do not exist in nature, even if the

mathematical and theoretical implications may be valid.4 Partch, however, does represent

another arm of the influence of Henry Cowell, who actually advocated the notion of

undertones, and who was a seminal influence on Partch.

4 Rehding, 2003. Hugo Riemann and the Birth of Modern Musical Thought.p. 16-17.

5

1.3.2. Representative Polytempic Works from the twentieth Century

Nancarrow’s actual physical representation of Henry Cowell’s ideas will be

explored as Nancarrow is quite possibly one of the greatest polytempic composers of all

time. His influence on Ligeti is legendary, leading to Ligeti’s own Piano Etudes, and his

sway is felt still today.

In addition to Nancarrow, Ives’s output features several works with polytempic

structures, which will also be examined, as polytempi seem to be a largely American

innovation in twentieth-century music, with a lineage from Ives, to Cowell, to

Nancarrow, and Elliott Carter. With regard to Stockhausen’s Gruppen, he was influenced

by the work of Cowell in relating rhythmic values to the overtone series. Although

Boulez’s Rituel: in memorium Bruno Maderna (1974) is composed of eight different

groups with their own tempos, it is not polytempo so much as it is antiphonal, since the

work is referencing Maderna’s Quadrivium (1969), which is antiphonal.

1.3.3. Literature and Works Suggesting Relationships Between Pitch and Rhythm

Henry Cowell’s manifesto New Musical Resources, written at the age of 20,

explores the heart of polytempic polymicrotonality by examining the overtone series

relationships by ratio, and then analogizing them to rhythmic structures ranging from

note duration, through meter, and ultimately to tempo itself, as evinced by Ben

Johnston’s string quartets, where his tempo relationships reveal the underpinnings of the

just intoned ratios already in use in his pitch vocabulary. Nancarrow also initially tested

these tempo relationships by way of Cowell’s erudition and suggestion.

6

Some of the integral serialists, such as Stockhausen, and Messiaen and tonal

composers from Riemann to Stockhausen have all tried in some measure to discover the

direct link between pitch and rhythm. The problem encountered by serializing rhythm

was the connect-the-dot nature of the compositional process, which would preclude

choice on behalf of the composer. The Enlightenment musicians Jean Phillipe Rameau

and Moritz Hauptmann found a relationship between the first few overtone series partials

and simple rhythms for tonal cadences. The serialists tried to pair pitch to rhythm

algorithmically, but it proved inconclusive due to the limitations of serialism, the

negation of choice, and the lack of ability to regulate timbre. Perhaps this idea of a direct

link is a mistaken one: how can you relate what is already the same phenomenon? Or,

rather, why are we thinking of them as different? Frequency is periodic. Frequency is the

periodicity of both rhythm and pitch. Where periodicity concerns rhythm, a slow enough

cycle, under 20 hertz, will be perceived by the ear as discrete “beats.” Above 20 Hertz,

the periodicity becomes blurred into a low frequency. Therefore, the psychoacoustical

nature of the human animal will perceive the same phenomena of periodicity as two

distinct and separate objects, when in fact, they are the same object at different states.

Karlheinz Stockhausen’s in How Time Passes presents a "new morphology of

musical time" used originally via Henry Cowell's insight, in his application of the

harmonic overtone series to musical rhythm, and predated Stockhausen's by 35 years.

Stockhausen’s failure regarding the serial control over timbre, however, was more about

controlling timbre and pitch to rhythmic compositional technique via serial methods.

Stockhausen was more interested in the applicability of this serial composition method to

duration. One can compare Cowell and Stockhausen in their use of overtone series and

7

use of instruments capable of performing according to their rhythmical ideas. So, the

notion of linking needs to be changed to a notion of correspondence, since it is not logical

to link something to itself, as that is a redundancy.

Pitch and rhythm, as the same phenomena, therefore can correspond to itself, and

create a relationship to itself, but the idea that they are separate is an illusion. This

relationship has been a riddle for composers for centuries because the quest has probably

been an erroneous one based on a lack of knowledge concerning basic acoustics that

could only have come about in the twentieth-century through technology. Nevertheless, it

is satisfying to have an organic union between these seemingly disparate parameters of

music. Henry Cowell, Ben Johnston, and Karlheinz Stockhausen have suggested the

systematization of the overtone series to rhythmic groupings and other structures, which

is inevitably arbitrary, since it is like linking water to ice to vapor. It is metaphorical.

Jeff Pressing, in his article “Cognitive Isomorphisms Between Pitch and Rhythm In

World Musics” presents yet another angle to the linking of pitch and rhythm, considering

human perception by relating perceptual space to one-dimensional arrays of rhythmic

number values as isomorphic, or structurally identical. 5 Pressing makes a case for the

correspondence between twelve note African Ghanaian rhythms from Ewe drumming, to

the twelve pitch classes in Western equal tempered tonality. The commonality here is

actually the number 12, which is an abstract construct. But this author’s point is that this

is still somewhat of an arbitrary pairing that does not apodictically necessitate a

relationship. I, on the other hand am seeking, quite literally, a direct relationship between

pitch and rhythm as a physical phenomenon. I maintain that pitch is simply a fast

5 Pressing, Jeff. 1983. “Cognitive Isomorphisms between Pitch and Rhythm in World Musics: West Africa,

the Balkans and Western Tonality,” Studies in Music 17: 38-61.

8

rhythmic cycle. On the other hand 2212221 does seem to arise as a type of morphogenic

resonance of human consciousness, a cloud of knowledge of a collective unconscious

nature that seems to bind disparate cultures and peoples.

1.3.4. Charles Ives’s Universe Symphony and its legacy: Polytempic Polymicrotonal Art

Music

Charles Ives (1874-1954) wrote a masterpiece of unprecedented vision in his

“uncompleted” Universe Symphony, a work that was actually completed, but just not

assembled. After examining the notes and pages of the Universe Symphony, Johnny

Reinhard acted as the individual agent Ives referred to in posterity, as the one who could,

and would, put together the scraps of paper and notes into a legible and audible form.

Reinhard’s realization, approved by the Ives Society,6 reveals the deep complexity of

Ives’s thought, replete with Pythagorean tuning, quartertones, eighth tones, and just

intonation, along with three competing global tempi and a subdivision of polyrhythmic

percussion into numerous divisions of the whole note, extending up to 43 beats.

The implications of this are astounding. Tantamount to total freedom, both in

pitch and in rhythm/tempo, the restrictive nature of 12-tone equal temperament could

conceivably explode into an infinity of pitch and rhythmic possibilities. While the

rhythmic structure and nature of time itself operate in an unlimited musical universe,

where the Einstein-Rosen bridge even suggests that nature itself is capable of time

manipulation, not to mention the Shapiro7 time delay, where gravity directly slows down

6 Reinhard. 2004. The Ives Universe: a symphonic odyssey. p. 6.

7 Shapiro delay is described in (Irwin I. Shapiro, 1964). Gravitational time dilation causes apparent delays

in radar signals.

9

time by proximity while other streams move ahead uniformly, also revealed by Einstein’s

general theory of relativity; just as it is in nature, so it is in art.

Rather than despair at the innumerable combinations of pitch and rhythm, the

twenty-first century composer can easily see new approaches to age old problems: form,

and the linking of pitch to rhythm. Aristoxenus of Tarentum adopted the analogy of

infinite space from Euclid’s geometry, more philosophically than practically, but his truth

is immanent in that there is infinite space between two points. Rhythm has not suffered so

much by this theorem: composers have had much more freedom with respect to new

rhythmic designs due to the limitless nature of rhythm, but even more to point—what

composers have allowed themselves to be limited by, by imposing rules and artificial

constraints on their ideas of musical materials. This is the same for intervals. The human

animal is capable of infinite development, including microtonal hearing and even though

there are always boundaries of perception, such as the Just Noticeable Difference, or the

JND, the human animal is capable of greater perception.

1.3.5. Compositional Resources: scale and tuning sources and computer music software

for the composition and realization of polymicrotonality

Limitations of this polytempic polymicrotonality will lie in the area of

instrumentalists’ resistance to practicing intonation and microtones. This is inevitable and

should be expected. A way around this is software that deals exclusively with microtonal

notation, and playback via MIDI interface.

Scala is a wonderful, free program that transcends the hardwired confines of

popular notation programs and it is open source.

10

Musescore is another freeware music notation program with microtonal plugin

capability, in which a composer can print microtonal and polymicrotonal scores for

performers, as well as export midi files that can be realized in more professional sound

synthesis software.

1.3.6. What is the Difference between Tuning and Temperament?

Tuning vs. temperament: what is the difference? A tuning is an unadulterated

system with untouched intervals, whether or not they exist in nature. If a mathematical

system invents a hierarchy of intervallic structures, then all attempts to modify the

structure result as temperament. It seems to be a matter of perspective, which draws fire

from all angles. The overtone series is a pure tuning. Pythagorean is also a pure tuning,

but meantone and equal are both temperaments, due to the tampering of the perfect fifth

from 702 cents, down to 696 cents and 700 cents for mean tone and equal temperaments

respectively.

I highly recommend William A. Sethares’s book Tuning, Timbre, Spectrum, Scale

as it has concise explanations of the various mathematics behind historical tunings and

temperaments and mathematics explaining the phenomenon of noise and the harmonic

spectrum in acoustics. Fast Fourier Transforms and spectra are discussed in a friendly

way that reaches the lay person, such as this author, as he is not a mathematician. This

paper will also not be going into deep mathematical formulas for tunings or

temperaments, but only to describe the nature of human hearing and the inclusion of

microtones through the ages.

11

CHAPTER 2

PITCH

HISTORICAL INFORMATION LEADING TO THE POSSIBILITY OF

POLYMICROTONALITY: REPRESENTATIVE FIGURES AND WORKS

2.1. ANCIENT GREECE

Central to this author’s ideas are the three Greek genera: the enharmonic, the

chromatic, and the diatonic. These three genera produce three different tunings of the

tetrachord spanning a perfect fourth; they constitute the basic vocabulary of ancient

Greek music, which was monophonic. There is the “characteristic interval,” or the IC,

which proscribes the boundaries of the main interval essentially defining the flavor of the

genus. It ranges from a whole tone to a large major third in size. The remainder of the

tetrachord is called the Pyknon and it literally means “packed-in space.” The pyknon

ranges in size from a semitone, or a limma (remainder), which is defined as an interval of

90 cents, to as little as a quarter tone of 50 cents. The small diesis, a term attributed to

Plato,was a fractional remainder of an intervallic space consisting of 63 cents (otherwise

known as third-tones). These are also used in some of the varying chromatic genera. The

largest pyknon, and the genus that sounds common to Western ears, is the diatonic genus,

which we hear as a perfect fourth with major sounding whole and half steps. It is the pure

12

Pythagorean major sound consisting of two 204-cent “major” seconds and a 90-cent

“minor” second.

In ancient Greek tuning systems, intervals have specific names. The apotome is

the result of the 90 cent limma combined with the 24 cent Pythagorean comma (see

below) and amounts to 114 cents. Then there is the limma, under which there are the

greater and lesser dieses, at 63 and 41 cents, respectively. There are two commas. The

Pythagorean comma, at 24 cents, is the remainder of twelve consecutive 702 cent

Pythagorean fifths at the octave; the Syntonic comma is the subtraction of the just intoned

major third from the Pythagorean major third, 408 c. – 386 c., equaling 22 cents.

Ex. 2.1.1. Ptolemy’s example of a diatonic tetrachord amidst three various sized whole

tones.8

Later in the Medieval and Renaissance periods, Zarlino discovered the senarius,

or the first 6 partials of the overtone series. René Descartes canonized these in his

Compendium of Music (1656).9 It is at this point in history that the overtone series was

understood to be demonstrable as a scientific reality, thanks to the work of Zarlino,

Mersenne, and Descartes. The senarius is important because it is the 5-limit justly tuned

version of Western tuning that incorporates both Pythagorean (3-limit just intonation)

8 Ptolemy.1999. Harmonics, p. 16.

9 Descartes. 1961. Compendium of Music. p. 17.

13

with 5-limit just intonation. This means that the fifth overtone, at 5/4, when converted to

cents (5/4 ln(1731.234)) equals 386 cents, which is the just major third that was

discovered by pipe organ builders in tuning pipe organs and thereafter manifested in

Western ears as pure consonance. Fokker went on to advocate for the “septenarius,”

which includes the 7th

partial, or the Bb, at 969 cents.

Ex. 2.1.2. From Ptolemy’s Harmonics showing individual differences in the leading

theorists’ intervals.10

10

Ibid. p. 50.

14

Ex. 2.1.3. and 2.1.4. Graphic display of the various sizes of Greek genera.11

11

Chalmers, John. 1992. Divisions of the Tetrachord. p. 18.

15

Ex. 2.1.4. (continued)

16

According to Canadian microtonal composer Siemen Terpstra, Anaximander, a

pre-Socratic philosopher who lived in Miletus in 560 B.C., believed in the infinite

division of unity (a concept that predates both Euclid and Aristoxenus), which in turn

applied to microtonal divisions of the whole tone in an Aristoxenian sense. Anaximander

said that all and innumerable worlds are infinite in possible harmonies, with respect to

tunings and scales. He believed in a matrix architecture with alternative choices, like the

three genera, and that we would pay the penalty for adhering to just one division, or

tuning, such as 12TET. Chronos the Greek Titan god “set the ball rolling and generated all

the tunings.” Anaximander believed in a matrix of alternative systems. This is a general

statement, but it certainly applies to tuning.12

Since then, there have been many

developments regarding tuning in Greece, of which the three genera are apparent.

It is quite possible that not just a few tragedies were set to music, such as

Euripides’s Orestes (set in the enharmonic genus13

) but that all of them might have been

set to music. This new spin puts the classical tragedians in a newly perceived role as

composers, in addition to their traditional standing as poets. Thinking of Euripides as a

composer is a rather new idea, but imagine also thinking of Sophocles, Aristophanes, and

Aeschylus as composers. Are these Greek tragic dramas an early form of opera? In any

case, the Greek genera are the most specific aspect of Greek music theory, due to

monody, which was the standard practice for Greek music.14

According to Pachymeres

(1242-1310), ancient Greek classical tragedy mainly used the diatonic genus, even

12

Siemen Terpstra. personal correspondence of Johnny Reinhard., referencing

http://faculty.washington.edu/smcohen/320/anaximan.htm 13

New Grove, Second Edition. 2001.Vol. 10. p. 341. And, West, E.P., (2001). Documents of Ancient Greek

Music. 14

Barsky. 1996. Chromaticism. pp. 2-3.

17

though Euripides was fond of the enharmonic.15

Why would this statement exist if there

were no standard performance and compositional practice? Horizontal music, linear

music, or monody, manifests its complexity in its intervals and rhythms, which can be

profoundly difficult, so as to effect the “subtler nuances of mode.”16

Greeks call melodies

ekmelik if they are not melodious, and melik if they are. Shades of microtones are called

chroai, and help enhance the melos and ethos of the modes, taught as morally edifying.

August Wilhelm Ambrose, in his Geschichte der Musik, 1881, discovered that melos

came in three varieties: common, mixed and non-mixed.17

Ambrose pointed out five

different accidentals, from ½, 1/3, ¼, 1/6, and 1/8th

tone shades of Greek micro-

intervals.18

According to the Montpellier Code,19

Greek music also has four types of

modulations, called metabolae: genera, tonoi, system, and melos. The genera are the

tetrachordal tunings, tonoi the tone position, system a type of mode or scale, and melos

the way it is sung. There is no right or wrong in Greek musical thought, but only good,

and not good, based on commonly accepted stylistic trends. According to Aristotle, the

function of genus is to indicate the essence of a thing; it is paramount.20

As information

about the Greek genera spread into Europe through Boethius, composers such as Nicola

Vicentino began to reanimate the Greek enharmonic genus, discarded in the Middle ages

as unsingable by the Catholic Church, which had embraced simple 3-limit Pythagorean

tuning. It is through Bulgaria that Greek tuning made its entry into Europe and it is in

15

Ibid. p. 5. 16

Ibid. pp 5-7. 17

Ibid. p. 8. 18

Ambrose. 1881.Geschichte der Music. p. 11. 19

Barsky, p. 19. 20

Barsky. p. 39.

18

Bulgaria, today, where there are still the three Greek genera, plus one more: pentatonic. It

is also Bulgaria that is the link between tunings of Europe and the Middle Eastern

maqam.21

The maqam is a traditional Arabic homophonic melodic system composed of

microtonal tetrachords. The Pythagoreans use of ratios include fractions called epimore22

ratios, where they are super-particular, in that they are of the form n+1/n, in an arithmetic

series.23

Non-super-particular ratios are called epimeres, and are of the form n+m/n.24

To

find intervals between two integers, the Greeks used Katapyknosis, which involved the

insertion of a epimore ratio by setting the lower and upper bounds by multiplication.25

All

these methods were anathema to Aristoxenus, who preferred to find intervals by ear, and

by inherent musicality, than by a mechanical method.

2.1.1 The Pythagoreans

Pythagoras’s student, Philolaus, ventured beyond the whole tone and semitone by

calculations of fifths and fourths based on instrumental tuning practices. Differences were

all relegated to the chromatic and enharmonic genera.26

The Greeks used tetrachords

spanning a perfect fourth, called tetraktys, where the first and fourth notes, the hypate

hypaton, and the hypate meson, were fixed; while the inner two notes, the hypate

parhypate, and the hypate lichenos, were moveable, just as their stringed counterparts in

21

Ibid. pp 40-41. 22

For further clarification on the difference between epimere and epimore ratios, consider, for example, 7/6

as epimore, where consecutive numbers define the fraction and 7/5, where a skip in numbers define the

fraction, as epimere. 23

Gibson, 2005. Aristoxenus Harmonics. p. 10. 24

Chalmers. 1992. p. 7. 25

Ibid. 26

G. Assayag. 2002. Mathematics and Music, A Diderot Mathematical Forum. p. 5-7.

19

the physical lutes upon which they were modeled. Pythagoras and Philolaus27

were

convinced there was a mathematical basis for consonance, expressed as ratios based on

nodes of a single string called a monochord. It is probable that Pythagoras did not invent

tuning by the multiplication of a tone by the fraction 3/2, but that he was introduced to it

during his period spent in Babylon, which is well known.28

Ex. 2.1.5 Ptolemy’s Greater Perfect and Lesser Perfect Systems.29

The Greek tetrachord based on the perfect fourth, which is not found in the

overtone series as an actual distance above the fundamental until very far up the series, is

in fact the “shadow” of the perfect fifth. It is the “undertone” inversion of the perfect fifth

27

Frazer, 2001. Development of Musical Tuning Systems, p. 1. 28

Pythagorean Knowledge in Ancient Babylonia. Accessed 3/5/12.

http://www.egyptorigins.org/babpyth.htm. Also, Pythagoras may have spent some time in Babylon, Iraq,

and had learned much of his mathematics in Babylon. 29

Ptolemy, p. 76.

20

in relation to the second octave of the fundamental and it is the main organizing principle

for all Greek music. The tetrachord is also used in the Middle East in the Arabic maqam

in much the same way it was spread by Ptolemy.30

Greek music did not make deliberate

use of the octave, so the scales, harmoniai, and modes, tonoi, seem to have come later

towards the first century C.E., under Ptolemy, from whom Boethius “rewrote” their

essential functions for use in European musical theory.31

The monophony of the Arabic

maqam comes much closer to the sound and use of the Greek tetrachordal system,

according to O Wright.

Pythagorean tuning is essentially 3-limit just intonation. The only overtone series

partials used are the fundamental and the second overtone, the perfect fifth (3:2), which,

at 702 cents, is pure. Archytas then improved 3-limit just intonation to include ratios

beyond the fourth partial, such as the fifth partial, which is the just major third, at 386

cents, making Pythagorean tuning 5-limit inclusive.32

Archytas also explored

proportional means and proposed three methods of octave divisions: arithmetic, 12:9:6,

geometric, 12:6:3, and harmonic, 12:8:6.33

What does this have to do with microtonality, let alone polymicrotonality? A

great deal, since the Greek genera, the diatonic, chromatic, and enharmonic, were divided

by these means to produce intervals of a profound diversity, ranging from the comma, at

24 cents, to a large whole tone at 231 cents. Each genus was a type of “tuning” that was

defined by the size of the pyknon, or remainder, of the tetrachordal divisions after the

major thirds, minor thirds, and whole tones had been designated. The pyknon was the

30

Wright, 1978, the modal system of Persian and Arabic music, 1250-1300. p. 32. 31

Chadwick. 1981. Boethius, the consolations of music, logic, theology, and philosophy. p. 89. 32

G. Assayag, 2002, p. 7. 33

Ibid.

21

densest region of the non-diatonic genera, the chromatic and enharmonic genera equal to

less than half of the span of the perfect fourth.34

The lichanos is the indicator, as the third

note from the bottom, of the genus in question and indicates the CI, or “characteristic

interval” within the pyknon, that tells the genus type.35

The CI width, therefore, indicates

the genus.36

According to Richard Crocker, Pythagoras, Archytas, and Aristoxenus all

had different views regarding the size of the lichanos. Archytas believed the lichanos to

be 28/27, or 63 cent third tones; Pythagoras believed it to be the limma, at 90 cents; and

Aristoxenus believed it to vary between 50-100 cents.37

In fact, Andres Barbera created a

formula called the “rate of change” for the lichanos: (lichanos2 – lichanos1/ parhypate2

– parhypate1).38

The Lyre was used to unequally divide the pyknon in Aristoxenus’s

time. Historical studies by classicists have organized groups of tetrachords and their

microtonal ingredients, placing them into categories and have also found that there was

modulation between the genera.39

The prototypical diatonic genus can be found in Plato’s Timaeus; called the

Timaeus scale, it contains the characteristic limma, at a ratio of 256/243, which is 90

cents.40

The Timaeus tetrachord contains the following ratios: 256:243:216:192, which

corresponds to 90 cents, 204 cents, and 204 cents, which is a half-step, and two whole

steps.41

This was given to Plato from Pythagorus to Philolaus to Archytas, who was then

a contemporary of Plato. The pinnacle of Greek scalar achievement lies within the

Immutable Perfect System, which contains 15 steps and covers two octaves, beginning

34

Chalmers, 1993. Divisions of the Tetrachord.p. 18. 35

Ibid., p. 47. 36

Schlesinger. 1970. The Greek Aulos. p. 161. 37

Chalmers. 1993. p. 48. 38

Ibid. 39

Ibid., p. 52. 40

Godwin, 1993. Harmony of the Spheres. p. 3-9. 41

Aristides Quintilianus, first century AD—1983. p. 161.

22

with the Proslambanomenos, typically at pitch class A, descending/ascending42

through 5

tetrachordal series: hypate tetrachord, meson tetrachord, synemmenon tetrachord,

diezeugmenon tetrachord, and the hyperbolaion tetrachord, each with specific steps:

hypate hypaton, parhypate hypaton, lichanos hypaton, hypate meson, parhypate meson,

lichanos meson, mese, trite synammenon, paramese, trite diezeugmenon, paramese,

paramete diezeugmenon, nete diezeugmenon, trite hyperbolaion, paramete hyperbolaion,

nete hyperbolaion, corresponding to: A,B,C,D,E,F,G,a,b-flat,c,d,e,f,g,a.43

All of these

Greek names refer to lyres from ancient Greece. The Immutable Perfect System is

composed of two smaller systems: the Greater Perfect System, and the Lesser Perfect

System, where the standing notes are the proslambanomenos, and the mese, and the

mutable notes are the lichanos and the parhypate, as discussed earlier.44

Nicomachus of Gerasa, first century A.D., took Plato’s Timaeus very seriously in

his The Manual of Harmonics.45

Nicomachus believed that the properties of musical

intervals were governed by number, as in Pythagorean thought, and that there was

something to be said of octave equivalence, where a tetrachord is paired with a

pentachord, disjunctly.46

The term diezeugmenon means “disjunct,” and synemmenon

means “conjunct,” where these terms indicate the combining of tetrachords. The disjunct

tetrachord is separated by a whole tone and the conjunct shares the same pitch. In

building the Immutable Perfect System, both techniques of tetrachordal combinations

were employed, as both the diezeugmenon and synemmenon tetrachords were the

42

It has been customary for Greek scales to be spelled downwards, starting at the top and climbing the

scale down; however, Xenakis claims this is erroneous, and has since reversed the opinion of this matter. 43

Schlesinger. 1970. p. 139. 44

The New Grove, 2001, pp. 663-665. 45

Nichomachus, 1994, p. 99. 46

Ibid.

23

distinctions used to determine the Greater from the Lesser Perfect Systems,

respectively.47

The heptachord is the result of the seven-note set made from combining

the synemmenon and the meson tetrachords, as they were conjoined, or conjunct; while

the octachord is formed by the diezeugmenon and meson tetrachords, which are disjunctly

separated by a 9/8 whole tone, thus creating the two species of the Perfect System. When

combined they form the Immutable Perfect System.48

Nicomachus reveals that Plato had realized all the scale degrees for the diatonic

scale in Timaeus: 1/1, 9/8, 81/64, 4/3, 3/2, 27/16, 256/243, 2/1, which is 3-limit

Pythagorean tuning, from Archytas.49

Plato had also canonized two of the methods for

string divisions: harmonic mean, b = 2ac/a+c, and arithmetic, b = a+c/2, which, along

with geometric mean, is the resultant root of the number of multiplicands.

Nicomachus also invented the term “identity” with respect to concordancy,50

or

consonance; this brings to mind Stockhausen’s Stimmung, in which the performers must

achieve equivalence by ear, which is ultimately an Aristoxenian principle (see section

2.1.2 below). As a side note, Plato’s Republic, at the end of book III, discusses the ethos

of the modes in feeling and character: Lydian, as mournful; Dorian, as cheerful; Ionian,

as happy; and Phrygian, as self-restrained.51

Nicomachus also helped pave the way to

polymicrotonality by his “combined genera scale,” whereby the Immutable Perfect

System is infused with all three genera.52

Examples are shown above. McClain notices

47

Ibid. 48

Ibid. p. 110-112. 49

Ibid. 50

Ibid., p. 182. 51

Plato, The Republic, pp. 157-59. 52

Nicomacchus, p. 176.

24

that Albert von Thimus had created the Thimus-Nicomacchian Table, as a model of

Pythagorean thinking, which yielded a 19-tone per octave scale.53

Aristides Quintilianus (2nd

century A.D.), incidentally, in his De Musica in three

books, invents the characters Florentius and Eusebius (Schumann’s “imaginary”

characters, possibly) who are found in a Platonic dialogue utilizing both Pythagorean and

Aristoxenian ideas. Aristides also borrowed the tri-generic Immutable Perfect System

from Nicomachus. Aristides was essentially a follower of Aristoxenus, who advocated

“six shades” in excess of the three genera, utilizing third tones.54

The third tones of

Aristides’s system come from Aristoxenus’s soft chromatic genus. Modulation of genera

is discussed in connection with alteration of the underlying scale and character of the

melody.55

Modulation also was accomplished by concordance of interval sizes within

corresponding genera.56

This type of modulation between genera is tantamount to

microtonal modulation, or polymicrotonal use of multiple tuning systems in one setting.

According to Aristides, melos, was divided into four areas: genus, scale, tonoi,

and rhythm, in which the three genera were the most important element of modulation.

2.1.2 The Aristoxenians

Vindictive and polemical in his book, Harmonics,57

Aristoxenus of Tarentum (4th

century B.C.) disapproved of the Greek harmonicists—a school of theorists who based

53

McClain. 1978. The Pythagorean Plato. p. 146. 54

Aristides Quintilianus, first century AD—1983. On Music. p. 77-88. 55

Ibid., pp.83-85. 56

Ibid., p. 89. 57

Aristoxenus. 2005. Harmonics. trans. Sophie Gibson, p. 1-4.

25

their music on one genus, the diatonic, emphasizing octave equivalence.58

Aristoxenus

derided the harmonicists’ notion of “scale” as mechanical and soulless, even though a

synthesis of scale and genera would yield many more than 7 species of mode, as shown

by Eratocles.59

Aristoxenus, however, claimed that scales were useless and that notation

was similarly pointless. Aristoxenus believed in the motion of the voice in actual

performance, where the tetrachord, alone and with modulations, was the basic musical

unit; the perfect fourth was its most essential interval. It is Aristoxenus who first hailed

voice leading as the most important principle of music.60

Aristoxenus did not resort to a

cosmology, as did Pythagoras, and also rejected the ratios of Pythagoras by stating that

“we do not perceive sound as ratios or relative speed.”61

Aristoxenus stated that the smallest melik interval to be sung was the quartertone,

at 50 cents.62

Anything smaller was considered ekmelik, or unsingable, even though the

comma was sung as embellishment.

For Aristoxenus, there were six shades of archetypal genera:

Table 2.1.2.1 Enharmonic 3 quartertone 50 c. 3 quartertone 24 large major third

Chromatic mild 4 third tone 63 cents 4 third tone 22 smaller major third

Chromatic hemiolic 4.5 4.5 21

Chromatic whole tone 6 6 18 minor third

Diatonic mild 6 limma, 90 cents 9 15 small minor third

Diatonic intense 6 limma, 90 cents 12 whole tone 204c. 12

58

New Grove, second edition, Volume 10, p. 336. 59

Ibid., p. 337. 60

Ibid. 61

Aristoxenus (Gibson). 2005. Harmonics. p.16. 62

Ibid., p. 339.

26

These tetrachordal divisions by 30 intervals were invented by Cleonides, a

follower of Aristoxenus, who tried as judiciously as possible to give an idea of

measurement to the divisions Aristoxenus envisioned (which were supposed to be

immeasurable approximations).63

The tetrachord, when extended into a full octave, would

yield 72 divisions of the octave by Cleonides’s divisions. Even though they were purely

theoretical, some have taken these measurements literally. Aristoxenus defined the

pyknon as three notes bounding two small intervals, where tonoi was the position of the

voice.64

The modulation of genera pivoted on the mese, an immovable note from which to

jump to other tetrachords.65

The modulation to other genera is considered melik, a

standard of Greek practice very similar to Arabic maqam. According to Aristoxenus’s

adoption of Euclid’s philosophy of geometry, there are infinite points on a line, and so

there are infinite lichanoi between two pitches. The enharmonic retuning of the lichanos

is called eklysis.66

The Delphic Hymns, according to Aristoxenus, featured modulations

of three different types: by genus, by system, and by key.67

Ethos, in Aristoxenus, was the

act of modulation, in which Cleonides managed four systems of modulation: by genera,

system, key, and melos. Additionally, there were “retunings” that Aristoxenus advocated,

which were the 13 tonoi, or modes, as progenitors of the modern day key. These were

also distinguished by genera tuning as well as by position in a system, or scale.68

63

Ibid., p. 339, and Chalmers, p. 7. 64

Ibid. 65

Ibid., p. 341. 66

Barsky, Chromaticism, 1996, p. 3-5. 67

Winnington-Ingram, 1936. Mode in Ancient Greek Music. p. 33. 68

Ibid., p. 74.

27

2.1.3 Ptolemy

Claudius Ptolemaeus (Ptolemy, 2nd

century A.D.) was born in Alexandria. In

addition to the Almagest, his famous treatise on mathematics and the cosmos, Ptolemy

was a brilliant music theorist who solidified all of ancient Greek theory in his writings.69

Ptolemy explored and defined Pythagorus, Aristoxenus, and Archytas, and combined the

ideologically opposing schools of thought regarding the historical divide between

Pythagoras and Aristoxenus.

Ptolemy further clarified the methodology for the divisions of intervals by

arithmetic: 4-8-12-16, a process of adding 4; harmonic, 6-8-12-18-27, a process of adding

1/3 the proportion of each number; and geometric, 2-4-8-16-32, simple multiplication by

2, including the notion of square roots, not available to Greeks without a mesolabium.70

Although each means of division allowed variance between the cents value of each

interval, the overall differences are slight. Consonance was determined as the smallest

and simplest ratio.71

Ptolemy also acknowledged modulation by genus, in which outer

notes, the hypate and mese, remained stationary at a perfect fourth, while the inner notes,

the parhypate and lichanos, were moveable, and therefore able to modulate, metabolae,

to other genera where a unity could be achieved.72

Ptolemy, with regard to Aristoxenus, created a chart of the latter’s genera in broad

strokes: the enharmonic was defined with quartertones; the soft chromatic, with third

tones; and the diatonic with semitones.73

Ptolemy cites Archytas as having equated the

69

Ptolomy, first century A.D. Harmonics. Trans. Jon Solomon. 2000. 70

The mesolabium was a tool for extracting roots and constructing geometric means in ancient Greece. 71

Ptolemy, p. 9-16. 72

Ibid., p. 39. 73

Ibid., p.41.

28

third tones to a ratio of 28/27, at 63 cents; however, in Archytas’s system, the third tone

was found in more than just one genus.74

The idea of polymicrotonality holds more

efficaciously if one follows a particular Greek theorist. In the case of a polymicrotonal

paradigm, Aristoxenus’s criteria for his genera hold very well; and since he was a major

Greek figure, aside from Pythagoras, in holding a theoretical stance with weight.

Ptolemy’s classifications are as follows:75

Table 2.1.3.1 Enharmonic 5/4 (just third) 24/23 (1/3 tone, 63 c.) 46/45 (38 cents)

Soft Chromatic 6/5 15/14 (apotome) 28/27 (third tone)

Intense Chromatic 7/6 (266 cents, third tone) 12/11 22/21

Soft Diatonic 8/7 (231 cent large whole

tone)

10/9 21/20

Tonic Diatonic 9/8 8/7 28/27

Intense Diatonic 10/9 9/8 16/15

These ratios are, as Ptolemy decreed, within both reason and perception for musicians in

second century Greece.76

Ptolemy himself recommended that the enharmonic genus modulate to the soft

chromatic and then to the intense chromatic by the congruence of pyknon sizes and

intervallic unity.77

This proves that genera modulations were not just supported by

Aristoxenus alone, but widespread enough that Ptolemy recognized the performance

practices of composers and musicians of his day. If we still agree that genera are the

Greek equivalents of tuned systems of pitch, then this is tantamount to poly temperament.

And since these genera involve microtonal intervals, they are polymicrotonal by

74

Ibid., p. 44. 75

Ibid., p. 50. 76

Ibid., p. 52. 77

Ibid., p. 58.

29

definition, despite their limitations to monophonic textures. For during the course of one

work, two or more genera will be used, thus allowing a species of polymicrotonality:

what Ivor Darreg called “immigration,”78

rather than modulation, since modulation refers

to a relative position within one tuning scale, as opposed to leaving the tuning altogether.

We can also see that Ptolemy was 11-limit inclusive in just intoned ratios that venture out

well beyond Pythagorean 3-limit tuning. Also in Ptolemy’s system, the renaming of a

pitch, in the Greater Perfect System, for example, changed its function, which would then

change its genus. “All genera can modulate.”79

In studying Ptolemy, we find that

intervallic re-spelling was not an original idea in Western chromatic theory. Ptolemy

termed modulation by genera to be “metabolic modulation.”80

In fact, a modulation in

genera, from the diatonic to chromatic genus, causes a change in melos.81

Table 2.1.3.2 Ptolemaic comparative table of the enharmonic genus from Archytas,

Aristoxenus, Eratosthenes, Didymus, and Ptolemy.82

Archytas Aristoxenus Eratosthenes Didymus Ptolemy

5/4 3 (Cleonides's

indications)

19/15 5/4 5/4

36/35 (perfect

quartertone)

3 39/38 31/30 24/23

28/27 24 40/39 32/31 46/45

Ptolemy, in fact, had raised just intonation to the 11th

limit, quartertones, and had never

espoused equal divisions for either the octave, the tetrachord, or the whole tone.83

78

From a personal correspondence with Reinhard: Johnny Reinhard knew Ivor Darreg, part of the San

Diego contingent of microtonalists, and Ivor referred to modulation of one tuning to another as

“immigration,” since it was akin to going into foreign territory. 79

Ibid., p. 75. 80

Ibid. 81

Ibid., p. 78. 82

Ibid., pp. 99-123.

30

2.2 EUROPE: THE MIDDLE AGES, RENAISSANCE, AND BAROQUE

The intervals in use in Gregorian chant did come smaller than the semitone,

according to the Montpellier ‘antiphonary’ codex.84

The diesis (63 cents), smaller than the

limma (90 cents) somewhere in the area of a third tone (also 63 cents) showed that the

Greek enharmonic genus was still in use. The Greek genera and the “new” European

diatonic flavor had overlapped during the dark and middle ages and classifications of

intervallic systems of the middle ages were based on the Greek genera and Greater

Perfect System.85

Eventually, the theoretical diatonic/chromatic systems of the later

Middle Ages, beginning with the Ars Nova, fluctuated between 12, 14, 17, and 19 pitches

per octave, all based on extended Pythagorean tuning.86

Carl Dalhaus called this variance

the lower and upper limits of available resources via monochord divisions by

mathematics.87

In fact, even Guido de Arezzo had insinuated the existence of the diesis.88

Philippe de Vitry’s Ars Nova in (1320) alluded to a 14-tone per octave system that

had double leading tone cadences. In fact, the term “leading tone” was first seen in the

writings of Ptolemy.89

The extended Pythagorean tuning of 17 tones per octave in the

system of Prosdocimus de Beldemandis involved dividing all scalar whole tones into two

83

Ptolemy, Harmonics, (first century AD), trans. Jon Solomon, 2000. 84

Barsky, p. 20. 85

Ibid. 86

Ibid. 87

Ibid. p. 20. 88

Ibid. 89

Ptolemy, p. 53.

31

parts so that each tone had upper and lower leading tones.90

But first we should explore

Boethius.

Boethius (Rome, 480-524), in his De Institutione Musica, which is an incomplete

and only a cursory translation of the works of Nicomachus and Ptolemy,91

was the last

European vector of Greek musical theory to the West until Guido de Arezzo.92

Boethius

called music musica mundana, an all-pervasive force. He believed in ratios, as in

Pythagorean theory, and Greek modal theory, which is the progenitor of scales in the

Western understanding, complete with misidentifications of Dorian and Phrygian as

reversed. The classical Greek modes are not discussed here because they are not issues

particular to tuning; nevertheless, they are a circumstance of the various diatonic genera

they contain. Moreover, they represent more of the beginnings of Western tonality. The

writings of Boethius were revived by the Carolingian renaissance, three centuries later.

The Greek theory and genera bequeathed by Boethius to the compilation of the French

codex by Cassiodorus was an ongoing process that extended through 150 codices during

the Middle ages. During the late 9th

century there was a Boethian revival.93

The

Pythagorean system espoused by Boethius was adopted by the Catholic Church, who

focused solely on the diatonic genus because it was readily adaptable into chant and

liturgy. Initially, there were discrepancies between Boethius’s interval ratios and the

chant melodies--- incongruities that arose from conflicting systems, yielding potentially

two simultaneous tuning systems of unintentional polymicrotonal effects.94

90

Barsky. 1996. Chromaticism. p. 21, this is referencing Carl Dahlhaus’s work on Johannes Ciconia. 91

Chadwick, The consolations, 1981, p. 89. 92

New Grove. Vol. 3, pp. 784-786. 93

Ibid. 94

Ibid., p. 785.

32

Musica Enchiriadis is a curious document due to its anonymous authorship and

its proposed use of just intonation up to the 7th

limit (chapter ix).95

The treatise was

written in Germany in the ninth century, by way of France’s Carolingian renaissance. In

an age that was completely dominated by Pythagorean tuning up to the third limit, or the

perfect fifth, this document stands in isolation against the prevailing logic of its time. It

obviously must have influenced a great deal of musicians in Germany, and therefore it

coexisted with the common Pythagorean tuning, leading to speculation that at least two

tuning systems were competing for wide spread use: Pythagorean and 7-limit just

intonation.96

Curiously, the document is filled with Greek intervallic terms: limma, diesis,

and colon, which are two commas (roughly equal to a quartertone). The tuning system of

Musica Enchiriadis is based on the 9/8 Pythagorean sesquioctava, the epogdoos, initially

referred to as the sonus.97

The official stance of the Catholic Church was that it had dispensed with the

enharmonic and chromatic genera in favor of the diatonic for its liturgical chant, due to

the small intervals.98

In France, musicians were training on the monochord, using the

three methods of means: namely the arithmetic mean; the geometric mean; and the

harmonic mean. These were in opposition to the Pythagorean methods of spiraling fifths,

so ratios were coming into focus.99

In Spain, Al Farabi (872-950) had spread his theory

based on the ratios of Archytas and Aristoxenian approximations,100

symbolizing the

simultaneous use of both arithmetic and geometric means, or rather the simultaneous use

95

Musica Enchiriadis, Ninth century---rewritten in 1976 at the University of Colorado. 96

Ibid., chapter ix. 97

Ibid., p. 10. 98

Ferreria, p. 13-16 (G. Assayag, 2002). 99

Ibid., p. 17. 100

Ibid., p. 18.

33

of the Byzantine method of discrete ratios and continuous approximations that Xenakis

valued.

Early medieval music theory, therefore, was about the simultaneity of competing

tuning systems around all of Europe, from just ratio, to Pythagorean, to the Aristoxenian

approximations by ear, until the Catholic Church settled on strict, simple Pythagorean

tuning, which remained intact until Zarlino. Johannes Ciconia, ca. 1400, had thoughts

about the fusion of certain tuning aspects of the Ars Nova with the Italian trecento, where

Pythagorean tuning would be fused with the 17th

harmonic, the sesquiseptima, at the

ratios of 17/16, and 18/17, semitone of 105 cents, showing already a desire to break away

from strict Pythagorean tuning.101

In fact, Marchetto of Padua, in his Lucidarium (1317),

had written about the justly tuned major third, or the fifth harmonic, in 5-limit just

ratios.102

In fact, the Luciderium was filled with divisions of the whole tone into nine

parts which was a carry-over from the theories of Al Farabi, who had brought the maqam

to the West. The maqam had been subjected to an octave division of 54 tones in Turkish

practice by the teachings of both Archytas and Ptolemy, except that Ptolemy did not

advocate equal divisions.103

Also, via Byzantine music, we get the chrysanthos, the

practice of the ancients, systematizing Greek modes with elements of Turkish music that

divided the whole tone into 9 parts, or commas, as seen in the writings of Marchetto and

Johannes Ciconia.

The confluence of the many different paths that by which Greek theory was

introduced into the West (and their concomitant inaccuracies) led to Renaissance

revivials of Greek practices that were based on spurious information; this incremental

101

Johannes Ciconia, de proportionibus and Nova Musica, p. 109. 102

Marchetto of Padua, Luciderium, trans. Joe Herlinger, p. 115. 103

Ibid., pp. 131-149.

34

digression inevitably happens over the course of centuries, yet this is how fundamental

change and growth occur in art—through mistakes. Nevertheless, it is on Marchetto’s

Luciderium that Ciconia based his work, incorporating the 17th

partial into Pythagorean

tuning.104

Also, the reduced set of diatonic tones had been increased to 12, by 1300, by

the use of multiple leading tones from the practice of Ars Nova and potential

polymodality, where clausulae were essentially polymodal, for example, by containing

both Phrygian and Dorian modes with their concomitant leading tones, as well.105

Although modes had infiltrated Europe through Boethius, it is not wholly true that mode

names originated solely with Glarean’s Dodecachordon. Polymodality can be seen as a

musical and ideological precursor to polymicrotonality. It appears in the music of Bartók,

for example, in the early twentieth century, alongside Ives’s Universe Symphony.

Around the sixteenth century, at the height of the Renaissance, chordal harmony

began to break away from the strict Pythagorean tuning found in organum, discant, and

motets. Musicians began looking back to the Greeks, a pattern of revival that has been

brought to this author’s attention, as well as to Barsky, prior to this research. Nicola

Vicentino and Gioseffo Zarlino were Greek revivalists, both of whom in the 1550’s used

just ratios, and rediscovered the three Greek genera, again, after Boethius had transmitted

them to Europe 1000 years earlier. The Frenchman Marin Mersenne, in his Harmonie

Universelle, 1637 had promoted the Greek systems of chromatic and enharmonic genera

as easy and necessary to musical composition.106

104

Ibid., p. 157. 105

Barsky, p. 80. 106

Egan, Marin Mersenne: Traite de L'Harmonie universelle: a critical translation of the second book ,

1962, p. 11.

35

Gioseffo Zarlino (1517-1590) brought Europe into just intonation by vociferously

leading tuning away from Pythagorean methods; thus it is odd that Pythagoras was to

become a tremendous influence on Zarlino.107

Zarlino expanded the overtone series to

include the fifth partial, or the justly tuned third, at 5/4, or 386 cents. In fact, it was

Zarlino who had established the senarius as the first six overtones to be the fundamental

components of Western tonality.108

Interestingly, it was Galileo’s father, Vincenzo, who

led the foray against Zarlino, in an effort to rid music of numbers, (as Aristoxenus had

tried to do in defiance of Pythagorus), when Zarlino proclaimed that Galilei’s relativism

was an assault on “God’s Plan.”109

Zarlino wrote his Le Istitutioni Harmoniche in 1558 and declared the major scale

as the following ratios: 1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, and 2/1, which is 5-limit just

intonation.110

Zarlino was also skeptical about ratio precision in intervals that necessitated

the use of geometric ortogonio for dividing intervallic space visually, using the

mesolabio, invented by Eratosthenes.111

In the early seventeenth century (ca. 1620),

Zarlino’s opponent, Vincenzo Galilei, proposed 12 pitches at the intervallic ratio of

18/17, or roughly, 99 cents, which would have predated Simon Stevin’s later discovered

12th

root of two (12TET). Even Descartes rejected equal temperament in favor of rational,

perfect intervals, if not for the sake of purity itself.112

Ultimately, Descartes discovered

the overtone series, which would unleash a third tuning approach to the already

disheveled system of tuning held by Renaissance music theorists, who were still

107

Isacoff, 2001-2003.Temperament: how music became the battleground for great minds of Western

civilization. p. 136-137. 108

Levarie, Levy. 1968. p. 30. 109

Isacoff. p. 142. 110

Zarlino. 1558, 1965. Le Istitutioni Harmonice. p. 51. 111

Ibid. p. 52. 112

Isacoff, p. 175.

36

grappling with just and Pythagorean tunings. Zarlino, Kepler, and Descartes all hailed the

overtone series as “nature’s divine plan.”113

Zarlino sought to improve the Pythagorean

thirds. At 408 cents he held they were too wide, rejecting them in favor of just thirds, at

386 cents, 22 cents narrower, and an interval itself referred to as the syntonic comma.114

Zarlino would come to deny genera modulation, advocating the coming hegemony of the

diatonic genus,115

free from polymicrotonal implications, as the diatonic status quo of

Europe.

Medieval and early Renaissance composers had fallen under the spell of the

Greek Perfect Immutable System, composed of the Lesser Perfect System, hypaton,

meson, and synemmenon tetrachords, plus the Greater Perfect System, hypaton, meson,

diezeugmenon, and hyperbolaion tetrachords, resulting in a 5 tetrachordal system. Many

medieval music treatises have this scheme, including all the Greek note names, from

Proslambanomenos to the hyperbolaion nete. Heinrich Glarean, in his Dodecachordon,

1547, had referred to the three Greek genera.116

In fact, in Europe, tetrachords still existed

well into the sixteenth century; Pythagorean tuning had admitted just intonation by the

use of the 5/4 major third, the splitting of the 9/8 whole tone, and the use of the 17th

overtone, all in order to round out and make certain intervals more consonant in

polyphony. As mentioned earlier, the whole tone divided into 9 equal parts was also

discussed in Glarean’s treatise, with weight given to certain parts of the nine divisions.117

Nicola Vicentino (1511-1576) can be considered the first microtonalist in

European music. He invented a 36-key-per-octave Archicembalo, which influenced

113

Ibid., p. 179. 114

Zarlino, The Art of Counterpoint, p. xx, terze part. 115

Ibid., p. 273. 116

Klein, 1989. Die intervallehre in Deutche Musiktheorie in des 16 Jahrhunderts. p. 32. 117

Heinrich Glarean, 1965. Dodecachordon. pp. 90-91.

37

Gesualdo and Frescobaldi, among others, on which one could play all three Greek

genera.118

Vicentino rediscovered Ptolemy and Boethius and their discussions of the

tetrachords. Archytas, Didymus, Aristoxenus, and Eratosthenes, again, became influential

in the early Renaissance, even though Vicentino was more Aristoxenian in his approach.

Vicentino desired more microtonal shading for text interpretation.119

The split key

Archicembalos of Vicentino were able to reproduce the enharmonic and chromatic

genera, and included the just ratios up to the thirteenth limit. Vicentino’s default tuning

was ¼ comma meantone, where 5.4-6 cents are shaved off four of the fifths to produce

roughly 8 useable keys.

In Vicentino’s L’antica Musica (1555) he diagrammed several keyboard layouts

for a 44-fret lute, a 19-tone per octave keyboard, a 17-tone per octave keyboard, and his

36/octave Archicembalo.120

He was a prophet of the nascent Baroque. In Libro Quinto,

chapter five, he transformed the notation of small enharmonic intervals from the Greek

genera, into a Renaissance ideology, thus transforming ancient Greek musical theory into

a Renaissance aesthetic. In Libro Primo he discussed a 38-tone per octave written system,

including “Quattro diesis” intervals, and also a 24-tone per octave system, though not

equal, but rational.121

118

Alves, 1989.The Just Intonation System of Nicola Vicentino. p. 1. 119

Ibid., pp. 1-4. 120

Vicentino, L’antica Musica, 1565—1959, index. 121

Ibid., pp. 12-13.

38

(Ex. 2.2.1 cited below)

39

Ex. 2.2.1 and 2.2.2 Nicola Vicentino’s 31-tone to the octave pitch and interval list122

Don Carlo Gesualdo (1566-1613), Prince of Venosa and the Count of Conza, has

been called a mannerist.123

Like Vicentino, he departed from the standard Renaissance

122

Alves. 1989. The Just Intonation System of Nicola Vicentino. pp.4-5.

40

aesthetic practice by leading into the Baroque creating his highly expressive madrigals

that employed some microintervallic tunings.124

Mannerism, according to Watkins, was

the act of producing art that did not represent nature, or natural logic, and was for its time

a type of maximalism.125

For this purpose, Vicentino was also a mannerist, and had

profound influence on Gesualdo. “The mannerist no longer sees the canon of nature as

immutable,” said Gesualdo according to Watkins, as he strove for the “unnatural.”126

Gesualdo’s use of the notes Gb, G, G#, Ab, A, A#, Bb, B, B#, Cb, C, C#, Db, D,

D#, Eb, E, E#, F, F#, which included no Fb, accounted for a circa 20-21/octave scale

from extended Pythagorean tuning. Again, the enharmonic genus was in full use, as no

spelling duplicated any pitches. In terms of hyperchromatic art, Gesualdo surpassed

Vicentino, even though Gesualdo used his 36/octave Archicembalo.127

From here, this

paper moves on to Marin Mersenne, who was the next historical figure in the

development of new microtonal frameworks.

Marin Mersenne, 1588-1648, was a scientist and a friend of René Descartes.128

Mersenne dealt with the overtones from the monochord and discovered up to the 27th

partial, noting the sequence of the first, second, and third partials, and their concomitant

significance in European tonality.129

Mersenne also found the same series in brass

instruments, thus leading to the first hypothesis of the overtone series, (even before

123

Watkins. 1973. Gesualdo: the man and his music. p. 294. 124

Ibid. p. 196. 125

Ibid. p. 97. 126

Ibid., p. 102. 127

Ibid., pp 201-294. 128

Ludwig, 1935, p. 11. 129

Ibid., pp. 41-42.

41

Descartes).130

Mersenne also had synesthesia, which led him to note specific colors to

specific pitches.131

Mersenne ratified Zarlino’s senarius, and also made a case for the 7th

partial, the septimal ratios, as consonant, years ahead of Adriaan Fokker. Mersenne also

discovered that multiples of ratios were also consonant, so the pitch palette could be

considerably expanded.132

Mersenne’s combination of systems led to his 24/octave

quartertone system in order to encompass the three Greek genera.133

In his Traite de L’harmonie Universelle (1627), Mersenne describes the Greek

genera, but misrepresents them. He claimed the diatonic genus had 9 pitches and 8

intervals; the chromatic had 16 pitches and 15 intervals; and the enharmonic had 25

pitches, and 24 intervals.134

These former figures, of course, are incorrect, and are based

on a heptachordal octave, plus accidentals, which the Greeks never used. Mersenne’s

format and style, for his treatise, adopted Euclid’s geometry, replete with theorems, and

axioms comparing intervals with geometry, a format also adopted by Spinoza in his

Ethics.135

The Timaeous Tetrachord, important as a prototype diatonic genus tetrachord,

also appears in his treatise, which was faithfully rendered: 256:243:216:192.

Finally, Christiaan Huygens (1629-1695), the Dutch genius, invented his

31/octave meantone system, described by Fokker as “the composition of the future.”136

Huygens created tuning from the overtone series, with ¼ comma tuning as his reference.

In the construction of his 31-tone system, Huygens used logarithms rather than geometric

roots in order to find consistent mean proportions, in order to preserve the 31-tone

130

Ibid. 131

Ibid., p. 44. 132

Ibid., p. 63. 133

Ibid., p. 75. 134

Marin Mersenne, trans. Egan, 1962, p. 11. 135

Ibid. 136

Hays, 1977, p. 18.

42

system—the first in Europe. He had made it perfectly circular, so that a full chromatic

modulation was possible, even in the mid 1600’s.137

He also advocated Vicentino’s

Archicembalo and had one built. Huygens, probably the first theoretical acoustician, as

well as microtonalist, also discovered that singers sing in a separate and distinct tempered

tuning that was not pure.138

On the polymicrotonal path, Huygens asked “how can we

reconcile multiple tunings?”139

Again, there is precedence for polymicrotonality and this

is also a part of Huygens’s legacy, just as well as Ives, Marie, and Carrillo.

In Huygens’s Le Cycle Harmonique, 1691, meantone temperament is the focal

point as the practical tuning system of Europe. He presents multitudes of meantone

tuning methods: 2/7 comma meantone, ¼ comma meantone, and equal divisions of the

octave as well. Huygens had drawn a design for a 19-tone keyboard in 1676, for

meantone transpositions between 12 and 19 tones per octave.140

Huygens had discovered

the meantone connection between both 19 and 31 tones to the octave, which became a

symbol of “Dutch tuning,” which is espoused by Fokker, as well. Huygens was the first

to break from the senarius.

Joseph Saveur and Jean-Phillippe Rameau both supported Huygens’s overtone

discoveries, instigating a new approach to European tonality via the overtone series.

Saveur was, in fact, Huygens’s true successor as advocate for the new science of

acoustics.141

Saveur had also discovered “beating” with respect to consonance and

dissonance, as scientific measurement of intervals started to become a source for a

genuine music theory. Like Huygens, Saveur had traveled up the overtone series to the

137

Ibid., p. 22. 138

Ibid., p. 27-29. 139

Ibid., p. 30. 140

Huygens, 1691-1986, pp. 46-76. 141

Farrar, 1956, p. 51.

43

32nd

partial, quartertones, and he gave numbers to what organ tuners had known all

along.142

On the other hand, Rameau had retreated from all the adventurous tuning methods

discovered by his predecessors by favoring the senarius, and consequently basing his

musical theories on only the first 6 partials in his “Nouveau Système.” The senarius is a

fundamental concept on which we all have become slavishly dependent, even until the

present day. Rameau called the first partial the “Fundamental Tone”143

in his reductionist

attempt at forming a legitimate music theory. Rameau was in favor of 12 equal divisions

of the octave and was opposed to both Well Temperament and meantone temperament.

Rameau criticized Zarlino, Jean Jacque Rousseau (who advocated larger tuning systems),

and even Ptolemy, the patriarch of Western tuning theory.144

Baroque tuning was based on a number of treatises, including Werckmeister’s

Musikalische Temperatur, Marpurg’s Anfangsgruende der Theoretischen Musik, Johann

Gottfried Walther’s Praecepta der Musicalischen Composition, the writings of Bach’s

student Kirnberger, and even Kelletat’s Zur Musikalische Temperatur inbesonders bei

Bach. Johnny Reinhard has written a very insightful work on tuning in the time of Bach,

who based his views on the tuning theories of Werckmeister and the German

Thuringians, of whom Reinhard feels were very microtonal by nature.145

Further

explorations of this topic will only produce redundancies in the quest for the most

practical tuning that facilitates chromatic modulations. But the addendum for the

Thuringian aesthetic is that it should reflect microtonal differences enough within the

142

Ibid. 143

Keane, 1961, p. 45. 144

Ibid., p. 176. 145

Reinhard, 2009.Bach and Tuning. p. 94.

44

modes that each mode has its own specific sound, due to the irregularity of the intervallic

structures. Hence, the notion of Well Temperament was not about the equality of the

sizes of the interval, but of making a workable 12-tone circle of modulation, while still

maintaining the integrity of each modal tuning.

Lewellyn Lloyd, the great acoustician, stated that “string quartets and choirs are

the acid test for chromatic shading that reaches for differences of a comma.”146

Lloyd

also stated that 12, 19, 31, 50, and 53-tone equal temperaments are all circular in design,

and all have a tempered fifth. They are irregular in that they are tempered, rather than

pure, at 702 cents per fifth.147

Lloyd also stated that “better players do demand a greater

degree of pitch accommodation,”148

rather than the poorer player. Lloyd is alluding to a

phenomenon that has actually been happening for a long time: the fact that we have

actually always been polymicrotonal. Strings always play in Pythagorean tuning. Singers

sing justly toned intervals. Brass plays the overtone series. These are three different

tuning approaches that, therefore, culminate theoretically into polymicrotonality, except

that it has been an unconscious phenomenon for centuries, at least until the primacy of

12TET, which took off in the common practice period. Another point of interest is that

singers, who initially sang in Pythagorean, began to sing justly intoned.149

Andreas Werckmeister (1645-1706), a Thuringian like Bach, wrote his

Musikalische Temperatur in 1691. In it he discussed his experience as an organist and

master tuner, Orgel–Probe (1681), which influenced Marpurg and Huygens conceptions

of tuning. Werckmeister also settled on the senarius of Zarlino and developed a

146

Lloyd, 1963, 1978, p. 109. 147

Ibid., p. 163. 148

Ibid., p. 164. 149

Ibid., p. 170.

45

classification for consonances in terms of disonance, where all his intervals came from

the monochord. Excluding the monochord, this is reminiscent of Paul Hindemith’s

intervallic classifications regarding his continuum of intervals from consonance to

dissonance.150

Werckmeister considered just intonation to be pure tuning, so he tried to

incorporate as much of just intonation into the meantone system at the time. There were,

in fact, circa 150 different meantone tunings before settling on 12TET. In terms of

microtonal intervals, Werckmeister approved of the syntonic comma, at 22 cents, the

diesis, at 41 cents, the Pythagorean comma, at 24 cents, and the diaschema, at 20 cents.151

Werckmeister is known for six major tuning systems: a 20/octave just system, a standard

meantone system, four “correct” temperaments based on his Well-tempered theories, and

lastly, a septenarius (based on the inclusion of the seventh partial) tuning. Even

Werckmeister utilized the 9-comma whole tone, introduced by Al Farabi via Turkey.

Commas and dieses were called “grad,” or degrees, and these systems were all tempered,

or “tampered,” as a more appropriate analogy. Werckmeister III is known as the “correct

tuning” and became the standard Bach used, as Werckmeister III was a closed circular

system, allowing a full chromatic tour of the 12 semitones.152

This is what is called well

tempered tuning. Werckmeister was against subsemitonia, which is the addition of

microtonal keys for the keyboard, as in the case of Vicentino’s Archicembalo; therefore

the Halberstaedt Plan, 12 notes per octave for keyboards, was adhered to.153

Werckmeister III was thus: C, 0 cents, C#/Db, 90 cent limma, D, 192 c., D#/Eb, 294 c., E,

150

Werckmeister.1691. Musikalisches Temperatur. pp. 8-19. 151

Ibid., p. 20. 152

Werckmeister, p. 26, for a full description of the six Werckmeister tunings, please see Rudolf Rasch’s

version of Werckmeister’s “Musikalische Temperatur.” Also, please download from the internet Johnny

Reinhard’s self-published “Bach and Tuning,” as it describes Werckmeister’s work in clarity. 153

Reinhard. 2009. Bach and Tuning, p. 53.

46

390 c., E#/F, 498 c., F#/Gb, 588 c., G, 696 c (the ¼ comma meantone), G#/Ab, 792 c., A,

888 c., A#/Bb, 996 c., B, 1092 c.154

Nevertheless, the Werckmeister III gamut produced a

total of 39 microtonal intervals to the octave, but only 12 are used at a time. Another

advocate for circular unequal tuning was J.N. Forkel, from Göttingen, where German

musicians post-Werckmeister preferred 1/6 comma meantone tuning, even with its “Wolf

fifths” 16 cents sharp.155

Telemann also had a tuning system of 55/octave, in equal

divisions, used for strings, winds, and singers, which equated with the 1/6th

comma

meantone in favor for these particular German composers. [Why is this not talked about

in academic theory books? Why is there a blackout of this vital information?] There is

literature out there now that discusses the variance of tunings during the Renaissance and

Baroque music of Germany and are listed in this paper’s bibliography. Nevertheless,

Bach in fact fully supported Telemann’s tuning system.156

Even Friedrich Wilhelm Marpurg, in his Beginning Music Theory (1757),

advocated for math exercises for musicians to calculate correct tuning ratios.157

Marpurg

shows the overtone series up to the 25th

partial as an arithmetic progression on page 26 of

his treatise, and does not deny the 7th

, 11th

, 13th

, 14th

, or even the 17th

partial tunings.158

Marpurg welcomes the microtonal pitches, and on page 33 of his manual, has a very clear

classification for all of them, including the syntonic comma, and the apotome, at 114

cents.159

Marpurg also advocated for “klanggeschlecht,” or pitch choice, for building

personal scales by employing the classical Greek arithmetic, geometric, and harmonic

154

Werckmeister, 1691.Musikalische Tempuratur,/Rasch, and Reinhard, p. 56. (also from same source). 155

Reinhard, 2009. p. 95. 156

Ibid., p. 99. 157

Marpurg, 1757. Beginning Music Theory. p.26. 158

Ibid. 159

Ibid., p. 33.

47

methods of means.160

Marpurg’s teachings can easily fit into the wide expanse of

polymicrotonality.

Johann Gottfried Walther, more conservative than Marpurg, in his Precepts of

Musical Composition (1708), predecessor of Bach, stated: “Schliesslich, dass ein Tonus

minor 8 commata übertieffe aber das 9te nicht erfülle—aber ein Tonus major halt 9

commata in sich und das 10th

macht er auch nicht voll!”(Lastly, that a minor tone

overtops 8 commas, but doesn’t include the ninth—but in major tone has 9 commas in

itself and the tenth also does not make it complete.).161

So, even Walther saw the

malleability of pitch and intervallic variance that is a feature of polymicrotonality, in

1708. Walther also stated that one can easily hear the difference between a chromatic and

an enharmonic tone, at 22 cents.162

Kelletat’s Zur Musikalishe Temperatur inbesonders bei Bach, is a comprehensive

survey of Baroque tunings. The treatise begins from Zarlino, Schlick, and Salinas (not

mentioned in this paper), who were all responsible for the invention of several meantone

temperaments.163

Kelletat states that well tempered tuning was a German enterprise. Bach

grew up with meantone, but fully embraced the well tempered system, even though he

composed in several temperaments simultaneously (polymicrotonality).164

In fact, is it

possible, that Bach may have composed harmonically in meantone, and linearly in

Pythagorean, negotiating between the two by Werckmeister’s III system?165

As a violinist

himself, Bach tuned his strings in Pythagorean tuning because that was the practice of the

160

Ibid., chapter IX, p. 51. 161

Walther, 1708. Precepts of Musical Composition, p. 88. 162

Ibid., p. 67. 163

Kelletat, 1960. Zur Musikalischen Temperatur insbesondere bei Johann Sebastian Bach. p. 21. 164

Ibid., p. 21-25. 165

It is my conjecture that Bach may have thought polytemperamentally, in both vertical and horizontal

directions.

48

day. As an organist and harpsichordist, Bach was accustomed to meantone and just

intonation. As a composer, Bach can not have been less affected by the sound and

function of these tunings in his ear. Pythagorean is best suited to lines, and not to vertical

structures because of the wide thirds, at 408 cents. But string players, even today, still

tend to play linearly in Pythagorean tuning, according to string players themselves. It is

not beyond the realm of possibility, therefore, that Bach thought linearly in Pythagorean,

and vertically in meantone, as well as well tempered tuning, which are all different

intervallic sizes.

Lastly, Kirnberger’s Kunst des reinen Satzes in der Musik, 1771, upon which Fux

based his Gradus Ad Parnassum, displayed the influence of Marpurg by the use of the

1/12 comma meantone system, which yielded him 194 pitches with multiple variations of

all 12 pitch classes.166

Simon Stevin and Huygens would seal the fate of polymicrotonality shut by their

joint contribution of logarithms and the twelfth root of 2 to produce the equally-tempered

scale that Western musicians have all come to know over the last two hundred years,

which has defined common practice tonality.

The reason the foregoing materials are so densely packed with tuning trivia is that

the nature of tuning and microtones, for various reasons, littered the Baroque musical

landscape. There was a heightened awareness of nuances of pitch never before imagined

in European history and it was the awareness that served as a prelude to the microtonal

explosion of the twentieth century, when composers again went into extensive detail

about the accuracy of pitch and the importance of microtones as both structural and

embellishing features.

166

Ibid., pp. 45-46.

49

2.3 THE TWENTIETH CENTURY: THE MAIN PROTAGONISTS

Perhaps the single most important work for twentieth century microtonality,

acting as a catalyst after a short period from a 2500 year history of quiescence in tuning

and microtones, is Hermann Von Helmholtz’s Die Lehre von den Tonempfindungen als

physiologische Grundlage für die Theorie der Musik (On the Sensations of Tone as a

Physiological Basis for the Theory of Music), translated into English by Alexander Ellis

in 1877.

Essentially a scientific inquiry into electro-magnetic theory and sound acoustics,

there is a profound wealth of tuning information and far-reaching ideas concerning

microtonality that moved many composers towards early microtonal experimentation in

the late nineteenth century and into the early twentieth century. George Ives is one

example.

Helmholtz deals with just intonation, extended Pythagorean tuning, Bosanquet’s

Manual for 53/octave organs, specially tuned harmoniums in just intonation, tuning tables

comparing all known tunings to the cent (an invention of Ellis himself), and an invention

called the “Harmonical,” which is a specially tuned harmonium using the overtone series

up to the 13th

partial.167

Additional personages from the age include Henry Ward Poole,

who invented an organ capable of both quartertones and eighthtones, as well as septimal

just ratios, including the septimal comma at the ratio of 64/63. The septimal comma at 27

cents is just slightly above the Pythagorean comma.168

167

Ellis/Helmholtz. 1877/1954. On the Sensations of Tone. p. 470. 168

Ibid, pp. 473-495.

50

The more notable composers affected by Helmholtz/Ellis’s work include Alois

Hába, Julian Carrillo, George Ives, Charles Ives, Ivan Wyschnegradsky, and Adriaan

Fokker. Alois Hába (1893-1973), shares the same birth year as Ivan Wyschnegradsky.

The Czech composer incorporated Slavic quartertone inflections into his works,

microtonality having come into Slavic music via Turkey into Bulgaria during the Middle

ages.

Hába tried to adapt quartertones into the Occidental heritage through the

prominent use of late Romantic chromaticism that began in the late Romantic period and

spilled over into the early twentieth century. In particular, Hába explored this synthesis

by way of his string quartets. Hába also made use of Forster pianos, specifically made for

the microtonal intervals of quarter and sixth tones.169

Julian Carrillo (1875-1965), from Mexico, created the Crusade for the 13th

Sound,

a microtonal club, or incorporation, dedicated to the “13th

” overtone, or microtone. His

theoretical work El Infinite Musical used the square roots of the intervals most commonly

found in Bach’s Preludes and Fugues, compressed into a range of half an octave, in

which the tritone is set to equal the “octave.”170

Carrillo’s Sistema General de Escritura Musical (1957) is an example of his

microtonal literature campaign, from his Mexico City office El Sonido 13.171

Carrillo is a

figure of incredible imagination who investigated history for alternative methods for

simplifying microtonal notation, since he was operating with 72 TET, and discovered

how unwieldy standard notation can be for advanced microtonal systems. Instead,

Carrillo sought to re-design the staff system using numbers, much similar to the way Jean

169

Starkhendon, 1962. XXth Century Experiments in Microtonalism. p. 2. 170

Ibid. 171

Carrillo, 1957. Sistema General de Escritura Musical. p. 12.

51

Jacques Rousseau’s use of numbers, rather than pitch and staff placement, to indicate

pitch. Register would be taken care of by moveable clefs172

(see Example 2.3.1). Number

would therefore represent the precise division of the octave, while clef would indicate

what range, or register the pitch is located in. The staff becomes reduced to a single line.

Carrillo also transcribed popular classical pieces to his new system, showing that it could

work.173

172

Ibid, p. 12. 173

Ibid, p. 32-45.

52

Ex. 2.3.1. Carrillo’s numerical notation process at work.174

174

Carrillo, p. 31.

53

Was Carrillo an advocate of polymicrotonal composition? Actually, yes, he was.

In his Toccata for violin, cello, guitar, cornet, and harp, he mixes quartertones with

sixteenth tones; the cornet and harp are tuned to sixteenths and the rest of the choir is in

quartertones.175

One possibility of this system having never caught on with composers

Ex. 2.3.2. Carrillo’s numerical microtonal notational technique.176

175

Ibid, p. 49. 176

Ibid.

54

may be the lack of melodic contour, or the absence of the visual element, that inhibited

the ease of vertical expression. Tepepan, is a work that also mixed sixteenth tones with

quartertones.177

Carrillo worked with a master tuning grid of 96 pitches per octave, which can be

subdivided into smaller microtonal systems, much like Wyschnegradsky, who had

subsystems within his 72-tone grid. The question whether or not polymicrotonality in this

context is nullified, based on the idea of subsumption, cancels out when one considers

compositional intent, and deliberate use of microtonal materials designated by the

composer.

Horizontes, for harp and percussion, is mostly in third tones. Although there are

no semitones, other odd divisions from fifth tones to sixteenth tones are present.178

177

Ibid, p. 50. 178

Ibid, p. 54.

55

Ex. 2.3.3. Carrillo’s use of third tones up to 16th

tones.179

In the example above, 2.3.3, Carrillo mixes tunings from a 96-tone master grid,

where all pitches are numbered accordingly. In the first system, the harp ascends and

179

Ibid. p. 51.

56

descends in eighth tones, followed by a 2/3 tone skip before a fermata. The harp then

descends in 16th

tones, followed by skips of a 1/12th

tone, followed by a fermata. Then

the second system features a harp descending in 1/16th

tones, followed by skips of 24th

tones in the third system.

Adriaan Fokker (1887-1972), famous for his Euler-Fokker genera, the Fokker

Periodicity blocks, just intonation, 31-tone equal temperament, and his work as a

physicist, working among such luminaries as Max Planck and Albert Einstein, wrote two

major treatises in microtonal theory: Just Intontation (1949), and New Music with 31

Notes (1975).

Ex. 2.3.4. The Fokker Periodicity Block.180

Let an n-dimensional lattice (i.e. grid) embedded in n-space have a numerical value

assigned to each of its nodes. Let n be preferably equal either to 1, 2, or 3. In the two-

dimensional case, the lattice is a square lattice. In the 3-D case, the lattice is cubic.

Examples of such lattices are the following (x, y, z and w, are integers):

· One-dimensional: 3-limit

A(0) = 1

· Two-dimensional: 5-limit

· Three-dimensional: 7-limit

181

180

Fokker, A. D., (n.d.), Unison Vectors and Periodicity Blocks in the Three-Dimensional (3-5-7-) Lattice

of Notes. http://en.wikipedia.org/wiki/Fokker_periodicity_blocks. Accessed 11/11/11

57

Fokker is known for rekindling the theories of Christiaan Huygens, whose tuning

of 31 notes to the octave with pure thirds, sevenths, and 5-cent flatted fifths are the same

as meantone tuning. Since 31 TET is circular182

, it also contains both 12 TET and

meantone tunings.183

Fokker advocated the adoption of the pure seventh (968 cents) incorporated as

standard tuning, adopting Giuseppi Tartini’s geometric angle symbol for a diesis flat

(resembling the number 7).184

This same symbol has been used by Ben Johnston, found in

the tuning legends of his scores.

Fokker termed his genera “complete contracted chords” in the form of (1, 3, 5)

whose “guide tones” were products of this formula, in this case: 15. Euler-Fokker genera

are essentially prime numbers 3, 5, and 7, or higher, where each number forms a

dimension in a multidimensional lattice, exploited to the fullest by microtonalist Erv

Wilson and also used by Ben Johnston for his “53 to the octave” tuning. In short, the

Fokker genera are products of multi-sets of prime numbers above 2.

Huygens’s tuning comes from extended Pythagorean tuning up to 31 tones, where

31 became the number where the spiraling fifths came close enough together to become a

181

Find n nodes on the lattice other than the origin such that their values are sufficiently close to either 1 or 2.Vectors from the origin

to each one of these special nodes are called unison vectors. A quantity n of unison vectors are enough to define an n-dimensional

tiling pattern. Let the n unison vectors define the sides of a tile. In 1-D, a tile is a line segment. In 2-D, a tile is a parallelogram. In 3-D, a tile is a parallelepiped.

Each tile has an area given by the absolute value of the determinant of the matrix of unison vectors: i.e. in the 2-D case if the unison

vectors are u and v, such that and then the area of a 2-D tile is

Each tile is called a Fokker periodicity block. The area of each block is always a natural number equal to the number of nodes falling within each block. Source: http://www.huygens-fokker.org/docs/fokkerpb.html. Accessed 4/2/12.

182 Cirularity is the condition that all keys are accessible within one gamut of pitches, e.g. meantone tuning,

if left at only 12 pitches is not circular since some keys will be grossly out of tune. 183

Fokker A. D. 1975. NewMusic With 31-Notes. p. 16. 184

Ibid, p. 19.

58

closed system, similar to Riccati’s diagram of extended Pythagorean pitches.185

For

example, E## and Gbb would be separated by only 10 cents, which is the limit that both

Fokker and Huygens each considered enharmonic.186

Interestingly, Fokker subscribed to

subharmonics and undertones, perpetuating an acoustic myth that probably influenced

Partch, regarding undertones187

. Fokker’s theory was triadic and included the 7th

,

previously considered too dissonant in Western tonality.

Calling polymicrotonality polytemperé, Jean Etienne Marie (1917-1989) would

become Charles Ives’s (to be discussed in Chapter 5) true successor in polymicrotonality,

following the lead made by Ives’s Universe Symphony.

Marie was a student of Messiaen and was influenced by Messiaen’s polytonal and

polymodal approach to composition.188

Marie is also an exponent of French color,

carrying on the lineage from Debussy. Marie thought that the most exciting music

contained microintervals and the confrontation of different temperaments.189

He felt that

the world of polytonality had “opened up” the world to polymicrotonality.

Marie wrote L’homme musical in 1976, in which he explicitly documents his

polymicrotonal system and gives a brief and particular survey of his predecessors: Hába,

Carrillo, and Wyschnegradsky. Oddly, he fails to mention Ives, except that it would be

Johnny Reinhard’s realization of the Universe Symphony, in 1996, that would bring

attention to its polymicrotonal characteristics, then unknown to Marie. Marie’s first work

in polymicrotonal systems was Concerto milieu divin (1969/1970) in third, quarter, fifth,

185

Ibid, p. 19-45. 186

Ibid. 187

Although undertones are theoretically possible, they are not actually possible, since they would fall

below the threshold of perception for the human apparatus. 188

Marie, 1976. L’homme musical. p. 20. 189

Ibid., p. 89.

59

and sixth tones for orchestra. There is a recording of this piece under the direction of

Lukas Foss from 1970.

Le Tombeau de Carrillo (1966) is Marie’s crowning achievement, featuring two

live pianos in 12TET and 18TET (third tones), with tape of two pianos tuned to fifth and

sixth tones. The tape idea helped alleviate portability problems by simply recording the

fifth and sixth tone tuned pianos to tape, whereby only one piano, in third tones would

need retuning on location. Marie was practical. The work is the second deliberate

polymicrotonal piece in history.190

190

Marie, 1966. Le tombeau de Carrillo.

60

Ex. 2.3.5. Marie’s graphical display of the Aristoxenian principle of polymictotonal

systems and equal division of the whole tone as ever divisible.191

191

Marie, L’homme musical. p. 90.

61

Le Tombeau is ternary: the A section uses live pianos in half and third tone

tuning, while the B section employs sound masses of the fifth and sixth tone tuned

pianos, performed in studio and stored on tape.192

In the previous example, 2.3.6, Marie demonstrates the multiple systems at work

in his Le Tombeau. All pianos, the strata break down thusly: the top system of 8 staves

are divided into 30 equal steps per octave; the middle system is 12TET; the second from

the bottom are divided into 18TET, and the last system is divided into 36 tones per

octave. Marie’s system is represented by a root 2 procedure, thus ensuring the equal

divisions within a 2:1 ratio, the octave.

192

Loc. cit.

62

Ex. 2.3.6. Marie’s Le Tombeau de Carrillo, 1966, showing the layers of different

microtonal systems by square root terminology.193

193

Ibid. p. 91.

63

Ex. 2.3.7. Polymicrotonal sets of tetrachords in Tombeau de Jean-Pierre Guezec, 1971194

In example 2.3.7, Marie demonstrates a tetrachord from his Tombeau de Jean-

Pierre Guezec (1971). The initial motivic cell, A, G#, Bb, A, becomes transposed

through four other equal divisions: third, quarter, fifth, and seventh tones. Each line

194

Ibid., p. 92.

64

begins with a different rotation of the cell, so there are serial procedures involved in the

displacement of the polymicrotonal materials. Each transformation of the cell results in

an audible compression of the intervallic structure. The lines also transpose down by one

semitone, for each layer downward.

Like Ives, Marie was not a tuning purist; they both evinced a predilection for

equally divided whole tones, rather than basing their pitch structures solely on the

overtone series, or just intonation.195

Marie, like Bartók in his String Quartet No. 4, was

interested in the compression and expansion of equally-spaced intervallic structures,

which necessitates equal divisions, rather than unequal just tunings. Schoenberg is also

credited by Marie in framing 12TET in terms of standard tuning as a series of the twelfth

root of 2, an octave divided into 12 equal spaces.196

Much in the same way as creating

any division of the twelfth root of 2—an objectification of our common materials seen

through new ways can be seen as a furtherance of Schoenberg’s bequest. A drawback to

Marie’s system, however, lay in his dependence and reverence for the piano, as the

talisman of musical hegemony.

Marie notes that Hába began to create scales from his tunings, including self-

limiting constructions as tetrachords, except over an octave, in third tones. Marie wanted

to avoid the possibility of sub-system subsumption (inadvertent inclusion of smaller

systems of intervals) as a potential threat to Hába’s, Carrillo’s, and Wyschnegradsky’s

systems, by deliberately creating layers of separate tunings, revealing the intent of the

composer to structurally oppose different divisions of the octave as individual and viable

tunings. According to Marie, Carrillo, Hába, and Wyschnegradsky offered a general

195

Ibid. pp. 41-44. 196

Ibid. p. 26.

65

approach and exploration of microtonal scales and new harmonies possible with artificial

limits.197

It appears that Marie’s attitude towards microtonality and tuning had an impact on

Xenakis, who also was a student of Messiaen. Marie also acknowledges the influence

Aristoxenus had on Xenakis, with respect to the Genera of Greek tunings.198

197

Ibid., pp. 52-65. 198

Ibid.

66

Ex. 2.3.8. Marie’s table of equal divisions of the octave.199

199

Ibid. p. 41.

67

Ex. 2.3.9. Marie’s polymicrotonal table of strings for the possibility of manufacturing

microtonal pianos.200

200

Ibid. p. 31. (Planche 3)

68

Ex. 2.3.10. from Marie’s Concerto milieu divin, for orchestra, with polymicrotonal string

section.201

Examples 2.3.8 and 2.3.9 are charts of microtonal systems to the exact cent value,

for 2.3.8, and for 2.3.9, for the potential construction of pianos, from third tones all the

way to twelfth tones. Each microtonal system on the spreadsheet is divided into numbers

of keys, to contain the system, size of the piano strings, duration of the keyboard

hammering mechanism, and number of strings per pitch.

Example 2.3.10 reveals the polymicrotonal layout of the strings in Concerto

milieu divin. First violins are in sixth tones centered around D above high C. Second

violins are incorporating both third and fifth tones centered on A . Violas are fluctuating

from third to fifth tones centered on E#. The cellos are fluctuating from quartertones to

third tones on B, while the basses are vacillating from third to quartertones centered on

pitch D below bass clef. Although microtones are present in other instruments as well,

such as a quartertone in the soprano clarinet, the bulk of the polymicrotonal usage is

focused on the strings.

201

Marie, Planche 21.

69

Marie felt that one can transcend the Occident by pushing its limits. Maximum

and minimum resolution, also, plays a role in limiting the reality of polymicrotonality: in

order to play third tones through sixth and seventh tones, there needs to be a realizable

electronic device, such as an Ondioline, by George Jenny,202

for example, that has a

tuning resolution of at least a 4twentieth tone. Our ears, according to Helmholtz hit the

JND at around 10 cents. Within the context of polymicrotonality, Marie felt it was

important for identities to become lost, and then rediscovered within the context of a

piece of music.203

In polymicrotonality, there can be transpositions and modulations that would

result in compression and expansion of intervallic structures, all of which could be

managed by small intervallic cells, like tetrachords, but in much smaller spans than a

perfect fourth.204

Marie’s system of notation, like Wyschnegradsky’s, was fairly

unwieldy, leading one to wonder whether or not accidentals are the problem, or perhaps

the staff.

Example 2.3.11 reveals a process of polymicrotonal compression of intervallic

structures. By retaining the same intervallic number of steps between all the tunings, one

notices a very gradual compression of the size of intervallic distances, resulting in a

powerful and new sonority. Since there are no major, or minor triads with respect to the

classical 12TET number scheme, new methods of chordal change become necessary,

such as intervallic expansion and compression. Persichetti also speaks of these processes

202

The Ondioline, from 1941, was an electronic synthesizer that could divide the octave by equal parts.

Also, Marie felt that by annexing additional keyboards and strings to a standard piano, one could also

achieve similar results in creating microtonal keyboards. 203

Ibid., p. 94. 204 Ibid.

70

in his discussions of twentieth century harmonic techniques, but this applies just as well

to microtonal systems.

Ex. 2.3.11. Marie’s theory and examples of proportional intervallic transposition via

microtonal steps from half tones through seventh tones. 205

205

Ibid. p. 88.

71

Iannis Xenakis (1922-2001) was polymicrotonal by way of the Greek genera of

Aristoxenus, and the Byzantine Church. Aristoxenus, as mentioned earlier, had

generalized a system from the Enharmonic, through the Diatonic by degree of size of the

Pyknon in the tetrachords: the Enharmonic, the smallest, were quartertone, at 50 cents;

the Chromatic, the middle sized, were third tones, at 66 cents; the Diatonic, the largest,

was the Pythagorean limma at 90 cents.206

Xenakis used tetrachords and materials from Greek music by going back in

history to his own genetic roots. Genera mixing and modulation did occur in Greece,

depending on the instrumentation involved and the performer’s erudition. Xenakis also

contended that Byzantine culture mixed the Pythagorean and Aristoxenian philosophies

of tuning. It therefore gave precedence to allow Xenakis to mix quartertones and third

tones in his sieves, while designing his “outside-time” scalar structures for some of his

works in the late sixties and early seventies.207

Xenakis used opposing tunings for the distinction between vertical and horizontal

effects, a type of counterpoint between quartertones and third tones, with no leaping in

the voices.208

Xenakis’s use of microtones, via sieves, or permutations, were unmasked

with respect to the differences of tuning and were intended to be obvious between the

third and quarter tones. The following table shows Aristoxenus’s genera, in terms of 30

divisions to the tetrachord of the perfect fourth, as used in ancient Greece.209

In addition

to thirdtones and quartertones, Xenakis also employed eighth tones, the comma.

Enharmonic 3-quartertone

50 cents

3-quartertone 24-large third

206

Xenakis had generalized these divisions as induced by his understanding of how the Byzantine culture

came into possession of the Greek genera as third tones, via Aristoxenian philosophy of non-measurement 207

Xenakis, Formalized Music, 1971. 208

Marie, L’homme musical, 1976, p. 65-71. 209

Ibid. p. 65.

72

Chromatic 4-third tone

66 cents

4-third tone 22-small major third

Hemiolic Chromatic 4.5- 75 cents 4.5 21-minor third

Diatonic 6-semi tone

90cents

12-whole tone 12-whole tone

Table 2.3.12. Xenakis’s microtonal pitch derivations via Aristoxenus.

Pythagoras 1/1 9/8 5/4 4/3 3/2 27/16 15/8 2/1

Cleonides 0 12 22 30 42 54 64 72

Table 2.3.13. Cleonides, a student of Aristoxenus is credited with the tetrachordal

division into 72 parts; here there is a one to one correspondence between Pythagorean

ratios and Aristoxenian approximations.

Here is Xenakis in a nutshell: “SILENCE IS BANAL.”210

A reason why

Aristoxenus appealed to Xenakis was due to the equal divisions of his tetrachords. In

creating a sieve, the resolution of the tuning can be infinitely small; however it will be

uniform and equal. Therefore, in Xenakis’s system where 1 equals the semitone, ½ will

be equivalent to the quartertone, and 2/3 will be equivalent to the third tone. The

quartertone will then be called k24 (since that is what Xenakis called them), or modulo

24 and modulo 18 for third tones.211

This paper will not discuss the particulars of Xenakis’s sieve theory and Boolean

logic, since they can be referred to in his book Formalized Music, of 1971. Nevertheless,

according to Xenakis’s ideas on sieve theory, a scale or tuning is an outside time structure

because it cannot be altered by vertical or horizontal implications.212

Xenakis also denied

octave scales in favor of the tetrachordal system, in particular the Immutable Perfect

210

Xenakis, 1985. Arts/Sciences Alloys: the thesis defense of Xenakis. p. 95. 211

Ibid, p. 107. 212

Xenakis, 1971.Formalized Music. p. 183.

73

System, as discussed earlier, because he favored the nesting of the pyknon within the

tetrachords. Aristoxenus took the tone and subdivided it into melik, or singable, divisions,

where the quartertone and third tone is considered performable. In fact, Xenakis’s entire

pitch theory revolves around the Enharmonic and Chromatic genera of Greek musical

theory. Xenakis accepted the perfect fourth diatesseron as a functional unit and the

mutable inner parts, the lichanos and parhypate, as the metabolae, or catalysts of

modulation and change.213

The outer hypate and mese remain fixed while the inner

pitches create the species differences. In Xenakis’s mind, the Byzantine style

amalgamated the two philosophical and mathematical differences between Greece and

Byzantium, the rational Pythagorean method, versus the intuitively approximated

Aristoxenian method. Another way of stating this relationship is the arithmetic versus the

geometric methodology of octave division, as canonized by Ptolemy in the first century

AD.214

According to Xenakis, Byzantine music focused more on scales and the octachord

and pentachord, rather than the diatesseron (containing the perfect fourth). One may even

mix the genera of tetrachords in the Byzantine style, as in the Selidia, a compendium of

mixed genera scales, of Ptolemy.215

The notion of montage comes to mind, or perhaps

even polymicrotonality, since Xenakis does, in fact, extrapolate the pyknon from each of

the enharmonic and chromatic genera for his own personal use. Was Xenakis looking for

a loosening of practice for his own vision, or seeking permission for his own use?

Ultimately, Xenakis saw tropes, the quasi improvised lines of solo singers in the

medieval motets, as a corollary to his own sieve formulation, with mixed genera and

213

Ibid., p. 184. 214

Ibid., p. 184-189. 215

Ibid., p.189.

74

juxtapositions of tetrachords within his metabolae, the ancient Greek method of

changing, or modulating, which he thought of as opening the compartmentalized

hierarchy. Metabolae is the underpinning to Xenakis’s philosophy as there is a free

circulation between the genera and their notes and subdivisions thusly leading to his own

brand of polymicrotonal usage.

Xenakis felt that Occidental music, or polyphony, had led to the reduction of the

outside time structures to the microtonal pitch palette. This reduced pitch selection, for

the sake of consonance in polyphony; negated microtones; and encouraged the

calcification of Western music that Xenakis called the degradation of outside time

structures,216

which many twentieth century composers have been desperately trying to

revive. The maqam of the Middle Eastern Arabic peoples has maintained its vitality due

to the monophonic nature of Arabic music. This concept was not lost on Harry Partch,

who also subscribed to the idea of monody, except for one thing: Partch said that

microtonal vertical structures are ugly(A Genesis of a Music)? I, for one, do not. Instead,

I believe that not only can single-system microtonality sound beautiful in its verticality,

but that mixed temperaments and tunings also sound beautiful in their verticality.

Xenakis also warned of suffocation underneath technology; in response I propose that

polymicrotonality remain an acoustic enterprise.

Xenakis felt that the beginning of the breakdown of this calcified Western system

was led by Debussy.217

I would suggest Charles Ives in addition to Debussy. The French

symmetrical pitch structures-- whole tone, octatonic, and modes of limited transposition--

were employed by composers, such as Messiaen, Debussy, and even Stravinsky, and

216

Ibid. p. 193. 217

Ibid. p. 208.

75

were all non-tonal materials.218

Conversely, there are theorists and composers, such as

Dmitri Tymoczko, and Stephen Taylor, who contend that they extend tonality. An

additional aspect to this predicament is the notion of compositional choice, where choice

in outside time structures is disregarded by composers beginning with serialism. One

must “go beyond them” with respect to the current state of Western pitch materials, and

the maximum impact that can be done is the area of polymicrotonality.

Anaktoria (1969) incorporates three systems: 12TET, 18TET, and 24TET. It is an eight-

part mixed octet with standard micro-intervallic voice leading, as Xenakis did not use

leaps or jumps in his microtonal intervals. Example 2.3.15 shows the polymicrotonal

chords, in particular at measure 39, where there are both third tones and quartertones

together in a cluster, as a counterpoint of sound masses as well as a counterpoint of

simultaneously competing microtonal pitch materials.219

Between measures 59 and 60

there is a sudden shift from third tones into quartertones, showing an instance of

metabolae (Ex. 2.3.16).

218

Ibid., p. 208. 219

Xenakis, 1971. Anaktoria.

76

(Ex. 2.3.14, cited below)

77

(Ex. 2.3.15, cited below)

78

Ex. 2.3.14-16. Anaktoria, Xenakis’s polymicrotonal piece in third and quartertones.220

Three more Xenakis works fall into the realm of polymicrotonality: Aurora,

(1971), Eridanos, (1972), and Akanthos, (1977). Again, all three works feature the

combination of Byzantine third tones and quartertones. Auroroa is linear in construction,

with different assignments for tuning: first violin is in third tones, second violin is in

quartertones, viola uses both, cello is in third tones, and bass uses both.221

Xenakis uses

polymictotonal detuning at m. 60, where a cluster centered on C is altered to C-1/4 sharp,

C-third sharp, and C#.

Eridanos was composed for orchestra and the work is more modulatory, or

metabolic, in conception; third tones appear from mm. 217 to 222, as a quasi cadential

220

From Anaktoria, Xenakis, 1971. 221

Xenakis, 1971. Aurora.

79

device as quartertones return just before the ending. The rest of the piece is tuned to

quartertones.222

Akanthos, also for quartertones and eighth tones, features a metabolae from

quartertones to eighth tones in the soprano voice at measure 86, where changes begin to

show from voice to voice linearly.223

In addition to the previous polymictotonal pioneers, there are others who also

made some significant contributions. Even though they made one or two pieces in this

style, they are worth noting.

György Ligeti’s Hamburg Concerto (1998-2002) for solo horn, naturally-tuned

horn choir and orchestra is a polymicrotonal piece mixing both overtone series tuning and

quartertones, alongside standard 12TET. The work exploits the natural overtones of the

horn, up to the 15th

partial, without tempering.224

222

Xenakis, 1985. Eridanos, pp 19-20. 223

Xenakis, 1977. Akanthos. 224

Ligeti, 2002. Hamburg Concerto.

80

Ex. 2.3.17. Ligeti’s legend for the Hamburg Concerto showing the exact tuning for

horn.225

225

Ligeti, 2002. Hamburg Concerto, Instruments page.

81

Ex. 2.3.18. Hamburg Concerto; quartertone and overtone chord.

82

Ex. 2.3.19. Ligeti uses many valve positions for a full range of natural tunings.

Not only is the lead horn in natural overtone tuning, but the entire horn choir is as

well, forming a block of overtone/just tuned intervals in direct opposition to quartertones,

83

as in Ex. 2.3.17. The harmony at this point, m. 11, is B,D1/4#, F1/4#, A, C1/4#, E in the

strings, and a standard tuned A with an 18 cent flat A, and a 35 cent flat B in the horns.226

Gardner Read’s 20th

Century Microtonal Notation is, so far as this author knows,

the first book to employ the word “polymicrotonalism.”227

The word was used in

reference to Jon Catler and Johnny Reinhard’s microtonal group where the two virtuosos

played in quartertones descending and 31TET ascending, in scalar fashion.228

Similarly,

Reinhard’s essay, Composing Polymicrotonally (1996), is in league with Jean Etienne

Marie’s “polytemperé.” For Johnny Reinhard, 1996 is also the same year that he realized

Ives’s Universe Symphony, featuring up to four competing microtonal tunings, in

contradistinction form Larry Austin’s which avoided polymicrotonality.

Reinhard states that he is not attracted to any singular tuning and that musical

communication improves by using unlimited set of meaningful intervals, harkening back

to Ives.229

Reinhard feels that since most microtonal music uses one tuning, such as

19TET, quartertones, or just, that the next step is polymicrotonality, looking back to

polytonality and polymodality of the early twentieth century composers Bartók and

Stravinsky. Polymicrotonality is a personal artistic choice. Like Xenakis, Reinhard

addresses a taboo twentieth century concept: composer’s druthers, which is the desire to

simply compose music without any particular system-- the ultimate faux pas.

Like Ives, Reinhard believes that the human ear’s acuity is capable of much more

than the mild prejudice of 12 equidistant intervals. Alexander Ellis’s conception and

226

Ibid. 227

Read.1978. 20th

Century Microtonal Notation, p. 120. 228

Ibid., p. 120. 229

http://stereosociety.com/jrpolymi.shtml , 11/11/2011.

84

division of 1200 cents to the octave has given us 98 new intervals between the piano’s

keys.230

Johnny Reinhard (born 1956), the director and founder of the American Festival

of Microtonal Music in New York, has led numerous microtonal festivals since the

1980s. He is a devout microtonalist and a proponent of polymicrotonalism, who has

known personally Jean Etienne Marie, and who has reams of scores from Julian Carrillo.

As a polymicrotonalist, Reinhard is convinced that the pitch continuum, as a tabula rasa,

can be incorporated into the full pitch continuum, revealing many more essential

intervallic relationships. As a composer and performer of polymicrotonality, Reinhard

states: “When a composer internalizes a relationship between point of pitch and uses it

compositionally, its affective logic transfers to the audience. Moving through different

tunings is exciting to listeners because audiences feel the intent and conviction of the new

material.”231

Reinhard got turned on to polymicrotonality while playing in Jon Catler’s

microtonal group Cowpeople, where both Reinhard and Catler performed quartertone and

31TET together. Jon Catler also composed for 19TET and 31TET, performing on electric

guitars while Reinhard played an electric bassoon. “Form takes on new meanings in poly-

organicism as fresh forms spring from imagination providing an ideal ‘gestalt’ for

presenting new semantics of poly.”232

Gestalt, in this sense, means essence, or its

complete undivided form, as conceived by the mind.

230

Ibid. 231

Ibid. 232

Ibid.

85

Ex. 2.3.20 Johnny Reinhard’s Dune, from 1990, for solo bassoon.

86

Ex. 2.3.21 Second page of Dune, by Reinhard, 1990.

Reinhard uses multiphonics and swirls the bassoon in a clockwise motion, as

though stirring up the air and speeding up. The piece is dedicated to Herbert Spencer’s

87

science fiction Dune, and is programmatic. Reinhard uses various divisions of the octave

in a quasi-improvisational way, while employing multiphonics for texture. The system

from example 2.3.21 appears to be in overtone tuning. The section dedicated to the

Fremen is when the additional polymicrotonal exploration begins.

2.4 Other Twentieth Century Microtonalists

David Doty’s Just Intonation Primer, although not in the direction of

polymicrotonality, is an excellent source for the nature of one of the more beguiling

tuning systems since ratios became the de facto standard for pinning down specific

pitches and intervals. Since polymicrotonality includes just intonation, Doty’s work

becomes relevant.

Doty explores the nature of limits, lattices and ladders, and other tricky subject

matters in a terse, to the point manual.233

For example, all of Western history, and some

of the Middle East, can be represented by tolerable limits in just intonation: Pythagoras

and 3-limit; Western tonality and 5-limit; the Middle Eastern maqam and 7-limit; Ptolemy

and 11-limit; Vicentino and 13-limit; Partch and 11-limit, and Ben Johnston has gone as

high as 13-limit justly intoned tunings.234

Limits in just intonation are the end points at which a new prime number, which

unlock new microtonal sonorities as divisors, stop. Each successive prime number added

to the limit includes all those before it. All other numbers, odd and even, are simply

multiples of overtones that are already represented in the tuning. Western tuning stops at

5-limit just intonation, since the ratio of 5 to 4, when logarithmically calculated into

233

Doty, 1993. A Just Intonation Primer. p.1. 234

Ibid, p.76.

88

cents, yields the just Major third, at 386 cents. Pythagorean tuning is set at 3-limit, since

the third overtone is the perfect fifth, at 702 cents. Primes are then paired with their

register: for example, the series 2/1, 3/2, 5/4, 7/4, 11/8, 13/8, 17/16, 19/16 contains all

successive primes positioned over the first, second, third, and fourth octaves, represented

as multiples of 2. Identities are therefore odd numbered and prime, which means they will

have a new microtonal meaning and position.

Among Doty’s subjects, the tetrachords of ancient Greece are mentioned as a

microtonal organizing principle.235

The Greek Genera can serve as a brilliant method for

creating smaller cells of manageable materials, much like the way they did in ancient

Greece. Tetrachords, also, do not have to conform to the span of a perfect fourth, either,

but can be whatever fits the needs of the composer.

Pythagorean tuning can be looked at as a flavor of just intonation. Pythagorean

tuning, again from the Greek viewpoint, works well for linear functions and melody, as

discovered by the Ars Antiqua for parallel fifths, fourths, and octaves of discant and

organum. Pythagorean tuning, due to its 408-cent wide major third, was and still is

considered too dissonant for harmony; so the introduction by Zarlino, of the just third

from the fifth overtone made its entrance into European polyphony, making 5-limit just

intonation, like Ptolemy’s Syntonon Diatonic genus.236

At the time Doty wrote this work in 1993, the 11 and 13-limit just intoned sound

was still considered beyond Western comfort. 11-limit is quartertone sounding, while 13-

limit is sixthtone sounding. Partch and La Monte Young both believed that we are still

climbing up the overtone series, to which I agree wholeheartedly. Young and microtonal

235

Ibid, p. 30. 236

Ibid, p. 35.

89

composer Glenn Branca have both traveled as high as the 128th

harmonic, employing all

the available primes therein, while Johnny Reinhard has recently written a soon to be

released article on his 8th

Octave Overtone Series Tuning, ascending to the 256th

harmonic, no less.237

One of the “complaints” against microtonal tunings of this

magnitude lie in root movement progressions, versus the static nature of such vast

harmonic possibilities.238

Doty suggests building one’s own arsenal of acoustic

instruments, but since we have seen the limitations of this approach through Partch, I am

not so sure this is a good idea. I think it is better to help our existing instruments find

their own ways of producing these tones, which are not beyond the realm of possibility.

Enrique Moreno’s thesis Expanded Tunings in Contemporary Music (1992) cites

Easley Blackwood, Joel Mandelbaum, and Joseph Yasser as exemplary microtonalists

with their own systems of tuning: Mandelbaum and Yasser with 19TET; and

Blackwood’s etudes on tunings in a numerical order, like the 24 keys of Bach’s Well

Tempered Klavier.239

Moreno argues for non-octave tunings in his treatise and classifies tunings into

two major areas: octave versus non-octave. The purpose was to redefine the octave

function as a “marker” organizing principle. The treatise is essentially a compendium of

octave and non-octave scales and systems of equal tunings. One of the drawbacks to this

system is that octave tunings will overpower, or psychologically induce false octaves in

non-octave tunings, if combined.240

237

Reinhard, personal communication. 238

Doty, p. 58. 239

Moreno. 1992. Expanded Tunings in Contemporary Music. p. 16. 240

Ibid., pp. 35-36.

90

Easley Blackwood (born 1933) composed an electronic set of 12 microtonal

etudes, Twelve Microtonal Etudes for Electronic Music Media (1980), which were

seminal in conception as a tool for comparative analysis between the tunings 13 through

24. Blackwood’s treatise, The Structure of Recognizable Diatonic Tunings, 1985, focused

on extended Pythagorean, just, meantone, and equal temperaments and tunings, which

was a rather math-intensive survey of the tunings in European history. For example,

Blackwood states that just intonation is simply a matter of tempered Pythagorean tuning,

by lowering the major third from 408 cents to 386, and that the ratio 4:5:6 is a pure

tuning.

Blackwood invented his “subscript system,” which entails the following: 0 =

same tunings; 1 = difference of a comma; 2 = difference of a small diesis, where 0 is

essentially Pythagorean tuning, seen as pure tuning by Blackwood.241

There are only two

pure tunings: Pythagorean, and the overtone series because neither tunings have tempered

fifths. Both follow mathematical series, but Pythagorean was created by man, and

overtone tuning follows nature’s infinitely mathematical arithmetic tuning, as an infinite

series. Blackwood raised the question, again, about the purity of the historical

methodology, except that even with his own treatise, there is always subjective

speculation. Music is both a science and an art: there are no steadfast rules for the

validity of tuning, unless one adopts nature’s model, the overtone series, which may as

well be the rudder for all arguments of purity.

Joel Mandelbaum’s 19TET, his dissertation of 1961, settled on this tuning, since

it represents meantone tuning and the inclusion of enharmonic subtleties.242

Mandelbaum

241

Tuning vs. temperament, what is the difference? Please see chapter 1.3.6. 242

Mendelbaum, 1961, p. xiii.

91

maintained that consonance was simply the conditioning of the ear, which I absolutely

agree with.243

He also noted that Aristoxenus and Partch, as well as Xenakis, claimed the

ear over number, reinstating the human being as the arbiter for intervallic meaning, rather

than pure number ratios.

The backbone to meantone tuning is Pythagorean tuning, and the best way to

represent meantone tuning in all its variance is through both 19 and 31TET, where all the

nuances of Pythagorean tuning are present. A good portion of Mandelbaum’s treatise is

on examples of Pythagorean tuning and its import. A key argument Mandelbaum makes

is that Pythagorean tuning cannot account for the thirdtones and quartertones in the

pyknon of certain genera.244

Therefore, other criteria were used in the divisions of these

pyknon, katapyknosis, probably, or the use of just ratio techniques.

Mandelbaum’s treatise on microtonality, even though it is specific and not

particularly related to polymicrotonality, nevertheless is an important agent in the

twentieth century microtonal experience, and it asks very good questions regarding the

nature of microtonality. Yasser, apparently, used polytonality as a phenomenon

anticipating a “new system,” except that there is no such thing as polytonality if the

enterprise is based on one tuning system: 12TET.245

In the end, Mandelbaum states that

there are only two intervals, period: consonant and dissonant, where dissonant intervals

are the smaller intervals.

This author recommends that the reader investigate Mikrotöne, 1985-1991, a

symposium and “journal” of the microtonal happenings in Europe, led by Franz Richter

243

Ibid., p. 5. 244

Ibid., p. 83. 245

Ibid., p. 130.

92

Herf, who has established Ekmelic Music, after the Greek term for microtones of an order

thought unsingable. The journals have covered Wyschnegradsky, Ezra Sims, Hesse’s

Grundlagedn der Harmonielehre des Mikrotonaler Musik, and all sorts of tuning schemes

and compositional ideas. The festival has been held at Salzburg, Austria.246

2.5 China and Tuning

In China, there is the history of the Lü, which has been described as a very small

division of pitch.247

“In Heaven and Earth, there is no boundary” seems to vindicate the

ideas of Aristoxenus, even in the far reaches of China, in 218 A.D.248

The Chinese,

independently of the Babylonians or the Greeks, had also come to the realization of

tuning by the perfect fifth, by the ratio 3:2. The Chinese concept of tone generation is

cyclic, spiraling fifths, and never ending, as opposed to the European concept of simple

divisions by 2.249

There are a number of pitch series in Chinese history, starting with the 60 Lü of

Jing Fang, 33-73 A.D., who used the gradations of pitch in relation to the horoscope of 5

sets of 12, totaling 60 “character” pitches, with no octave equivalents, at 1203 cents to an

“octave.”250

There is also the 360 Lü of Qian Lezhi, 581 A.D., a pitch for each day of the

year, with an octave at 1201 cents. All of these were figured by Pythagorean

multiplication. There was also the 144 Lü of Wan Baochang, who chose 12 pitches from

246

Herf, 1985-1991. Mikrotöne I-IV, 1985-1991. 247

Cho, 2003. The Discovery of Musical Equal Temperament in China and Europe in the sixteenth

Century. p. 100. 248

Ibid., p. 150. 249

Ibid., p. 152. 250

Ibid., p. 159.

93

Lezhi’s 360, and then proceeded to divide the semitones, limmas at 90 cents, into 12 parts

of their own.

In just intonation, Wang Po created the pentatonic 1/1, 9/8, 5/4, 3/2, 5/3, 2/1, at 5-

limit, in 959 A.D., circa 500 years before Zarlino had introduced the just third into

Europe’s Pythagorean tunings.251

The shu’bas and tarkibs in Arabic modal music all require modulation from

maqam to maqam with fluency, where many modulations are changes of genera, as well

as mode.252

In fact, the word shu’ba means “branch” of a tree that metaphorically shoots

off in a new, but related, direction, the way music does when it modulates.

Kevin Jones, in his paper about Xenakis and Ancient Lü of China, also mentioned

that multiple temperaments were common in Chinese music and that it was characteristic

of ancient Chinese music and contemporary Chinese folk ensembles to use simultaneous

tunings derived from open cycles of fifths, just ratios, equal temperament, and the higher

overtone series tuning.253

To a Western listener, this “sweet and sour” combination is

bewildering and distressing, but is consistent with the broader axioms of Chinese

thought.254

Polytemperament (polytuning) is an important trait in Chinese music,

providing an out-of-tune flavor and is essential to Chinese music.255

So, are polymicrotonal pitches perceptible to the ear? Yes, and by framing

polymicrotonal structures in relief by putting them in differing tempos, the independence

required in order to hear is met with more drama, and efficiency, by deliberately

251

Ibid., p. 169. 252

Wright, 1978, pp. 194-216. 253

Kevin Jones/Kathleen Wong, The Architecture of Scales and the Notion of Lü, from the International

Symposium Iannis Xenakis, Athens, May 2006, p. 8. 254

Ibid. 255

Ibid.

94

misaligning the vertical structures in such a way that individual pitches can be discerned

within the context of seemingly dense textural implications.

2.6 Chapter Summary

This chapter is lengthy because it is the workhorse of this essay setting a solid

foundation for what follows. In other words, it lays down the foundation of all that

follows. The Greek genera were methods of tuning derived from stringed instruments,

such as the Kithera, which found their way and influenced Greek music, ethos, and

philosophy through hundreds of years. The modes that were built from these tetrachords

then made their way into Europe influencing most of European music, from Gregorian

chant, to modern string tuning. Since the Greek tetrachords were all tuned differently

within the space of a perfect fourth, modulation was possible. This modulation essentially

was a modulation of tuning, since the three genera were three different classes of tuning.

Modulating tuning within a work is the definition of polymicrotonality. One tuning with a

set of microtonal pitches modulating, metabolae, to another is polymicrotonal in scope.

The tuning conflicts that emerged during the early Renaissance are important

historically because of the juxtaposition of several tuning systems having been

unwittingly, albeit, in just and Pythagorean tuning. Later, the discovery of the overtone

series led to overtone tuning, and began to manifest into the European consciousness and

its music. From that point on, music theory as we know it emerged from the works of

Zarlino and Descartes, to Rameau, who matched pitch, the perfect fifth, with rhythmic

necessity: the cadence to the octave. All the while, a standard tuning system was

broached by many composers and theorists all over Europe, who were looking for a

95

panacea for a singular tuning, which frankly, has never been found. At the peak of the

Baroque, there were approximately 150 different mean tone temperaments. What does

this tell us? First, it tells us that no one tuning system satisfied everyone; that people

could all hear more than one way to tune, allowing for even the tiniest shades of

microtonal pitches to make the difference between similar tunings. If they were all

capable of being substituted for one another, then why were there so many variants? The

answer is that human ears can tell the difference. The human ear has the capacity to hear

microtonal shades that lay beyond systems of notation. Europe was inadvertently

polymicrotonal without being fully conscious of it. For example, lute players played

historically in just intonation. String players tuned to Pythagorean. Singers sang in just

and Pythagorean tuning, and keyboards were in meantone. They all performed together.

They therefore played in several tunings simultaneously, and they probably knew it and

accepted it as standard performance practice.

The twentieth century served as the playground for microtonality in a more

conscious way. The acoustician Helmholtz paved the way for a new exploration of

microtonality with his book, On the Sensations of Tone. This book provoked the Ives

family, Carrillo, Wyschnegradsky, and many others to explore alternate tunings to further

the pitch palette which had become stale by 1945. Charles Ives was the first composer in

history to deliberately make music in two or more tunings, let alone four. The composers

and theorists mentioned in chapter two all have contributed to an atmosphere of a

possible viability of polymicrotonality: something considered absurd by many, but which

is actually a logical extension of the Western pitch palette. American composers have, so

far, led the way.

96

The section on the Chinese is essential, due to its relative and cultural objectivity,

while still attending to the same ideas about tuning as the West. This section provides

proof that this phenomena is not just a by-product of Western thinking, but is a universal

musical concern that has manifested throughout the history of humanity.

97

CHAPTER 3:

RHYTHM:

REPRESENTATIVE TWENTIETH CENTURY POLYTEMPIC WORKS

Although this paper adopts an essentially theoretical approach to

polymicrotonality, the topic of polytempo will be dealt with by score examples and

representation in a less intensive manner due to the burgeoning popularity of the topic

and papers concerning it. Jake Rundall has recently written a paper on polymeter, which

touches on the subject of polytempo, but in a purely analytic fashion: POLYMETER:

disambiguation, classification, and analytical techniques (2011).256

Although Rundall

makes a case for polymeter, and not polytempo, the two are, nonetheless, sufficiently

interconnected that much of what he describes can be inferred here. Rundall’s conclusion,

however, is that a clear and complete framework has yet to be developed, with respect to

a complete polyrhythmic theory.257

One of Rundall’s points of contention in the area of polymeter and polytempo is

the differentiation between rhythm as being either polytactic or polytempic. Polytactus

rhythms occur when independent pulse tempos exist in non-fully coincident meters. In

other words, neither the meter nor the pulse link up. Polytempo involves the simultaneous

use of different tempi independently assigned to different parts.258

To the listener, what

would be the difference? To the author, the answer lies in the intent of the composer.

256 Rundall, J. (2011). Polymeter: disambiguation, classification, and analytical techniques. University of

Illinois at Urbana-Champaign. 257

Ibid. p. 1. 258

Ibid., p. 59.

98

The reason there is so much ambiguity concerning polyrhythm, polymeter, and

polytempo, is that they all involve the same phenomenon, but in varying durations of

time. A protracted use of polyrhythm can be measured in metric terms, as for example, a

seven-beat figure occurring in 4/4 meter could simply be written out as 7/4; this would,

however, require an adjusted tempo increase, since the seven beats within four beats

would necessitate a faster surface speed. A protracted use of a polyrhythm that becomes

a new meter, with a new tempo, if it is of sufficient duration, then can be assigned its own

tempo. This situation is analogous to psychology, in which a sensation becomes a feeling,

which then becomes an emotion, whereupon all extend through longer increments of

time. Therefore, a polyrhythmic motive (local) can grow to become a larger metrically

(regional) structured part of a composition, to its own tempo (global). Once a separate

tempo has been established, either implicitly written, or explicitly indicated by a tempo

marking, the part develops its own life, like a character in a novel, and has its own

trajectory and identity, while maintaining some relationship to the entire composition in

which it is contained. This happens to be the concept behind polytempic music—giving

life to more independent parts beyond standard polyphony and its concomitant

homogenous rhythms, and even tunings.

Charles Ives is probably the original polytempo master of all time, employing

polytempo in works such as Central Park in the Dark, Unanswered Question, In Re Con

Moto, et al, Fourth Symphony, Theater Set, Putnam’s Camp, and Universe Symphony.

However, to save space, I will focus more on Ives and Polytempo in chapter 5, rather

than give thorough explanations here, even if I do write briefly about some of these

pieces.

99

According to Jonathan Kramer in his book The Time of Music, temporal

multiplicity was a symptom of the movement of early twentieth century music, due to

culture and the current developments in technology.259

This is apropos for today, in what

is now the early twenty-first century, with our current computer-nano-technological

revolution. Kramer’s assertion, with respect to Ives, appears to be the case.

Ives, in his 2nd

Orchestral Set, has instrumental polyrhythmic divisions: trumpets

in triplets; flutes in four groupings; zither and viola in 11-note groupings; harp and bass

in half notes; and cellos in sextuplets. There are also antiphonal devices involving an off-

stage choir while first violins play in quintuplets in a 16th

-note grid, an ostinato, while

second violins play in four note groupings. Chimes play in half notes and basses play in

groups of fives. All of this is set against the main orchestra, where the clarinets play in

seven note groupings, while trumpets play in 5, solo piano in 3, 4 and 7 simultaneous

groupings, cellos in 3, and violas play in sextuplets. Although these are surface rhythms,

they do carve different surface speeds out of a block of a sound mass that is still

temporally divided by the off-stage choir traveling at their own speed.260

All of these

techniques come to fruition in Ives’s Universe Symphony, but before 1915, he was

already thinking polytemporally about rhythm in his music. Putnam’s Camp, from Three

Places in New England, is famous for its polymetric/polytempic section, at mm. 74-82,261

where there is a 4:3 ratio, but again, at what point does the polymeter disambiguate from

the polytempo? Since the main orchestra is indicated at 100 beats per minute, and the

brass band is at 133 beats per minute, we can see that this is a short snippet of an example

259

Kramer, 1988. p. 166. 260

Ives, 1957, 2nd Orchestral Set. 261

Ives, 1935, Three Places in New England.

100

of polytempo. These earlier compositions of Ives’s, nevertheless, all lead to the Universe

Symphony, where true polytempo, implicit and explicit, reigns supreme.

Henry Cowell based a substantial amount of his theory of rhythm on the works of

Ives and he discussed it in his book New Musical Resources, which will be addressed in

chapter four. However, Cowell, via Ives, had influenced the polyrhythmic/metric/tempic

Conlon Nancarrow to such an extreme that Nancarrow literally took Cowell’s suggestion

to write for player piano, in Mexico.262

For Conlon Nancarrow (1912-1997), the correspondence of pitch interval to cross

rhythm was literally composed out in the piano rolls, where 3:2, a perfect fifth, became

three against two in rhythm.263

Ironically, even though Cowell had abandoned his

rhythmic ideas early on, his Quartet Romantic wound up being performed for the first

time in 1978 by use of head phones and individual click tracks,264

which shows that not

all ideas need be discarded: some are ahead of their time. In terms of lineage, it is clear

that there is a contiguous descent from Ives, to Cowell, to Nancarrow.

Nancarrow, like Cowell, took the hypothetical fundamental “C” tempo from

which to begin manipulations. If tempo 1 is at 120 BPM, and there is a 4:7 ratio, then the

other remaining tempo will be 210 BPM. Nancarrow began at the beginning of the

overtone series and worked his way up, from simple ratios to 12:15:20, a minor triad, for

example, G-B-E, in first inversion, just as Cowell had prescribed in his New Musical

Resources. Study No.s 32 and 36 are in the ratios 5:6:7:8, and 17:18:19:20, respectively,

spelling out E-G-Bb-C, (if C = fundamental) and Db-D-D#-E. Study No. 33 is about the

tritone, at 2 root 2, or 600 cents, in 12TET. The nearest simple ratio would be 7:5,

262

Gann, 1995. p. 1. 263

Ibid., p. 5. 264

Ibid.

101

yielding 582 cents.265

Nancarrow ultimately raised the bar to include irrational numbers

as ratios, expressed as tempo relationships, as measured out meticulously over a long

piano roll—a physical manifestation or representation of time itself. Nancarrow’s

irrational transcendental numbers recall Ives in a way, due to Ives’s interest in the

Transcendental New England writers who were also concerned with boundlessness.266

Study No. 1 begins on a common downbeat, with just two tempi: 120 and 210, in

a 4:7 relationship. There are both polymeter, different simultaneous use of time

signatures, and polytempo, two different simultaneous tempos, present, but rhythmic

groupings within the meters are additive. Nancarrow is both additive and divisive,267

rather than choosing just one approach. It is as though we merged Messiaen with

Schoenberg, since the former was purely additive, while the latter was purely divisive in

his rhythmic structures. In a formula, one could say thusly: additive rhythms are n+1,

n+2, n+3, etc., while divisive rhythms are Whole-Note/n+x.268

Study No. 5, for example, intersects both as 35 integrates both meters of 5/16 and

7/16, and isorhythmic patterns 14, 7, 14, 21, 7, 14 mapped to 15, 5, 10, 5, 10, 20, where

the 16th

note grid acts as a unifier. Nancarrow curbed Cowell’s weakness of periodicity

by inventing a-periodic rhythms via additive processes. Nancarrow ultimately eliminated

the barline and used pure tempo markings, allowing each part of the composition to

“come alive” and become purely polytempic. Study No. 24 reveals “hypermeasures” of

great extended lengths, also indicating tempo as a primary structure of organization.

265

Ibid., p.7. 266

Loc. Cit. 267

By divisive, this author contends that the whole note is divided into smaller equal segments, in

contradistinction to the additive rhythmic process, where rhythms are added, resulting in asymmetrical

structures. 268

Ibid.

102

Nancarrow’s open-mindedness allowed him to procure from other seminal influences in

order to achieve these polytempic features of his music. Study No. 24 also uses a tempo

system related to the harmonic series, of the type that Johnston used.269

Kyle Gann asks whether or not Nancarrow solved the Messiaen/Schoenberg

dilemma between additive and divided rhythms. The answer—yes—is plagued by

problems concerning human performability, to which Nancarrow seemed impervious,

and left alone for the player piano. Today’s version of the player piano is the computer

sequencer, except that there exists no lengthy piano roll on which to draw enormous

physical tempo relationships—relationships that in Nancarrow’s language eventually

involved transcendental numbers, which are completely counter intuitive. Also, as human

performance involves many idiosyncrasies that a mechanical object cannot duplicate,

Nancarrow circumvented this problem by introducing “jazz feel” and broken triplets into

his piano rolls.270

Nancarrow never intended for humans to play his music. The

acceleration studies, in which each voice changes tempo independently, are proof of that.

269

Ibid., p. 8. 270

Ibid., p. 9.

103

Ex 3.1 Study No. 9, Nancarrow’s first true polytempic work with explicit tempo

indications.271

271

Nancarrow, 1985. Studies No. 4, 5, 9, 10, 11, 12, 15, 16, 17, and 18 for Player Piano, Vol. 6.

104

Ex. 3.2 Page 2 of Study No. 11, Isorhythmic canon with 8 repetitions of a 15-note

ostinato.

105

Ex. 3.3 Page 11 of Study No. 11, Isorhythmic canon on a 120 note melody showing

clearly the different qualities of differently moving tempos and polymeter.

106

Ex. 3.4 Study No. 17, for player piano, Canon 12:15:20.

107

Ex. 3.5 Study No. 16, canon in 4, 5, 6, and 7 additive rhythms.

Examples 3.1 through 3.5 show Studies 9, 11, 16, and 17. Study No. 9, an ostinato

canon, can be considered Nancarrow’s first true polytempic work with rhythmic ratios of

108

5:4:3 in an eighth note setting.272

Study 9 is a clash of 3 ostinati, one per ratio part. Study

No. 11 is composed of a 120-note long melody which is isomorphic, beginning in C and

becoming more atonal. The 120 notes comprise eight repetitions of a fifteen-note

isorhythm which itself is divided into three segments of twenty 8th notes each: 5 5 6 4 5 5

3 4 3 5 4 3 3 3 2.273

Study No. 16, another isorhythmic canon, is based on four sets of

four-note rhythmic schemes: 3+4+5+4, 4+5+6+5, 5+6+7+6, and 6+7+8+7. Nancarrow

uses these rhythms in 2 cyclic blocks with a tempo modulation from 84 BPM to 140

BPM, ending in both played simultaneously.274

Study No. 17 is a canon in three tempi

with three different canonic subjects in a ratio of 12:15:20, where each subject is a 336

beat total of the same isorhythmic patterns of the type seen in Study 16.

In general, the isorhythmic and ostinato studies feature polytempo implicitly,

while the isorhythmic studies also have a great deal of polymeter. Nancarrow made

tempo thematic, rather than his pitch materials, of which Boulez had complained due to

their “banal” Blues and Boogie-Woogie American cultural associations.275

History also provides examples of this area of rhythm in the mensuration canons

of Josquin de Pres, at ratios of 2:1, during the Ars Nova and Renaissance. Mensuration

canons also had prolations of different lengths and proportions as well, such as the 3:2

hemiola in the Agnus Dei canon in Missa L’homme arme.276

Ockeghem’s Missa

prolationum employed all four of Philippe de Vitry’s prolations simultaneously, allowing

for four different tempos of each rhythmic structure in canon, and covering progressively

272

Nancarrow, Studies No. 4, 5, 9, 10, 11, 12, 15, 16, 17, and 18 for Player Piano, Vol. 6, 1985. 273

Gann, p. 98. 274

Ibid., p.117. 275

Ibid. p. 10. 276

Ibid., pp. 111-114.

109

the unison to the octave in imitation.277

In fact, multiple time signatures had occurred as

early as the eighteenth century. Fux’s Concentus musico-insturmentalis has a movement

with a simultaneous Italian Air in 6/8 and a French Air in stile antico, in cut time. Also,

Monteverdi madrigals had some conflicting time signatures as well, in an edition by

Leichtentritt.278

Nancarrow, in determining length, used the Lowest Common Multiple principle,

where for example, a canon in the ratios 16:20:24:28 = 1680, as in the number of eighth

notes.279

For Nancarrow, the logic of his music centers around the theory of convergence

points, where a major chord is used in a quasi cadential fashion for relief from the

polytempic dissonance. The whole point concerning Nancarrow’s irrational

transcendental numbers such as pi and e in his tempo relationships was to allow non-

convergence, where two or more points of tempo never meet for a resolution. Gann

believes that Nancarrow’s transcendental pieces are more conceptual, rather than literal,

since only approximations can suffice for pi, or e. For instance, if e/pi is meted out into

real numbers, we get 2.717: 3.141, as a crude approximation. As the resultant is equal to

.865, we could also take a fraction at 13/15, which is equal to .866.280

Finally, Nancarrow’s Study No. 37, a “chromatic scale” of tempos, concerns a 12

voice chromatic canon of 12 contiguous pitches,281

straight from the ideas of Cowell, and

serving as an example for Ben Johnston. This work also has no convergence points; if

one were to graph the tempos one would see a curve, in an arithmetic series, of each of

the entrance points of each voice in seconds: 10, 9, 9, 7, 6, 5, 4, 3+33/56, 2 + 33/56, 2 +

277

New Grove, Vol. 18, p. 318. 278

New Grove, Vol. 18. p. 153. 279

Ibid., p.114. 280

Ibid. 281

Nancarrow, 1982. Study No. 37 for Player Piano.

110

4/7, and 3 + 1/21 seconds apart. The tempi are as follows in BPM: 150, 160 5/4, 168 ¾,

180, 167 ½, 200, 210, 225, 240, 250, 262 ½, and 281 ¼.282

The antiphonal composer Henry Brant (1913-2008) composed Verticals

Ascending (1967),283

in which there are two fundamental tempos for two groups of

musicians. Group I is in 4/4 time at 94-100 BPM, while Group II is in ¾ time at 72-80

BPM. According to Brant, antiphony was a solution to polytempo by dispersing

musicians around a hall, a technique Xenakis borrowed, where “space is an essential

aspect of music composition.”284

Additionally, Brant noted that any difficulty in

combining contrasting textures in music can be aided by the use of antiphony. The

deterioration of rhythmic coordination, as separation, would ultimately enhance both

polyphony and contrapuntal clarity. Antiphonal techniques also permit the simultaneity

of contrasting meters and tempos while being easily controlled by assistant conductors.285

Brant has stated that the spatial-contrapuntal-polytemporal principles so brilliantly

exemplified in “Ives’s Unanswered Question are the basis for the more complicated

spatial superimpositions present in all my own recent large scale works.”286

Karlheinz Stockhausen, (1928-2007), composed Gruppen in 1958 for three

orchestras with three conductors, similar to his Carré, for four antiphonal orchestras.

Gruppen begins uniformly in 6/4 at 120 BPM, but at p. 7, there is a tempo indication of

113.5, which is an indicator concerning rational tempo constructions based on the

overtone series, as in Cowell’s New Musical Resources. Thorough examination of the

tempi indicates that there is a possibility of this technique, even though there are more

282

Ibid., p. 103-108. 283

Brant, 1969. 284

Schwartz, 1998, p. 221-242. 285

Ibid., p. 234. 286

Ibid., p. 236.

111

than 12 tempi written in the score. Stockhausen was aware of the overtone series with

respect to his other works, for example Stimmung, which is based entirely on overtone

principles.

The first large scale tempo breakdown occurs at score number 28, where there is

also a metrical split, into 5/4 and 2/4 at tempos 113.5 against 90.287

The texture is

polyphonic and multi-stratified as three orchestras are now involved with the polytempo

structure, pitting the third orchestra at 90 BPM against the first two, at 113.5 BPM.

Although the tempos are serialized, there is a possibility they are related to overtone

series procedures, since a fractional tempo indicates the product of two multiplied

numbers.

Example numbers 3.7-3.9 show rehearsal numbers 12, 36, and 51, where different

configurations of tempos are grouped together between the three orchestras, where no

singular orchestra dominates, but all compete for the listeners attention by their tempo

fluctuations. Stockhausen is very conscious with respect to texture not to overload the

listener with too much information. There are many cases where only one orchestra plays

at a time vis-á-vis polytempic changes.

287

Stockhausen, 1958. Gruppen.

112

Ex. 3.7 Stockhausen--Gruppen, at score marking 12, where the tempo splits into 90 and

67 BPM.

113

Ex. 3.8 Stockhausen--Gruppen, at marking 36, showing three tempi for each of the three

orchestras.

114

Ex. 3.9 Stockhausen--Gruppen, marking 51, showing continued polytempo, as

structurally significant.

115

Luciano Berio’s Tempi Concertati, (1962), is also antiphonal, and in score pages

47-54 there is an ametrical section in open form, where all individual instruments have

notated and visual starting and stopping points.288

This piece, however, is conductorless,

and is comprised of 4 groups, plus solo flute, where the groups are mixed ensembles. The

“polytempo” is more or less asynchronous as all performers enter an individual tempo

space, without indications or markings.289

Does this mean the work is not polytempo?

No, because in this case the listener qualifies the textural disparity by noticing the lack of

uniform forward motion. Whether or not the tempos are different by way of accident or

design is irrelevant.

Girolamo Arrigo’s Fluxus (1961) is explicitly polytempic with eighth-note based

meters of 5/8, 3/8, and 2/4 (4/8), with the following tempi: 190, 152, and 114. Group one

consists of flute, trumpet, and harp; group two has 2 clarinets and a bassoon, and group

three consists of viola, cello and bass. The section is short and is only found at the

beginning and the ending of the work.290

Brian Ferneyhough’s Agnus Dei (1969) splits the chorus into three divisions

where each division has its own tempo: tempo I is at 56 BPM, tempo II, at 60, and tempo

III, at 50 BPM. The formal scheme is open form where the divisions are notated in boxes,

from which the vocalists refer for performance. The tempi converge at score mark 10, at

60 BPM.291

288

Berio, 1962, Tempi Concertati. pp. 47-54. 289

Ibid. 290

Arrigo, 1961. Fluxus. 291

Ferneyhough, 1969. Agnes Dei.

116

Magnus Lindberg’s Action-Situation-Signification (1982), for mixed quartet,

features polytempo during the section called the Sea, where at the beginning of the

section, the piano is in 6/4 meter, at 59 BPM, the bass clarinet is in 5/8, at 56.2 BPM, and

the cello is in 6/8, at 52.8 BPM. All four instruments are in four different tempi and

meter, by the 6 minute mark: piano at 89.9, bass clarinet at 71.5, cello at 88.3, and

percussion at 60.4 BPM.292

Each part of the quartet seems to swell and surge, just like a

body of water, showing Lindberg’s talent for auditory metaphors. The meter changes

within the individual parts reveal that each instrument’s tempo and “life” is fully

independent, yet connected.

292

Lindberg, 1982. Action-Situation-Signification.

117

Ex. 3.10 Lindberg’s polytempic section, the Sea, in Action-Situation-Signification.

118

Ex. 3.11 Polytempo in Lindberg’s music.

119

Elliott Carter (born 1908) is more known for metric modulation than for

polytempo. Eve Poudrier, however, has shown that Carter uses simultaneous pulse stream

speeds, if not tempos, subdivided from a tempo grid, in two or more parts, in much of his

music.293

Carter calls these different strata “character patterns,” where metronomic

speeds, rhythmic groupings, and tuplets are used to dramatize the musical personalities of

instruments and instrumental groups.294

Poudrier discovered tempo ratios of 23:21:18 in Carter’s piece 90+, at deeper

structural levels. If the sixteenth note is taken as a grid, there are three types of rhythmic

patterns that form a chord pulse, the 90+ pulse, and the “rogue” pulse, that when all put

together create polytempo, where the explicit tempo of the 90+ pulse is staccato and

stands out.295

In short, there are three groupings of rhythms: 16 eighth note triplets, chord

pulses at 17 sixteenth note intervals, and staccato quintuplets at 23 sixteenth note

intervals. When all of these are factored into 96 BPM X 60 minutes and divided by these

quotients, 18, 23, and 21 BPM, respectively, result.296

Carter makes each “character” perceptible to the ear by way of his personal

vocabulary of intervals and tonal materials, such as the “all-interval” sets, (0137) and

(0146). Underlying logic and continuity leads Carter to make deep structural and global

polyrhythms that can extend up to ratios such as 75:56, for solo guitar in Changes,

(1983)297

Carter’s compositional method is to use graph paper to construct the

polyrhythm (like Nancarrow and his piano rolls) before composition. The constructed

293

Poudrier, 2009, p. 205. 294

Ibid. 295

Ibid., 206. 296

Ibid. 297

Ibid., p. 207.

120

time line engenders organicism, with its own inherent patterns of tension and release and

reduced textures, the clarity of which reveals Carter’s neo-classical ideals. Carter

maintains that the overlap and interchange versus divergence is the dialectic of materials

that create organicism in music. The interaction between foreground polyrhythms and

pulse streams, hidden structurally, create tension for the listener, who wants something to

grasp in order to hear it unfold.298

Polymeter and polytempo are most noticeable when

other musical elements, such as register, articulations, dynamics, intervals, and tunings

are used to reinforce the independence of layers of rhythmic strata. Due to this textural

complexity, it could be highly beneficial to use different tunings and micro-intervallic

systems to help create distinct identities between different polytempic parts, all

competing for the listener’s attention.

In Carter’s other works, such as String Quartet No. 2 (1959), metric modulation

appears while the first violin plays quintuplets against the second violin’s triplets over the

general pulse of eighth note values. There is a compound ratio of 5:3/2, over held tones at

variable durations.299

String Quartet No. 3 (1971) features a split quartet into two meters

with two tempos: 6/4 at 105 BPM, and 12/8 at 70 BPM, essentially a hemiolic 3:2, and a

favorite ratio of Ives. Since the quartet is split into two duos—second violin and viola

versus first violin and cello—rhythmic tension is created within each duet, where the first

punctuates at 16th

note rhythms against the second duet’s 5:6 ratio. The structural levels

of rhythmic complexity are layered, as the stratification articulates subdivisions of

rhythms within the competing tempi of 105 BPM and 70 BPM. The surface polyrhythms

298

Ibid., p. 209. 299

Carter, 1961. String Quartet No. 2.

121

invert, like rhythmic counterpoint, as meters change via metric modulation.300

(Please see

below for Ex. 3.12 and 3.13)

300

Carter, 1971.String Quartet No. 3.

122

Ex. 3.12 Carter’s polyrhythmic articulated surface in two tempos.

123

Ex. 3.13 Polytempo, metric modulation, and polyrhythm, multilayered rhythmic strata.

124

Carter’s String Quartet No. 5 (1993) is in the ratio 10:7:8:9, with 12 movements

and 5 interludes; however, the surface rhythms vary so much that there is no detectable

underlying pattern. 327 measures in length, the piece has considerable metric

modulations; the thickest part of the work where the textural density is at its

polyrhythmic maximum occurs between mm. 149 and 160. Carter’s work with regard to

his character continuities in their individual rates of speed, whether considered

polytactus, or not, still fit under the auspices of polytempo.

Lejaren Hiller’s String Quartet No. 5 (1962) displays polytempo in its variation 12,

where the first violin, in 4/4 time, is played at 120 BPM, second violin, in 11/8 meter at

110 BPM, viola, in 5/8 meter at 50 BPM, and cello in ¾ meter at 90 BPM.301

The

movement is nonisochronous and has no convergence points. In addition to its polytempo

section, variation 12 is also in quartertones, making it a “sleeper” in the polytempic

microtonality category. In addition to Ben Johnston, the University of Illinois at Urbana

Champaign has produced some adventurous composers with respect to this inquiry.

301

Hiller, 1968. String Quartet No. 5.

125

3.1 Chapter Summary

This chapter intended to focus on twentieth century works, but the Renaissance

crept in anyway through the mentioning of Josquin and Ockeghem. Ockeghem framed

the canons of his Missa Prolationum in four different mensural signatures to create as

much attention to his voices as possible, and for the art of rhythmic differentiation by

organizing canonic pairs that included renderings of the same melody in both duple and

triple meter simultaneously. In some respects the composers of the late Middle ages and

early Renaissance were more adventurous than today’s composers. It was a period of

experimentation before any rigorous standard for meter had been set by a “common

practice.” This author feels that the music of the twentieth and twenty-first centuries

demonstrates the same predicament, except for existing on the other side of the common

practice period. Periods like this that are open are very similar to the Middle ages and

early Renaissance. New perspectives in art arose, as in the 4-dimensional art of the

Cubists. Today’s computer art programs allow for myriad manipulations that go beyond

the Cubists, into an other-dimensional territory. These developments have also affected

music.

Music tempos began to split apart and run in concurrent streams in the early

twentieth century. The resurgence of early music practices, such as isorhythms, and

antiphony, allowed composers to experiment with musical time. The classical period and

its static rhythms from the eighteenth century proved too constricting.

Polytempic music had surfaced in the early part of the century, but went

underground and did not resurface again until Carter, Nancarrow and Stockhausen had

126

brought it into their music. The polytempo segments of composers like Lindberg, and

Ferneyhough tell us that this aspect of musical rhythm is still in its infancy, and needs to

be explored further.

127

CHAPTER 4:

PITCH + RHYTHM:

LITERATURE AND COMPOSITIONS SUGGESTING RELATIONSHIPS

BETWEEN MICROTONAL PITCH AND RHYTHM, IN TERMS OF TEMPO,

LEADING TO THE UNION OF POLYMICROTONALITY AND POLYTEMPO;

Henry Cowell, Ivan Wyschnegradsky and Ben Johnston

4.1 Historical Methods and Thought Concerning Connections of Pitch and Rhythm by the

Overtone Series

Moritz Hauptmann, 1792-1868, in his Die Natur der Harmonik und Metrik, is the

only music theorist prior to the twentieth century who tried to explain all harmonic and

metric phenomena on the basis of a sole universal principle, called “One Law.”302

Based

on Hegelian dialectics, thesis, antithesis, and synthesis, Hauptmann saw this as the

generator of all music.

Hauptmann was the first theorist to raise the level of rhythmic theory to the same

level as harmony. Hauptmann proclaimed that temporal and tonal phenomena came to be

by the same dialectical laws. Since harmony involved chords and tones of resolution, two

classifications emerged: those in “determinate” locations, such as dominant chords,

leading tones, and suspensions, and those in “indeterminate” locations, such as resolved

triads.303

Hauptmann established the concept of “Unity,” as expressed in the octave and 2/4

meter, where a fifth and triple meter represent “Opposition;” his ideas seem to come from

302

Caplin, 1981, Theories of Harmonic-Metric Relationships from Rameau to Riemann.p. 235 303

Ibid, p. 238.

128

the discovery of the overtone series, occuring between the time of Descartes and Rameau,

in terms of its application to musical theory.304

The Unity, a 2/4 meter, is equated with the octave, and the ¾ meter is equated

with the perfect fifth, thus revealing the direct influence of the overtone series.305

The

major third, however, is not included Hauptmann’s Hegelian dialectic since a 5/4 meter

was too abstruse for the time, even though they did know about the fifth overtone and its

relationship as the Just major third. Hauptmann’s system of positive and negative meters

based on harmonic logic is a prescient and early indicator for the future of music theory.

Riemann had initially carried on Hauptmann’s work equating rhythm to pitch, but

Riemann abandoned the primacy of rhythm and relegated it to an inferior and dependent

status, even though he had established even further the Dialectical process of moving

away from tonic as antithesis, and moving toward tonic, as thesis. 306

Meter, rhythm, and tempo were second class as a permanent basis until Leonard

Meyer broached the matter by way of prosody, the study of meter in poetry.307

Further methodologies in the inclusion and connection of rhythm and tempo into

microtonal works rest on Ivan Wyschnegradsky, Ben Johnston, and Iannis Xenakis,

whom will follow.

304

Farrar, 1956. The harmonic series, where the first 5 overtones are considered the foundation of

Occidental Harmony: the fundamental, the fifth, the fourth, and the third. 305

Caplin, p. 239. 306

Ibid, p. 289 307

Meyer, G. C. (1960). The Rhythmic Structure of Music. Chicago: The University of Chicago Press.

129

4.2 MODERN THOUGHTS CONNECTING PITCH AND RHYTHM

Jeff Pressing has formulated a theory of pitch and rhythm that bind together by

isomorphic structures. If the number 2 equals a whole tone and 1 equals a semitone, this

pattern then is equivalent to our major scale. If the number 2 is set to a quarter note and

the number 1 is set to an eighth note, then this structure forms the bell pattern of Ewe

drumming music of Ghana, in west Africa.308

Although Pressing’s inquiry concerns the cognitive phenomenon regarding the

simultaneity of this mathematical structure in two different cultures, there is a method for

combining pitch and rhythm within isomorphic techniques. Gestalt psychology,

according to K. Koffka’s Principles of Gestalt Psychology, indicates that people tend to

see both a figure and its complement, such as in visual ink blot Rohrschach tests,

suggesting that perhaps the same condition applies to sound and rhythmic patterns.309

The

issue is about negative space and its presence in human perception. Pressing also points

out that Schoenberg used the idea of complementation in his ideas of his 12-tone music

and combinatoriality. Similarly, this author argues for the possibility of negative space

inherent between the semitones at 100 cents, such that at least 100 further divisions are

possible in that negative space.

Regarding Pressing’s inquiry, this author would like to propose an answer

regarding the nature of the mathematical structures inherent in world musics. The reason

for this paradigm behind both the African and European mathematical structures of pitch

and rhythm is, perhaps, the Fibonacci series. There is a 2-group separated from a 3-group.

In other words, there is this mathematical grouping: 22 1 222 1. Each group is a

308

Pressing, 1983. pp. 38-39. 309

Ibid., p. 42.

130

Fibonacci number. If the series were to continue, it would possibly be 5, then 8, and then

13. This series seems to be embedded in human consciousness and surfaces around the

world in various musical structures, regardless of space and time.

Therefore, between the senarius, or even the septenarius, or the first six or seven

overtones for pitch, and the lower level Fibonacci series numbers 2 and 3, all human

music might be contained, structured and repeated, from culture to culture. This therefore

can work as a singular method for combining pitch and rhythm: mathematical

isomorphism.

Henry Cowell (1897-1965) is among the few who have had rhythmic insight into

the harmonic series allowing him to logically extract pure rhythmic structures from the

overtone ratios themselves. In doing this, the arithmetic nature of the overtone series

becomes a model of subdividing the whole note into equal durations, just as the

fundamental is multiplied by each succeeding partial. This is an inverse relationship: as

pitch is multiplied, rhythm divides the whole note. Cowell, not a microtonalist, did not

see the far-reaching effects of his work with respect to polymicrotonality, although he did

see the connections with standard tuning by 12TET. Ex. 4.1 shows Cowell’s thought

behind matching pitch to rhythm: if C is the fundamental, then each

131

Ex. 4.1 From Henry Cowell’s New Musical Resources.310

succeeding overtone will have a concomitant rhythmic value: the octave at 2:1 will be a

C higher, and 2 beats in rhythmic value; a perfect fifth, at 3:2, will have three beats and

its pitch will equal G, and so on, theoretically into infinity.311

The problem with this system is that it begins to automatically write itself

algorithmically. This is more or less the same problem the integral serialists faced with

respect to systematizing all parameters of music: akin to connecting the dots, the system

becomes self-generative. Since all notes are determined by their overtone series position

and number, all rhythms become pre-assigned, and the issue of choice becomes obsolete,

as it did with the serialists, since all compositional parameters are predetermined, like

Calvinism.312

Facing this problem, Cowell cheated his own system by not finishing the

algorithmically assigned length of each of the series positions durations. Cowell, also,

used free will and choice in determining pitch positions for a grouping of quintuplets,

which would correspond to the fifth overtone, (or the Major third, E above C), for

example, where five E’s would have to be grouped in accordance with this system. One

could say that Cowell compromised, but the composer would say happily that choice

saved the day.313

For further clarification, please see example 4.3.

In an alternative attempt, Cowell’s assignment of overtone position to meter also

produced failed results, since his methods of fractional meters resulting from his peculiar

310

Cowell, 1969, New Musical Resources, p. 47. 311

Ibid. 312

Nicholls, 1990, p. 148. 313

Ibid, p. 149.

132

understanding of the overtone ratios led to incomprehensible metric problems, such as a

meter of 5 and ¾, or 3 and 3/8 beats in length. Again, I stress, for his day, and this was

1920, Cowell did not have the resources of musicians to overcome these metrical

problems, that would eventually become playable later in the century. Cowell, however,

did succeed in his ideas for tempo, where tempi based on overtone series positions do

make a good deal of sense. Ben Johnston, a student of Harry Partch, who in turn was a

student of Henry Cowell, made use of fractional tempi based solely on the overtone

series, or Just ratio numbers. Nevertheless, after the Rhythm Harmony Quartets, Fabric,

and Rhythmicana, Cowell abandoned this system completely and eschewed any further

rhythmic complexity.314

314

Ibid, p. 150.

133

Ex. 4.2 Cowell’s ratio to tempo based on the 12 chromatic pitches.315

In example 4.2, if we set C to 60 beats per minute, and use it as the foundation for

the tempo structure, we can derive the series indicated above. (For example, the “semi-

tone” at 15/14 X 60 = 64 2/7, etc.)

315

Ibid. p. 107.

134

316

Ex. 4.3 Beginning example of rhythmic disposition of overtones, Quartet Romantic.

316

Cowell, 1964, Quartet Romantic, and Quartet Euphometric.

135

Ex. 4.4 Cowell’s Quartet Euphometric.

136

Cowell’s Quartet Romantic for string quartet, and Quartet Euphometric, for

mixed quartet are both pure examples of Cowell’s rhythmic overtone positioning system.

From examples 4.3 and 4.4, it is easy to see that Cowell’s ideas, though revolutionary for

his time, were also unperformable due to the awkwardness of his system. Quartet

Romantic, by rhythmical duration, and Euphometric, by metrical process, were also

powerfully influential. Both these quartets influenced most of Conlon Nancarrow’s works

for player piano. Ironically, it is the awkward (considered awkward by musicians in

Cowells day, who were unwilling to play these rhythms) mixed fractions of Cowell that

piqued Nancarrow’s curiosity. In addition to influencing Nancarrow, Cowell’s unique

rhythmic geometric noteheads and non-tuplet third, fifth, sixth, and seventh notes

(divisions of the whole note) also apply to time signatures where a measure in 4/6

appears in Quartet Euphometric, influencing directly Brian Ferneyhough’s use of non-

standard time signatures.317

Instead of ratios, Cowell preferred “vibration cycles,” which are multiplicands of

a simple low frequency pitch, such as 16 cps, Cowell’s foundational vibration of choice.

Cowell uses these multiplied values rather than explicit ratios from the overtone series.318

Another popular idea Cowell helped propogate was the notion of undertones, which have

since turned out to be am acoustical falsehood, even if they are mathematically and

theoretically valid. Unfortunately Harry Partch had adopted this idea as the basis of his

unique brand of tonality, called U-tonality, for undertones, an erroneously perpetuated

physcial phenomena that accompanies the overtone series, but human hearing cannot

detect them. Undertones, therefore, can only work in relation to overtones in a

317

Personal Correspondence with Brian Ferneyhough, 2002. 318

Cowell, New Musical Resources, 1969, pp.3-9.

137

mathematically practical way, as a rational reciprocal, or numerical inversion. But

Cowell’s largest misconception is his notion of secondary overtones, akin to secondary

dominants, where, for example, the third partial, or the perfect fifth, has overtones of its

own. This is also false, but has been a boon in the mixed fractional reductions of

Cowell’s rhythms from the overtone series.319

Aside from these errors, one cannot discount the importance and revolutionary

ideas that Henry Cowell, extending those of Charles Ives, helped elucidate. Time as

duration, and tempo as a rate of speed, and their relationships to the overtone series

cannot be underestimated. Tempo canons based on Cowell’s ideas are now in use, due to

the innovations of Nancarrow and Johnston, who both directly linked their tempo

structures to the overtone series and Just intonation.

Cowell also suggested rejecting the simpler ratios of the overtone series in favor

of those larger, higher in the series, that produce microtones, as in the classic quartertone,

at 33/32, or the eighth tone, at 68/67. Keep in mind that although Cowell was not a

microtonalist, he did advocate their use, even if he himself did not use them. In fact,

Cowell thought the next step up the series after Schoenberg’s chromaticism would

involve quartertone harmony, as acceptable to the human ear.320

Ultimately, Cowell’s

tight grip on his ideas produced a symbolic notion of division, in dividing the octave, and

in dividing the whole note, an approach that characterized the Second Viennese school,

as opposed to the additive approach of Messiaen and Bartók. Bartok is also known for

319

Undertones, or Riemann’s Dualism, was an accepted premise well into the twentieth century in German

music theory, as well as the rest of the continent. (Rehding, p. 17) The tones one hears are in actuality, sum

and difference tones which can be detected one octave below the sounding pitch, but these are not

undertones, since undertones are an acoustic impossibility. Karl von Schafthäutl, in Allgemeine

Musikalische Zeitung, 1878, declared that the laws of mechanics preclude the possibility of undertones,

thus disproving Riemann, and for that matter, Aristotle, who was the progenitor of the undertone/overtone

dualistic conept. 320

Cowell, 1969. New Musical Resources. p. 17.

138

equal divisions of the octave within the 12TET medium. Example 5.9 shows Cowell’s

Fabric for piano, from 1922, and demonstrates the use of his overtone series derived

rhythms and pitch materials. But the static nature of pitch repetition, if his system is

perfectly adhered to, is missing, in favor of his musical ear. In Fabric, Cowell is arguing

for equal and odd divisions of the whole note, much like modern day microtonalists argue

for equal and odd divisions of the octave. In Fabric, Cowell also explores his rhythmic

ratio theory with his amended noteheads, and he also makes the connection between

overtone ratios and tempo relationships, a model both Johnston and Nancarrow carried

out in their works. Cowell’s system was extensive and included whole note divisions up

to 15th

notes.

139

Ex. 4.5 Cowell’s Fabric, with geometric noteheads and their rhythmic meanings.321

321

Cowell, 1922. Fabric for Piano.

140

Cowell abandoned this system due to the difficulty in carrying out cross rhythms

and accounting for subdivisions of odd groupings, which would result in fractional

rhythms, an issue that today’s computer music composers can cope with easily. Cowell

wrote: “…these rhythms could be cut on a piano roll for player pianos,” to which

Nancarrow agreed.322

In addition to new meters, such as 2/3, or 3/5, based on third notes and fifth notes,

Cowell began the notion of “Tempo Scales.” Metric harmonies, and metric ratios, all

based on the quarter note, led Cowell to the formulation of a correspondence between

pitch and rhythm.323

“Meter and Time combinations,” otherwise known as polytempo,

could be achieved by the reconciliation of accents and meter of different values.

Tempo itself, an exact rate of speed relative to one minute as in BPM, can

therefore coexist just as chords, or pitches coexist, as tempo relationships. Cowell,

therefore invented the tempo scale by systematizing the relationships in terms of ratios

324

Ex. 4.6 Cowell’s metrical duration scales.

322

Cowell, 1960, p. 65. 323

Ibid. p. 67-70. 324

Ibid, p. 105.

141

from the overtone series. Large scale rhythmic structures, such as tempo, therefore work

congenially with this system, as evinced by Johnston in his string quartets.

In Ex. 4.6, Cowell is using the chromatic scale to adjust measure lengths. If C is

set to one measure of 2/4, then all the ratios represented by the chromatic interval become

a multiplier for the multiplicand 2/4 single measure. Also, the multiplier will be reflected

as the time signature as well.

As an aside, Lucytune.com is an example of a current website that also

exemplifies the relationship between tempo and pitch, even using extended Pythagorean

tuning to 21 places, like Ives used in his Universe Symphony.325

Ivan Wyschnegradsky, born in 1893 in St. Petersburg, Russia, a lawyer,

composer, and polymicrotonal pioneer, was similarly influenced by American composers,

such as Henry Cowell,326

as well as Messiaen. In one word, Ultrachromaticism,

Wyschnegradsky’s music also found its way into tying together a microtonal system with

a logical extension into rhythm, still under the influence of Henry Cowell and Ives.327

In Europe, Schoenberg’s liberation of twelve tones put the issue of “tonality”

back into the hands of tuning and temperament as the logical extension to what Wagner,

Strauss, and Lizst had started in the area of total chromaticism.328

In other words,

Schoenberg’s liberation of the twelve tones can be seen as the re-clothing of the common

practice tuning as the first legitimate 12 equally divided octave, from the perspective of a

325 Lucy, C. E. (2001). Lucy Tuning. Retrieved 2011, from Lucytune.com:

http://www.lucytune.com/midi_and_keyboard/tempo.html.

326

Barthelmes, 1995, p. 49. 327

Barthelmes, 1995. p. 105-06. 328

Ibid, p. 34.

142

microtonal framework. Nevertheless, in the twentieth century, pansonority and

metaphysics and their representation in sound found its legitimacy in not only

Wyschnegradsky, but Ives, Cowell, and others.

Regarding Ivan Wyschnegradsky’s contributions, both of which were in

polymicrotonality and rhythmic structures, I will divide his work into pitch in this

section, and rhythm on page 144.

Wyschnegradsky’s ultrachromatic system was essentially about 72-tone equal

temperament, divided into subcategories: third tones, quartertones, sixth tones, and eighth

tones, from a broad spectrum template of twelfth tones. Some may argue that the 72TET

palette subsumes the smaller contingents, and therefore denies the polymicrotonal intent.

This author disagrees with this stance for the following. Since quartertones, used in the

Arabic maqam and in Greek genera, constitute their own identity as a microtonal system,

I argue in favor of the viability of 24TET still remaining intact as a distinct and

identifiable system even with respect to a larger system behind it. Just as Reinhard

identifies both quartertones and eighth tones in Ives’s Universe Symphony, without

recourse to critics scolding him for the possible subsumption of quartertones by eighth

tones, then I have at least one ally in this regard. In any case two systems, 12TET and

24TET, are not mutually interchangeable. Yes, one can find the 12TET within the 24TET

system, but the converse is impossible. If a piece is written in both 12TET, obeying the

laws of the common practice, and quartertones, perhaps adding inflections and extensions

to the common practice chords and voice leading, then the compositional intent includes

both temperaments into one system. But if the quartertones, 24TET, are composed in a

different manner in contradistinction to the common practice behavior, then the tunings

143

seem to occupy two different sound fields simultaneously. This case can be argued with

respect to polytonality, as well, with the exception that as the temperament remains the

same, the intent is different and the outcome sounds differently. Additionally,

quartertones do not sound the same as eighth tones, which do not sound like twelfth

tones. If the composer chooses to exclude pitches that overlap, then the integrity of each

system remains. If one takes the alphabet and writes in Latin and in English, we still have

two different languages. How does subsumption, then, make any difference? This author

believes it is the intent of the composer.

Chant douloureux et etude, Op. 6 (1918) is Wyschnegradsky’s first

polymicrotonal work in five microtonal tunings: third tones, quartertones, sixth tones, and

eighth tones, plus a piano in standard 12tet. The other tunings were also performed by

piano, manufactured by August Forster pianos, in the Czech Republic.329

Wyschnegradsky had synesthesia, like Skryabyn, and felt a great comeraderie

with him as a fellow Russian.330

In addition to the metaphysics of the early twentieth

century, theosophy was also en vogue and much of Wyschnegradsky’s ideas were in

league with that school of thought, especially with regard to his color theory as related to

pitch, again, also in step with Skryabyn. Wyschnegradsky also made use of Cowell’s

cluster technique (from Ives), establishing the link between himself and Cowell.

Wyschnegradsky knew about Cowell’s book. In terms of systems, Wyschnegradsky was

adept at isolating microtonal systems one at a time, either using quartertones or third

tones, for example.331

329

Ibid, p. 35. 330

Ibid, p. 211. 331

Ibid, p. 37.

144

The philosophy of the system Wyschnegradsky used entailed terms like volume,

for microtonal density, and type, for the size of the microtonal step, with respect to his

polymicrotonal hierarchy, to which he would use color theory to mix pitches: “das

Volumen und die Dichte einer Klangaggregationen.”332

Dichte, or type, would involve

twelfth tones, sixth tones, or quartertones. The pansonority wheel (shown in Ex. 4.7)

depicts his 505 gamut intervals, stretching over 7 octaves in a resolution of twelfth tones.

332

Ibid, p. 38-39.

145

Ex. 4.7 Wyschnegradsky’s pansonoric wheel.333

333

Ibid. p. 500.

146

Ex. 4.8 Wyschnegradsky’s color assignments to pitch materials.334

334

Ibid. p. 501.

147

Ex. 4.9 Marie’s example of Wyschnegradsky’s microtonal accidentals.335

Ex. 4.8 and 4.9 shows the multitudes of microtonal symbols capable of cluttering

up a musical line. This is one reason for this author’s fattened staves (Section 5.3) as a

solution to monstrosities like the 1/12th

tone microtonal symbol. Marie notes that

335

Marie, 1976. L’homme musical, p. 49.

148

Wyschnegradsky also had plans to color code sets, or chords, audio-visually, by

inventions such as the crescent, for broad ultrachromatic clusters.

Ex. 4.10 Arc en ciel, Op. 37, polymicrotonal piano piece for 6 pianos, where clusters of

microchromatic chords engage in a process of cellular contraction and expansion.

149

Arc en ciel (Ex. 4.10), Wyschnegradsky’s polymicrotonal masterpiece, features 6

pianos tuned in twelfth tones, all utilizing a subgrouping, or Dichte, of the gamut.

Colored notation visually tells the performers what happens within the work: red for half

tones, orange for twelfth tones, yellow for sixth tones, green for quartertones, and blue

for third tones.336

As far as I know I have not been able to locate a recording for this

piece. The Zykeln, or cirlces, are analogous to the circular 12-tone system. In addition to

the above, Wyschnegradsky’s Klangmillieus are 5 and 7 equal divisions of octaves,

similar to pelog and slendro Gamelan pitches. He also built scales based on 2/3 tones, 66

cents, in modes of limited transposition—a reciprocity existed between Messiaen and

Wyschnegradsky.337

Also, there was a quartertone harmony manual, written as a

continuation of chromatic harmony, for which he is most known, rather than the

polymicrotonal works he actually produced. Nevertheless, even Marie has noted the

influence of the overtone series in the ideas of Wyschnegradsky, as upper partials were

rounded off to the nearest quarter, sixth, and twelfth tones. For Wyschnegradsky,

ultrachromaticism was based on acoustics and resonance.338

Marie notes that there was an

epic battle between just acoustic resonance, and intuitive ultrachromaticism, where

octave equivalence is important, but ultimately disposable.339

Ivan Wyschnegradsky is not only useful to this inquiry due to pitch alone; he also

worked on rhythmic ideas. There is his notion of a time continuum, wherein

Wyschnegradsky ties in rhythm to his ultrachromatic theory based on ideas of the

overtone positions, just like Cowell. One must induce this as Cowell’s influence, since

336

Ibid, p. 38-39. 337

Ibid, p. 43-48. 338

Marie, 1976. L’homme musical, p. 50. 339

Ibid.

150

the idea had only been addressed once before, by Cowell himself in his New Musical

Resources. Wyschnegradsky’s Klangkontinua, a sound continuum, of both pitch and

rhythm, can be equated quite easily to polytempic polymicrotonality, since there is an

element of infinite division, as in Aristoxenus, inherent in the idea. In a sense,

Wyschnegradsky is a type of amalgamation of both Henry Cowell and Olivier Messiaen.

Also, what lends credence to Wyschnegradsky as a polymicrotonalist is his stratification

of layers, rather than a holistic and interwoven texture.

As for a harmonic analysis of Arc en ciel, there is a Bartókian approach to the

waxing and waning of the texture and intervallic compression and expansion. A 5/12 tone

interval grows to a half step, and then from a third tone, down to a quartertone, all in tight

clusters of microchromatic harmony. (Please see Ex. 4.10)

“Merkmale des rhythmischen Ultrachromatismus” marks the area of

Wyschnegradsky’s thought that parallels metaphysical ideas of space in music that is

realized in his ultrachromatic harmony, vorstellungsraum.340

Wyshnegradsky was

concernced about the imprecise nature of musical directions for tempo and came upon a

type of metric modulation similar to Carter’s, in order to gain more control over time and

tempo within a composition. From a “time continuum,” Wyschnegradsky devised a

uniform system as equal divisions of the octave[s], he also thought of equal divisions of

the whole note, harkening back to Cowell. This would be a type of rhythmic

temperament.341

La loi de la Pansonorité is an example of polyrhythmic counterpoint in four

categories : I applying 3:2, II applying 2:3, III applying 6:7, and IV applying 7:6 to

340

Barthelmes, 1995. Raum und Klang: das musikalishce und theoretische schaffen Ivan Wyschnegradsy.

p. 65-69. 341

Ibid.

151

different ratios in a system of inversion and retrograde techniques.342

The system is quite

remarkable in that it is a system based on limits (please see examples 4.7 and 4.11), and

four contrapuntal techniques found in serialism: prime, inversion, retrograde, and

retrograde inversion. The system is dictated by the intervallic ratio in question, as it

appears alongside the rhythms of the rows. If the fraction is less than 1, it slows down. If

the fractional interval is greater than 1, it speeds up.

342

Ibid., pp. 86-87.

152

Ex. 4.11 Wyschnegradsky’s rhythmic system in four categories, P, I, R, and RI. 343

343

Ibid. p. 74.

153

Ex. 4.12 Wyschnegradsky’s rhythmically derived system from his pitch system.344

344

Ibid. p. 74.

154

Ex. 4.13 Marie’s admiration for Wyschnegradsky’s system of combining pitch and

rhythm as a solution to an age old problem.345

In Ex. 4.13, Wyschnegradsky has paired the functionality of the ratio of pitch to

the ratio of rhythmic value. The 5/4, for example, is shown as a 5 in the space of 4 tuplet,

345

Marie, p. 52.

155

The 5/4 ratio is the just major third at 386 cents. So the realm of microintervallic ratios

are now controlling rhythmic behavior.

Wyschnegradsky’s ratio-based rhythmic system (Ex. 4.12) involved natural

numbers up to 13 in a grid of 169 numbers represented as fractions. After removing

duplications, there remained 115 different values he applied as rates of duration and

overtone positions. For example, if Eb is the octave at 2/1, then the numbers 27-45-75-

125 correspond to C, A, F#, and D#/Eb, when broken down into their factors with respect

to the overtone series for Eb. Similarly, the number 125, 5*5*5, is three consecutive

major thirds away from Eb, which is, again, Eb, or D#; but when assigned as rhythm, the

prime factors of the numbers become rhythmic values, such as 27th

notes, as 3*9,

correspond to triplets, and 45, 5*9, corresponds to prime numbers five and three, and 125

becomes a complex of quintuplets.346

Wyschnegradsky’s system can be represented as a formula: a+1/b+1, where the

coefficients represent rhythmic values, such as an arithmetic series ½, 2/3, ¾, 4/5, etc.

When put into rows, each row can then represent divisions of the octave. (please see Ex.

4.14)

Row 1 in

whole tones

½ 2/3 ¾ 4/5 5/6

Row 2 in half

tones

1/3 2/4 3/5 4/6 5/7

Row 3 in third

tones

¼ 2/5 3/6 4/7 5/8

Row 4 in

quartertones

1/5 2/6 3/7 4/8 5/9

Table 4.14 Wyshnegradsky’s pitch to rhythm system based on the overtone series.

346

Ibid., p. 73-75.

156

The four categories can then be generalized as follows: category I, the “ground”

row, slower; category II, inverted row, a little faster; category III, retrograde, still faster;

and category IV, retrograde inversion, fastest. The influence of the Second Viennese is

obvious, as well as the influence of Cowell. Wyschnegradsky was unique in bridging the

most unique schools of both Europe and America. In addition to the above,

Wyschnegradsky chose “C” as the ground note to base his system on, again, just as

Cowell did, but it is probably a universally convenient starting point. Larger fractions in

Wyschnegradsky’s system are used for larger rhythmic constructions, not for

pitch/duration relationships. Although this system is fairly comprehensive, it is still

arbitrary, and it is also mostly unknown in America, as Ivan Wyschnegradsky’s name is

rarely mentioned in any graduate textbooks on music. I found out about Wyschnegradsky

in year 2000 while trying to inquire into a quartertone theory of harmony and finding his

book written on an extension of chromatic harmony in terms of quartertones.

Ben Johnston (born 1926), although not a dedicated polymicrotonalist, is an

extended just intoned, singlularly microtonal composer who uses up to 53 pitches per

octave. He has also engaged in the use of Cowell’s tempo scales, and to some degree,

polytempo. In a polytempic setting, including microtonality, Johnston comes very close

to the auditory experience of Ives himself.

In the area of tying pitch relationships to rhythm, Johnston brought the two

together elegantly in his Knocking Piece, 1978, a percussive piece derived from the just

157

intoned chords from his A Sea Dirge.347

Knocking Piece features a good deal of metric

modulation, like Carter, and tempo and just intonation are related by just ratios.348

Johnston’s metric modulations are also based on an elaborate system of tempo

realtionships via ratios from his just intonation scales. This is the keystone, or the master

stroke of Johnston’s system, which ties everything together: pitch and tempo are related

by tuning, essentially through the overtone/just rational system, pioneered by Henry

Cowell, from 1922.

Johnston does have an example of polymicrotonality: his Sonata for Microtonal

Piano, 1962-67.349

Johnston is serial in this piece, where there is a tuned piano spanning

7 octaves and encompassing 81 separate microtonal pitches, based on a 4:5:6 ratio of a

major triad.350

Similar to Hindemith, Johnston has a consonance continuum, of a very

large collection of intervals, due to an 81/octave scale. Additionally, there is an element

of polymicrotonality to this work, where 5-limit Just intonation and

347

Von Gunden, 1986. The Music of Ben Johnston. p. 140. 348

Gibbens, 1985, p. 1-12. 349

Ibid. 350

Ibid.

158

Ex. 4.15 Table showing Johnston’s ratios to tempos.351

extended Pythagorean tuning coexist.352

In this elaborate pitch scheme, even the

Pythagorean comma, at 24 cents, is utilized, written as Db- -, vs C#++.353

351

Ibid. p. 73. 352

Ibid, p. 27. 353

Ibid. p. 38.

159

Ex. 4.16 Ben Johnston’s tempo scale from Sonata for Microtonal Piano.354

Johnston, like Nancarrow, was influenced by both Ives and Cowell (via Harry

Partch), and used ratios from his tuning table to apply to note durations and eventually

tempi. The 2-dimensional lattice Johnston uses for Sonata for Microtonal Piano consists

of two arrays: one Pythagorean, based on the Major triad 4:5:6, represented as x, and y is

5 limit Just intonation. For future purposes, Johnston added a new dimension per newly

introduced prime number, with respect to Just tuning, for example, adding 7 would result

in a 3 dimensional lattice, 11, a four dimensional lattice, and so on. Thus, this particular

piece by Johnston is polymicrotonal; even if it is considered a Pythagorean gamut of 81

pitches, it is still relying on prime numbers 2, 3, and 5, meaning that there really are two

354

Ibid. p. 35.

160

systems at work: 5-limit just and Pythagorean tuning. Ex. 4.15 and 4.16 are both

Cowells’s ideas applied, by correlating tempo to intervallic ratio.

Ultimately, Johnston’s philosophy is about tuning purity, rather than

polymicrotonality. He maintains that scales depend upon cultural conditioning by

intervals of negligable amounts.355

Johnston’s 53 tone tuning is based on interlocking

triads from Fokker and Euler via Pythagorean extensions.356

Johnston’s lattices similarly

derive from Leonhard Euler’s theories.

Johnston credits Schoenberg by stating that the emancipation of dissonance

helped extend the range of consonance.357

In league with Euler, Johnston throws out

octave equivalences and replaces it with other areas of consonance, such as fifths and

thirds. Ultimately, Johnston is triadic, and tonal, even if hyper microtonal, even using an

interval 2 cents wide, known as the schisma. Within Johnston’s tonality, there are also

serial procedures, quotes, and Elliott Carter’s metric modulations, as seen in his Knocking

Piece. Like Ives, Johnston sees microtones as teasing the listener with a type of “out of

tuneness.”

For Johnston, laws of harmony and tempo come from ratios of overtones and

proportional tuning. Johnston’s ratio based proportional organization creates complete

unity between pitch organization and rhythmic structure by using a tuning scheme in

ratios and then applying those particular ratios as multipliers to a general base tempo,

355

Johnston, 2006. Maximum Clarity and other Writings on Music, p. 13 356

Euler Fokker genera are based on prime numbers. Euler genera are generated from the prime factors 3

and 5, whereas an Euler–Fokker genus can have factors of 7 or any higher prime number. The degree is the

number of intervals which generate a genus. However, not all genera of the same degree have the same

number of tones since [XXXYYY] may also be notated [XxY

y], "the degree is thus the sum of the

exponents," and the number of pitches is obtained adding one to each exponent and then multiplying those

((X+1)×(Y+1)=Z), according to HuygensFokker.org. 357

Johnston, Maximum Clarity, p. 34.

161

creating a tempo scale, through which he used metric modulation from tempo to tempo.

Again, the work of Cowell, Carter, and Nancarrow comes to mind.

Amazing Grace, String Quartet No. 4, is also polymicrotonal since it uses both

Pythagorean and Just tuning. Johnston uses limits of Just intonation to create a family of

tuning lattices, which reflect each piece’s tuning. Johnston has gone well beyond 7-limit

4.17 Johnston’s meticulous nature and uncompromising attitude towards tuning.358

to include 11-limit Just intonation, creating “gestalts,” or triadic gestalts for the

comparison and inclusion of new tunings.359

The symbol denoted for the seventh from the

358

This is a tuning legend leaflet in all of Johnston’s String Quartets.

162

above tuning legend from String Quartet No. 5 is borrowed from Adriaan Fokker’s

adoption of Tartini’s flat diesis symbol from geometry; Johnston uses it as a number 7

indicating the seventh harmonic.360

The thirteenth is self-explanatory.

According to Johnston, tonality is the organization of pitches by ratio.361

Tempo,

similarly is also the organization of rate of speed by ratio. Johnston also thinks that pitch

is tempo, or rather that tonality is merely a system of proportional tempi, where meter is

but a simplistic example. Therefore, a succession of pitches is a fluctuation of rapid

tempi, where the pitches themselves are hyper-rhythms, or tempi, beating so fast that they

become tones in themselves. It is basic acoustics, if we think about it, but we rarely do,

and it comes as a sudden shock when we realize that tempo equals pitch: polytempo

equals polymicrotonality. Polytempo is polymicrotonality. They are the same

phenomenon, but only at different relative speeds. Since pitch is rhythm, then does

matching them become a moot point, or a philosophical redundancy? It is akin to the

three states of matter: water, ice, vapor. All is water, yet we are trying to find a way to

relate one state to the next, as though we were ignorant of the fact that they are the same

thing.

William Sethares offers the perspective of tuning from a psychoacoustical point

of view. He maintains that pitch and rhythm may be related, but are perceived differently

by listeners. Pitch and rhythm may be the same phenomenon, but are seen initially

differently due to their disparate natures. Pitch and rhythm are seen as polar opposites in

the psyche in terms of human perception. In fact, they are polar opposites because rhythm

359

Ibid, p. 62-68. 360

Fokker D. A., 1949. Just Intonation. p.18. Also this is the Tartini sign for indicating the flat seventh

partial. 361

Johnston, 2006. Maximum Clarity. p. 99.

163

is pitch slowed down to 20 cycles per second, or less, which are perceived as beats, or

rhythm. 21 cycles or more fall within the domain of pitch, since the jnd (just noticeable

difference) of the periodicity is blurred. Sethares maintains that perception research and

psychology have proven that there is a limit to the discreteness of beats blurring into

pitch, and that is fundamentally based on human limitations. Similarly, humans cannot

hear beyond 20,000 cycles per second, as that is too far out of our ciliac nerve range in

our inner ears.362

According to Johnston, the translation of techniques of rhythmic organization

from one time scale to another is characteristic of contemporary composition.363

Tonality

and harmonic thinking imply microtonal pitch distinctions, and it is due to the

sumpremacy of the ear—Ives, Aristoxenus,Varèse, and Xenakis all agree. Ben Johnston,

for instance, is an example of a just intonation composer who translates his ratios of pitch

intervals into tempo relationships, but only in terms of tempo modulations, and not

polytempo. Just intonation is also not polymicrotonal, since it is a singular system. Just

can, however, be divided into subsets of competing limits, e.g. 5-limit vs. 7-limit, just as

just intonation lattices of Euler-Fokker elucidate. Nevertheless, these are only extensions

of one type of system, called just intonation.

Elsewhere in Ben Johnston’s oeuvre, relationships between pitch and tempo

abound. String Quartet No. 2 features a 3:2 rhythm congruent with the literal perfect

fifth, while the 5:4 Major third equates to a 5:4 rhythm, as well as tempos following the

same ratio patterns. In order to make logic this strict for each piece, Johnston spends a

great deal of time composing them: “I work very slowly, with much care and

362

Sethares. 1998. Tuning, Timbre, Spectrum, Scale. Forward, and p. 11. 363

Johnston, Maximum Clarity, p. 97-100.

164

computation.”364

Johnston also believes that “any interval is possible,”365

which increases

the likelihood of the potential for polymicrotonality as a new genre of music.

In String Quartet No. 5, 1979, Johnston uses the following ratios: 3:2, 4:3, 5:4,

and 6:4 both in intervallic and rhythmic acuity. There is polytempo, beginning on page

15, in the last measure and continuing for 6 measures with the following scheme: violin I,

at 160 BPM, in 4/4, violin II at 135 BPM, in 9/8, viola at 150 BPM, in 5/4, and the cello

at 120 BPM, in 4/4. All derive structurally from the following chord: tonic doubled, Just

major third, and a Pythagorean major second (Please see Ex. 4.18). Example 4.15 shows

explicit polytempo, both in written instructions and in time signatures. The tempos are a

result of multiplying the fundamental tempo, by the just ratio tunings, 160, 135, 150, and

120 beats per minute. This author’s written notes indicate the characteristic intervals of

the pitch content of the measure, 16/15, and 9/8, which translate to notes B and D.

364

Ibid. p. 199. 365

Von Gunden, 1986, p. 60.

165

4.18 Ben Johnston’s String Quartet No. 5 and polytempo.

166

CHAPTER 5:

THE PROGENITOR

CHARLES IVES’S UNIVERSE SYMPHONY AND ITS LEGACY: POLYTEMPIC

POLYMICROTONAL ART MUSIC

Larry Austin technically was the first person to put together Ives’s Universe

Symphony, from 1974 to 1993. Austin’s realization was premiered by conductor Gerhard

Samuel and the Cincinnati Philharmonica, combined with the Cincinnati Conservatory’s

Percussion Ensemble, in 1993.366

Austin maintains that the Universe Symphony was Ives’s largest and most

compelling and visionary work. That opinion seems to be held by many, including

Johnny Reinhard, who also realized Ives’s Universe Symphony a few years later.

Austin divided the Universe Symphony into four parts: the Life Pulse Prelude

orchestra, the Heavens orchestra (itself in four parts), Rock Formation orchestra, and the

Earth orchestra. All orchestras follow essentially one tempo, as indicated in Ex. 5.1.

366

Lambert, 1997. Ives Studies. p. 179.

167

Ex. 5.1 Larry Austin’s arrangement of Ives’s Universe Symphony.367

Although Austin does a fantastic job with respect to the LPP, or Life Pulse

Prelude, Ives’s “Basic Unit” percussion orchestra, with all of its ratios in the percussion,

Austin did not pick up on the extended Pythagorean tuning indicated in Ives’s score by

deliberate enharmonic notation, where D#, for example, does not equal Eb (please see

below). Nor does Austin incorporate eighth tones, also implied in Ives’s score (please see

below). Austin does incorporate a “perfectly tuned overtone machine”368

in his section B.

Austin’s section B also introduces 12 violins tuned in quartertones, a stretched octave

367

Ibid. p. 211. 368

Ibid. p. 206.

168

scale, “equal tempered” tunings (unspecified), and the just/overtone tuned machine.369

This author’s reason for preferring Reinhard’s realization, is that the arrangements of his

Heavens and Earth orchestras feature the various tunings much more clearly. Reinhard

also discusses in much more detail the nature of the polymicrotonality, and the specifics

in terms of cent values. Another reason I personally prefer Reinhard’s realization is the

polytempo nature of his particular realization, where three, rather than four, orchestras

are in three different tempos, as will be explained below.

For this author’s intents and purposes of this paper, it was more logical for me to

choose Reinhard’s version over Austin’s without any bias towards either gentleman’s

aesthetic, but only for helping me to establish this author’s aesthetic of combining both

polytempo with polymicrotonality, which is much more explicit in the Reinhard version.

5.1 Johnny Reinhard’s Universe Symphony Realization

Microtones are an Ives family tradition.370 George Ives, Charles’s father, is

effectively America’s first potential microtonalist. A naturalist, obsessed with bells and

natural sonorities and figuring out pitches and chords found in the real world. George

opened up Charles’s mind to allow him in becoming the great pioneer he was.371

George Ives gave Charles several ideas: small intervals on slide cornet, water

filled glasses for eighth tones (commas), an overtone piano, non-octave scales, and

makeshift monochords with weights, for tuning in just intonation—a contraption made

369

Ibid. 370

Reinhard, The Ives Universe: a symphonic odyssey, 2004, p. 99. 371

Burkholder, 1983. The Evolution of Charles Ives’s Music.

169

from a clothes press and violin strings.372 Ives later employed these experiments in his

Universe Symphony, the score of which includes a just intonation machine and a justly

tuned harp as well. “Why can’t the ear learn 100 other intervals if it wants to try?” was a

question Ives took seriously, in true trail-blazing fashion.373

Ives’s experimentation led to non-octave scales, Pythagorean tuning, and eighth

tone tuning, not to mention standard quartertones. Ives also possessed a keen interest in

inharmonic timbres including non-tuned percussion and metallic sounds such as anvils

and brake pads. These sounds represent his adopted family interest in representing the

natural world and its infinite continuum in music. Ives’s open-minded approach is again

summarized by his whimsical, yet vitriolic, personal expressions: “Why tonality should

be thrown out? I can’t see. Why tonality should be kept? I can’t see. It depends as clothes

upon the thermometer—on what one is trying to do.” 374

Ives agreed with John Cornelius Griggs: “The tempered system is not conducive

to correct and vigorous musical thinking, as has been the violin and voice training of

earlier centuries.”375

It was also through his father that Charles came into possession of Helmholtz’s

On the Sensations of Tones, which argues for the case of extended Pythagorean tuning.

Helmholtz had, in fact, devised his own notation, which Ives borrowed (or rather co-

opted). Alexander Ellis, through his attitude of questioning, had opened the door for

young Charles about the speculation of notation and the possible alternatives of tuning.376

372

Ibid. 373

Ibid., p. 102. 374

Taruskin, 2000, p. 292. 375

Op. cit. p. 103. 376

Ibid.

170

Just as Helmholtz scaled the Pythagorean spiral of fifths to 26, thus avoiding

enharmonic tones, Ives used extended Pythagorean tuning to 21 pitches in the Universe

Symphony, so that all enharmonic equivalences would be absent: A-Bb-A#-Cb-B-E-B#-

Db-C#-D-Eb-D#-Fb-E-F-E#-Gb-F#-G-Ab-and G# correspond to: 0-90-114-180-204-294-

318-384-408-498-588-612-678-702-792-816-892-906-996-1086-1110 cents.377

Ives also

did not favor small super-particular ratio consonances, and liked harsher dissonances

from equal divisions of the octave and larger ratios—“if they hear anything but do-mi-

sol, or a near cousin, they have to be carried out on a stretcher.” 378

(Please see Ex. 5)

Ives’s acoustical plan is based on fifths and octaves, and reinstates the Pythagorean major

third at 408 cents. Ives liked to poke fun at the temperament limitations of the piano (and

12 tone equal temperament in general) by breaking out of the harmonic series by

featuring competing tunings: Pythagorean, Just, quartertones, and eighth tones.379

In the Universe Symphony, at measure 78, there is a D flat and F# in the cellos,

and a B in the double bass. This chord is spelled purposefully, as it otherwise would have

been re-spelled with C#, or perhaps, G flat. The B-D flat interval is 180 cents (two

limmas, if we are talking Pythagorean terminology) and the D flat to F# is 522 cents, a bit

larger than a perfect fourth, showing Ives’s penchant for dissonance. The F flat and E

together in the Orchestra Unite, measure 112, shows an intervallic difference of a comma,

or 22 cents, while there is a B flat-C flat in measure 224, showing a difference of 90

cents, a limma. All these reveal the nature of Pythagorean tuning.

Charles Boatwright once said that in the last analysis, “finer distinctions between

pitches rests on Charles’s ear,” (reminiscent of Aristoxenus) “in fact, on the judgment of

377

Reinhard, p.103. 378

Ibid, p. 104. 379

Ibid, p. 105.

171

his ear, and in his case ‘a very sharp’ ear.” 380

Intonation is fully tied in with Ives’s higher

level of thought, where the Universe Symphony represents Ives’s physical manifestations

of the possibilities of the sounds of nature.

380

Ibid, p. 110.

172

Ex. 5.1.1 The legend from Ives’s Universe Symphony. (above)

173

Ex. 5.1.2 Polymicrotonality in Ives’s Universe Symphony.

174

In Prelude 3 of the Universe Symphony, there are 24 quartertone chords, but the

tonality is also modulated to quartertone tonality, a brief respite from the overall extended

Pythagorean tonality underlying the work. The Wusta Zal-Zal, named for a dark ages

Middle Eastern music theorist, comes to mind, at 350 cents, representing the influence of

quartertones in early Arabic maqam, and that quartertone tonality has existed

independently from 12TET, Pythagorean, just, and meantone. The Wusta Zal-Zal is the

neutral third that exists as the midpoint between the classic tunings of the just major and

minor third, between 316 to 386 cents, used by Persian and Arabic musicians. 381

Ives tuned the harp to Just intonation, which improvises, alongside the Just

intonation machine (Ex. 5.2), and the trumpets are tuned to eighth tones, reminiscent of

George’s influence on his slide cornet. According to Joe Monzo, Ives created a non-

octave “stretched” scale in eighth tones, of seven pitches: C-G♭-E 1/8th♭ -B 1/8

th ♭-E

1/8th

♭-D- and A 1/8th♭, which perform a free cadenza for the percussion entrance at

measure 131. (Ex. 5.2)382

In fact, the very first measure of the Universe Symphony

features an E-A#- E♭ chord, where the E-A# tritone is 25 cents larger, an eighth tone,

and the A#-E♭ is 25 cents smaller, also an eighth tone. The polymicrotonal intent is

evident from the beginning of the work.

In terms of tempo, the Universe Symphony is divided into three strata: the Basic

Unit, at 16 seconds, or roughly the metronome marking of 15, an Orchestral Unit of

30BPM, and the “Earth Orchestra,” and the “Heavens Unit,” at 45 beats per minute. The

381

Wright, 1978, p. 41. 382

Ives, 1911-1926. Universe Symphony: realized by Johnny Reinhard, 1993-1996.

175

three divisions of the orchestra represent possibly the trinity, but more concretely, the

Harmonic Series, where the bottom strata, the Basic Unit, is composed entirely of

percussion and inharmonics and is itself divided into 25 layers of polyrhythms, from

prime numbers up to 43 (Please see Ex. 5.1.3). The three tempi coincide at measure 27,

Ex. 5.1.4, showing Ives’s masterstroke and his innovation in terms of vision and concrete

composition.

Actually, there really is an additional microtonal resource at work in Ives’s

Universe Symphony: inharmonicity. At measure 137, there is an instruction for

percussion to play on a marble surface, along with the Just intonation machine. Since Ives

is already exploring the limits of tuning, it also appears that he was also exploring tuning

in terms of timbre, as well (Please see Ex. 5.1.5).

Measure 199 is the point where polymicrotonality begins, as quartertones are

played in the brass while the orchestra plays in Pythagorean tuning. It is not the case that

polymicrotonality is an isolated “effect.” The tonality of the work reduces to a simple

layer of tuning: there are mixtures of tunings and there is a deliberate pairing of tunings

within each orchestral division.

The ending of the Universe Symphony reveals distinctly the Pythagorean

influence of the work as the last chord is spaced on fifths, C-D-A-E-B. As mentioned

earlier, this paper is not an analysis of form or harmony, but the tonal plan behind the

Universe Symphony is also multileveled. Pythagorean tuning focuses on the fifths

relationships, the just intonation machine and harp focus on improvisation, the eighth

tones are featured linearly in the trumpets, in particular, and the quartertones are chordal,

in texture, and in harmony. There is a conflation and melding of harmonic purpose within

176

each tuning that is sculpted out of the density with rhythmically identifiable motives and

register.

Ex. 5.1.3 Ives’s Universe Symphony showing the polyrhythmic divisions.

177

Ex. 5.1.4 The polytempi of the Universe Symphony.

178

Ex. 5.1.5 An example of Just intonation in the Universe Symphony.

The Heavens Orchestra, the highest of the three divisions, features nine flutes,

glockenspiel, and other high-frequency timbres, revealing further the overall overtone

179

series plan in Ives’s conception. The fixed register of frequency is counterbalanced by the

freedom of the improvisation of the harp and just intonation machine.

The harmony can be said to be ultra-chromatic, or polymicrochromatic, where

one can see sets of intervallic cells based on (01). The orchestra itself is sub-grouped into

chamber ensembles for the convenience of the hemiola (3:2) tempo relationship. The

Basic Unit, composed of percussion, expands and contracts, much like modern theories

of cosmology, and is represented by 25 layers of polyrythmic percussion, like small

planets cycling a star, or even stars encircling a black hole.

The Heavens Orchestra, the highest registrally and timbrally, as well as the fastest

moving tempo at 45 beats per minute, has five groups of rhythmic subdivisions, all in

turn subdivided into small chamber ensembles: group one moves in half-note triplets and

has 3 flutes, 3 violins and a viola; group two moves in common time and is composed of

2 violins, 2 violas, and one flute; group three moves in quintuplet rhythmic subdivisions

and has 3 flutes and 2 violins; group four is a general movement of septuplets by any

instrument, and group five is composed of the glockenspiel and celeste. All these timbres

represent the highest overtones possible as they reach into “Heaven.” Ives called these

cloud shapes, where each group is a particular code that represents a chordal

counterpoint.383

Is there an actual correspondence between pitch and rhythm in Ives’s Universe?

In other words, if we find triplets, are they related to the perfect fifth, as the 3:2 ratio

relates to the fifth in the harmonic series, or is there, per chance, an elaborate system for

pairing pitch to rhythm, or tempo, in Ives’s Universe Symphony? No, there is not;

however, the tempo relationships are differentiated more by texture and timbre than by

383

Reinhard, 2004. The Ives Universe: a symphonic odyssey. p 38.

180

pitch, whereby there is similarly no system at work. There are only a few generalizations

based on circumspect analysis: the Basic Unit consists of percussion, mostly non-pitched

and metallic sounds, but does have a hierarchical rhythmic structure within itself; the

Earth Orchestra, the middle level tempo moving at 30 BPM., has mid-ranged and low

frequency instruments; and the Heavens Orchestra, the highest ranged frequency division,

has the faster moving elements composed of five subgroups, as discussed previously.

There is no correlation between polymicrotonality and polytempo, even though they both

coexist within this one work, albeit separately. Ives appeared to be working more in the

direction of timbre towards the end of his compositional career.

Example 5.1.6 shows the minutiae of the divisions of pulse time by Ives’s Basic

Unit of 16 seconds, with rhythmic divisions from 1 through 19, 21, 23, 29, 31, 37, 41,

and 43, mostly prime numbers, totaling 25 divisions of the 8/4 double whole note.

There are ten cycles of the expansion and contraction of this ensemble,

representing the pulse of the cosmos. It is also the articulated surface of the Universe

Symphony under which there still operate three divisions of structural tempo, from 45, 30,

to 15 BPM, showing both 2:1 and 3:2 ratios, which happen to be the two intervallically

significant structural scaffolding underneath the work.

There is a recording of the Universe Symphony, on The Stereo Society’s label, from

2005, recorded by James Rosenthal and Mike Thorne. Conducted by Johnny Reinhard,

and performed by the American Festival of Microtonal Music Orchestra, it spans an hour

and five minutes in length.384

Please note Ex. 5.1.7, an original page from Ives’s own hand-writing, showing the

precarious nature both Reinhard and Austin were placed in. Ives is known for his bad

384

Universe Symphony, 2005, CD liner notes.

181

hand-writing. The challenge to pick up his work where he left off, was steep; in addition

to the complexity of Ives’s musical architecture, there was the difficulty in deciphering

the notes themselves.

182

Ex. 5.1.6 This author’s notes on the exact percussion and rhythmic subdivisions in the

Basic Unit.

183

Ex. 5.1.7 Original sketch from Ives’s Universe Symphony.385

385

Austin, p. 226.

184

5.2 IVES’S LEGACY: POLYTEMPIC POLYMICROTONAL ART MUSIC

The possibility of this notion of polytempic polymicrotonality has been hinted in

music in the examples provided in this paper. This author inquired about music in

different tunings from several composers and musicians who thought the idea absurd.

After researching the internet for dissertations, theses, and other research papers on the

subject, Johnny Reinhard, a fierce polymicrotonal warrior in New York, appeared on the

internet, through whom he discovered Ives’s Universe Symphony. An open and unbiased

mind capable of connecting ideas in new ways is helpful when considering something

this far outside the box of the status quo. Polytempo is not new anymore, but

polymicrotonality is still an outlandish concept, and when coupled with a one to one

corresponding tempo, or rhythmic reassignment, there seems to be a wall, conceptually,

into which musicians collide.

After discussing various theories and methods of combining pitch and rhythm

through the ages, the immensity of the infinite possibilities of music, when all is said and

done, is staggering. There is a similarity that occurs in art and culture that also occurs in

the individual: growth. There are times in history where different ideas of tuning have

overlapped, competing for supremacy, at a point in time where one tuning system was

thought to be the way music ought to be, in direct violation of the thoughts of the Greeks

and of Anaximander. This same thing occurs in humans unconsciously, when opposing

aspects of the personality compete for dominance, when ultimately one or the other wins

out. This author’s point is that when Ives deliberately put together four different tuning

systems, he did so consciously, not out of overlapping tuning philosophies, as we have

185

seen, particularly in the middle ages and Renaissance, and even the Baroque. Yet, all

along, composers were becoming more aware of tuning differences as aesthetical and that

at some point they could be used together in an amalgamation, or used simultaneously.

This is akin to an individual, over time, accepting and coming to terms consciously with

their inner aspects that once were so secret and hidden, but through constant reworking,

came to light and became consciously integrated with the personality. This is the same as

art music in the twentieth and twenty-first centuries coming consciously to the point

where tuning is not so dark and mysterious, but can be integrated, amalgamated, and used

in a deliberate, conscious, and artistic way. This is essentially what this author is arguing

for. He believes that setting each part in a different tuning in a separate tempo is a similar

process of using relief in art, by elevating a theme off and above the canvas and framing

each part contextually in the ultimate polyphonic texture, polytempic polymicrotonality.

186

5.3 POLYMICROTONAL NOTATION TECHNIQUES

This author has had ideas with respect to notating multiple systems of tuning and

temperament by fattening the staff and allowing a visual representation of the microtonal

pitch without the encumbrances of additional accidental notation. As microtonal systems

become smaller, and more varied, the need for accidentals increases, forcing the

performer to learn whole sets of symbols for each piece. This is burdensome and can

easily deter performers from even considering microtonal works. But perhaps a new staff

can offer a solution, rather than additional microtonal symbols, which also inadvertently

lengthen the measure, particularly when there are complex rhythms at the 32nd

-note level,

which can draw a measure out to a full page. A visually based fattened staff approach,

eliminating all microtonal symbols, could be a solution to this problem and provide more

visual continuity with respect to score reading (Please see Ex. 5.3.3, 5.3.4, and 5.3.5).

One can argue that learning a new staff is also asking the performer to learn new

nomenclature, too, but far less when dealing with polymicrotonal systems and their huge

concomitant array of accidentals that clutter up the line.

As a futher demonstration of this new proposal, this author’s piece Jove Defeats

Saturn, for polymicrotonal saxophone solo (Ex. 5.3.2-5.3.5), was written using this new

staff method. Nathan Mandel, the saxophonist, was the first person to try out this author’s

new system. He had some corrections for me and helped me by redrawing the staff larger

and with greater clarity. There was to be no confusion to identifying any pitch laying

anywhere in the staff, even without the accidentals. The lines flowed visually. Nathan is

now willing to use this system and has helped me to edit and improve the system.

187

Another more practical way for polymicrotonal notation is to simply use what

symbols we have, the quartertone symbol, but to also combine them with cent numbers

above the note in question. Therefore numbering from 0-49 cents, since 50 cents is the

quartertone, would allow perfect precision. This technique is used by Johnny Reinhard

and has proven to work for him and the AFMM for a number of years.

Either method cuts down on the plethora of microtonal symbols, which have

appeared in many treatises but have proven incomprehensible. As a polymicrotonal

composer, this author wants people to want to play his music, where fancy and extensive

microtonal symbols seem a bit contrived and elitist, not to mention difficult to memorize.

Fattening the staff, an idea this author picked up from artist Agnes Martin’s artwork,

seems the most beneficial in the long run, but there are problems to work out, such as

accurately identifying the pitch at first sight by definitively allocating the spaces, lines,

dashed lines and borders to gradations of pitch. As mentioned earlier, this author

currently has a piece for polymicrotonal solo saxophone, Jove Defeats Saturn, in 12, 19,

31, and 53TET, utilizing fattened staves in different colors, for the enhancement of visual

acuity, which thus far has proved better than “51% advocacy,” which means in its test run

it can be useful after making edits and adjustments. One of the ways Nathan Mandel

improved this author’s Poly staff, was to actually draw it larger, with more clearly

defined edges to the colors and dashed lines in the center of the spaces for the natural

pitch. The staff remains the standard EGBDF, but expanded for visual acuity.

(continued below. please see Ex. 5.3.1)

188

Ex. 5.3.1 An example of artist Agnes Martin’s work that inspired me to incorporate larger

staff systems that help reduce linear space cluttered up by unlimited microtonal

symbols.386

386

Agnes Martin, Canadian artist, b. 1912, http://www.abstract-

art.com/abstraction/l3_more_artists/ma57a_agnes_martin.html.

189

Ex. 5.3.2 Jove Defeats Saturn legend with color scheme for different tunings.

In Ex. 5.3.2, the staves are changed, rather than the inclusion of excessive

microtonal symbols that will ultimately conflict with each other for the performer, so I

chose to by-pass microtonal symbols and fatten the staff, instead. Color codes drape the

staves of different tunings as the main visual indication of a tuning change.

190

Ex. 5.3.3 Section B of the rondo form Jove Defeats Saturn in 19TET.

Ex. 5.3.3 is the B section of Jove and is color coded in pink highlight to designate

the tuning system of 19TET. The colors do not indicate synesthesia, but are randomly

chosen from any pack of highlighters for study purposes. The legend, Ex. 5.3.2, shows

clearly which shade of pitch from each system lies in its delineated area of the staff. The

staff is kept at the common practice EGBDF, so the basic shape remains the same for

familiarity.

191

Ex. 5.3.4 Section C of Jove, in 31TET.

Ex. 5.3.4 is color coded blue for 31TET and its shades of pitches are also found in

the legend. This fattening of the staves will accommodate a full 31-note vertical chord,

but not much more, depending on how thick the staves are drawn. The linear flow is kept

as horizontal distances are minimized by the absence of extraneous and confusing

microtonal accidentals. As a very rhythmic composer, I find that the visual impact of the

line maintains its integrity, instead of being haplessly pulled across the page by the clutter

of accidentals.

192

Ex. 5.3.5 Section D of Jove, in 53TET, but in a restricted pitch set spanning a perfect

fourth tetrachord.

Ex. 5.3.5 is color coded in orange, for 53TET, which is not fully employed. Only

a small set is used form the gamut of pitches in 53, encompassing a set of 21 pitches.

Again, the horizontal space is saved and the rhythmic linearity and continuity is kept

intact, due to the lack of excessive symbols.

This author’s string quartet Hypercube (2012) uses Reinhard’s system of

numerical cents above the note and a written out legend showing the pitch to cents

relationship (Please see Ex. 5.5.1-5.5.2). I chose to use both systems of polymicrotonal

pitch notation in order to investigate the expediency of each system.

One drawback to the fat staff technique is that it must be composed free-hand,

since a professional notation software program will not be able to duplicate it. On the

other hand, composers now-a-days are falling into the habit of composing for the notation

program, and not for themselves. In other words, the limitations of Finale and Sibelius

are beginning to restrict composers and dictate their terms and limitations onto the

creative process. Free hand composition is probably something that needs to return.

193

The open-source free ware MuseScore is probably the only graphical notation

program in existence that can allow any tuning system by way of plug-ins, aside from

Lilypond, or other programming language-based music notation software.387

But in terms

of employing polytempo, with different measure lengths, there are no acceptable

professional notation programs capable of achieving this.

5.4 POLYTEMPO NOTATION, APPROACH, AND ORGANIZATION

Though there is no particular notational technique over and beyond what is

currently examined in this document, there is an approach that needs to be discussed.

Polytempo can be explicitly notated by a metronome marking, or it can be implied within

the score itself. Both Nancarrow and Ives provide examples of this. Nevertheless, there is

the whole notion of what polytempo really means, in the deeper sense, with respect to

composition.

For this author, polytempo is the furtherance of part identity coming alive within

the scope of a musical work. By coming alive, the part develops its own identity, as in

Carter’s polyrhythmic structures that are also polytempic (Third String Quartet), but it

can also develop its own tempo, or life, independent and intrinsically related to the piece

from which it grows. Polytempo is akin to the techniques of high relief in densely textural

artwork, such as Rembrandt’s impasto technique, which is stratified above the two-

dimensional canvas to project a three-dimensional character and play of light.

387

MuseScore, 2012, http://musescore.org. (Accessed 11/11/11).

194

One can begin by setting, or framing, different polyrhythmic motives in different

time signatures with a common downbeat which then, over time, develop their own

intrinsic tempi which then begin to “give life” to the musical voice in question, where it

can then do whatever it wants to do, as it has established its own identity.

The error Cowell encountered in the clumsy recurrent downbeat alignments of his

polymeter ideas can be overcome by skillful part writing (which is among the primary

values in the art of composition). Does not 4/4 time, similarly, have downbeat

convergence points every four beats, and haven’t composers adapted to writing

“downbeatless” music that perpetuates momentum and avoids the banal nature of the

absence of surprise? Then how is this any different, unless one is expecting the differing

time signatures to compose in lieu of them, instead of the composer creating within and

“without” them? Of course it is always possible for music in 4/4 to be free as well, and

there are many examples of it. Free Jazz, for example, is an excellent illustration of that.

The point is to not allow boredom to manifest. Nevertheless, it is the will and artifice of

the composer that makes the difference, which when applying polytempo, can push the

art of polyphony even further and by developing truly independent “characters” in the

narrative of time and music.

Again, notation programs preclude the use of competing tempi and meter, so it is

an arduous task to overcome via commercial notation programs. One must go into the

staff and discontinue meter signatures and “write” them in by special time consuming

procedures within the program, as in Finale, for example. Today’s reality is that scores

must be printed, and unless one has fantastic drawing skills, scores must be processed and

195

printed by computer, as most of us act as our own publishers and editors in this day and

age.

Tempo organization strategies, however, can stem from Cowell’s correspondence

alluding to the overtone series with tempo ratios equating to the overtone series, as has

been demonstrated by Ivan Wyschnegradsky, Ben Johnston, Karlheinz Stockhausen, and

Conlon Nancarrow. Tempo canons, based on the pitch structure, can provide cohesion

and logic to the piece overall. One can invent a series of different tunings to which,

numerically, tempi can be ascribed, such as a four-voice quartet, which can be set in the

equal temperaments of 12, 19, 31, and 53 tones per octave with corresponding tempi at

either exact numerical values, or their multiples, such as 144, 57, 93, and 53 BPM,

respectively. Even if there is some arbitrariness in choosing these musical parameters,

does that negate the value of the work, particularly if the uncertainty is drastically

reduced by direct mathematical relationships?

There can also be perfectly exact relationships between pitch and rhythm, as

described earlier in this paper, where rhythms multiplied by certain constants will yield

the pitch content, showing the very same acoustical phenomenon, yet, this will lead to the

type of connect-the-dots integral serialistic compositional techniques that deprive any

intuition, choice, and thought. Since 1950, choice has been equated with a lack of logical

rigor; in other words, having aesthetic druthers is a bad thing. Nevertheless, even Xenakis

would interrupt his own processes of stochastic music and interject his own musical

aesthetic in his works and he did not deny that, either.

Both Nancarrow and Carter would draw out their large scale tempos onto either

piano rolls, or graph paper, in order to give a physical perspective on the scale of the

196

relationships. The exact mathematical objectification of numbers gives solid scaffolding

of the pure structure of the work.

5.5 POLYMICROTONAL PITCH ORGANIZATION AND FORM

In considering pitch organizational methods, John Chalmers has suggested the

construction of new genera by using the CI, the characteristic interval, which can vary

from 13/10, at 454 cents, to 10/9, at 182 cents, yielding approximately 73 different

classes of CIs.388

Also, as the Greeks employed katapyknosis, which is the procedure of

linear division, composers can further produce intervals of ever diminishing size, in the

spirit of Aristoxenus.

David Doty, similarly, has suggested using tetrachords spanning the perfect

fourth, as the Greeks did, as a method of organization for microtonal pitches. Ben

Johnston has used Alexander Ellis’s duodenal method of dimensional grids for each

prime number in constructing just intoned systems, leading to vast architectural lattices,

such as those by Erv Wilson.389

Enrique Moreno and Easley Blackwood have used the process of equally dividing

the whole tone into a continuum of equal tempered infinitude, akin to what Aristoxenus

may have thought had he lived today. Also, the division of pitch materials into octave and

non-octave equivalents can be another area of pitch polarity. Xenakis used sieves and

stochastic methods to generate pitch materials and to order them.

388

Chalmers, 1993. p. 25. 389

Doty, 1993, p. 30.

197

The best way to approach polymicrotonality is to investigate the exact cents of

each tuning system and make a diagram about near unities between pitches within 10

cents, for modulation purposes. For large systems, cutting down the materials into

manageable sets, or cells, seems only common sense. Josef Straus’s book on atonal

theory is an excellent resource for investigating music created by way of pitch sets, which

would work quite well in polymicrotonality.

Tetrachordal approaches are already within this realm of organization, so having

boundaries of a particular interval within which to work is a good start. One of the

reasons this paper dwells a good deal on previous historical theory, such as the Greek

genera, is that that information can serve as case histories, providing much needed

tutorials for the newly initiated.

Just as in Greek tetrachords, the bounded or fixed intervals can serve as “tonic” or

convenient reference points, or can even be considered a microtonal centricity. Each

tuning, or temperament, in a polymicrotonal work can have its own “tonic,” or tonus. One

could even use chance operations to choose notes randomly from many different tuning

systems, to employ in tetrachords, pentachords, or any interval, or set number one wishes.

The doors are wide open.

Below, in Ex. 5.5.1 and 5.5.2, one can utilize boundaries and tempo

differentiation with respect to pitch organization.

198

Ex. 5.5.1 Legend from this author’s polytempic polymicrotonal string quartet Hypercube.

199

In Ex. 5.5.1, there are four temperaments. They are temperaments, and not

tunings, due to the fact that the fifths are tempered down by a few cents, so they are not

pure. I am careful to introduce this mixed temperamental, and polymicrotonal, piece by

allowing the 12TET first violin to serve as the lead instrument using the foundation

tempo, set to 96 beats per minute. Each part of the string quartet has its own tuning,

tempo, and behavior, so that when the texture is at its thickest and most dense, the parts

can still be perceived.

200

Ex. 5.5.2 First page from Hypercube,in four temperaments and four tempos.

201

5.6 COMPOSITIONAL RESOURCES: SCALE AND TUNING RESOURCES

AND COMPUTER SOFTWARE FOR THE COMPOSITION OF POLYTEMPIC

POLYMICROTONAL MUSIC

We live in the computer age; all things seem possible with respect to tricky

performance issues, such as tuning and rhythm. Some of us are not computer savvy, with

respect to pure computer programming, yet there is software that can help with the

production and notation of polymicrotonality. As far as polytempo, there are really not

many programs that can do this, in terms of sequencing due to the availability of only a

singular global tempo. Due to this limitation in the software, composers would have to

map out the tempo relationships vis-à-vis a global tempo separately on scratch paper. In a

way this limitation defeats the whole purpose. The ultimate timeline, of course, is time

itself, set to 60 beats to the minute.

MuseScore, an open-source music notation program free to the public, features

downloadable plugins that can work with the minutiae of microtonal pitches. There is

also a playback function for auditioning the sound, which is very important. Open-source

software is Scala, from the Huygens-Fokker organization.390

Scala has a bank of literally

hundreds of tunings that can be imported to any program, for midi tuning playback, and

is perfect for polymicrotonal usage. One can store, edit, compare, build, analyze, and

convert tuning for midi instruments, both hardware and software.391

Old synthesizers are an excellent source for polymicrotonal music. E-mu

synthesizers, for example, feature five built-in tunings, 12TET, Just, Pelog/Slendro,

Valotti, and 19-tone meantone temperament. Each channel, 1-16, can be programmed to

390

Scala, 2011. 391

Ibid.

202

have its own individual tuning, not to mention the user tuning, which can be saved to

RAM.392

This author’s intent, though, for polytempic polymicrotonal music, is for acoustic

instruments and the invitation, or cajoling, of instrumentalists to expand their extended

technical means by learning to produce microtones of a great many shades. This author’s

piece Saturn Defeats Jove features 12, 19, 31, and 53TET for the saxophone family, for

solo performer. Nathan Mandel, who premiered the work, agreed to work with me in the

development of new fingerings to achieve the tunings. As a composer, one must be more

than willing to admit faults and learn to overcome obstacles to this new music by learning

from the players and their instruments, since they are the true source of all knowledge

concerning orchestration. The composer must have humility, and be willing to make

changes, and to admit when things do not work out favorably.

In terms of scalar resources, Lydia Ayers’s Exploring Microtonal Tunings: A

Kaleidoscope of Extended Just Tunings and Their Compositional Applications, Vols. I

and II, 1994, is a magnificent compendium of scales, tunings, and documentation for

anyone interested in microtonality. Volume II contains a collection of systems, scales,

and tools from around the world, including Javanese Gamelan, and the Indian Shruti

system, for composers to discover, in cents, the exact nature of these exotic tunings.393

O. Wright’s treatise The Modal System of Arab and Persian Music, A.D. 1250-

1300, 1978, is also a useful compendium of modal maqam, from tetrachords, to

392

E-mu Corporation, Scotts Valley, CA, 1991. 393

Ayers, Exploring Microtonal Tunings: a kaleidoscope of extended just tunings and their compositional

applications, Vol. I, 1994.

203

octachords, comprised of Safi al-din’s classifications of 84 modes, and their notation.

There is also Qutb al-din’s compendium, based on combinations of modes.394

394

Wright, 1978.

204

CHAPTER 6:

CONCLUSION

Much material has been covered in this thesis. The reason for this is because this

author wanted to show that microtonality, and possibly polymicrotonality has been a

normal state of affairs since ancient Greek music. The tetrachords made their way into

Europe during the dark ages, and into the middle ages. The Greek humanist revival in the

Renaissance brought back the Greek genera again, where composers such as Vicentino,

Gesualdo, and even Claude le Jeune,395

engaged in Greek humanism, and the three

genera, including the enharmonic genus and its quartertones.

Werckmeister and Bach and the Thuringians, who preferred their unequal tunings,

lived in a time when there were approximately 150 differing meantone tempered tunings,

where there was probably some overlap, or mixing. This mixing could have resulted in

potential microtonal shades of different tunings by different musicians resulting in

inadvertent polymicrotonality in performance practice.

The twentieth century had its own Greek revivalism, from the work of John

Chalmers, and Kathleen Schlesinger, to David Doty, all incorporating aspects of Greek

tuning theories in the Greek revivalism of the current times. Ives, too, by his discovery of

Ellis’s translation of Helmholtz’s magnificent treatise On the Sensations of Tone, had

been influenced by extended Pythagorean tuning, as explored extensively in that book.

Ives’s Universe Symphony, and its mix of tunings and temperaments, as shown by

Reinhard, and Austin, have led this author into a potentially new paradigm. A paradigm

that is ready to be accepted because musical consciousness has risen to it. It has pointed

to it in order to embrace it. Musicians can hear it.

395

The New Grove, 2000. Vol. 14. p. 532.

205

What is tuning? Tuning is a set of intonations for intervals or pitch classes,

established by a culture, a group, or a composer. Tuning includes sub-systems and

modes, or sub-sets, of a gamut, or a palette of pitches, which are scales extracted from the

tuning system. There are five constraints that must be adhered to with respect to scalar

structure: pitch sets of a fixed number, the repetition factor (such as the octave), intervals,

hierarchy, and key, or centricity.396

Some tunings have stretched octaves with no octave

equivalence, such as Gamelan tuning. Some will say that a tuning is only that which is

pure as it is derived either by an a priori mathematical law, or from the overtone series of

nature. These two conditions yield both Pythagorean tuning and unlimited ratio overtone

tuning, both being untempered. Temperament is about the manipulation of intervallic

integrity so as to fit into the mold of the aforementioned scale definition of octave

equivalence, for identity. Cultural considerations have also determined tuning around the

world for centuries.

What is polymicrotonality? Polymicrotonality is akin to an omnivorous approach

to the pitch gamut, by incorporating all systems. Polymicrotonality has the potential to be

universal. Polymicrotonality, however is not commonplace, and this is one of this

author’s key points. Being out of tune by way of error is absolutely not the same as

polymicrotonality and can never be construed as such. One must be able to deliberately

play and hear pitches to the nearest 10 cents or better. The schisma is 2 cents wide, and

Ben Johnston allowed this narrow interval into his music. This false notion of

polymicrotonality precludes high school bands and orchestral warm-ups, or any other

such amateurish endeavor. Polymicrotonality and the information in this document

396

Polansky, 2009. “A Mathematical Model for Tuning Systems.” Perspectives of New Music. p. 71.

206

represents a most serious attempt at the ultimate control over intonation and inner

hearing.

To expand the role of the expressive was a late Romantic goal, but is it any less

relevant today? Ferrucio Busoni, in his Sketch for a New Aesthetic(1919) had divided the

octave into third tones in Italy, while Julian Carrillo was dividing the octave into 96 parts

in Mexico. Hába and Wyschnegradsky were dividing up the octave into quartertones and

beyond in Czechoslovakia and Russia. Ives would have said that this was the atmosphere

of the progressive development of humanity, as he devoured his Transcendental New

England writers and philosophers. There seemed to be a new consensus between the

composer and the listener, but somewhere between the early twentieth century and the

early twenty-first century, Rock’n’Roll and the big record companies happened. They re-

affirmed 12TET and put it squarely back on the public, reinforcing the tyranny of the

piano, once again, for the sake of profits and selling “ear-worms.”

Then, the personal computer came along in the 1980’s and shifted control over

tuning away from the piano, making possible a panoply of tunings immediately

accessible to the public for the first time in human history. When the internet boom

occurred in the late 1990’s, compressed digital audio files became small enough to send

via email and post on the internet. An already growing number of microtonalists around

the world suddenly had access to one another via computers and the proliferation of

microtonal art music once again began to skyrocket. There are numerous online

microtonal musicians, composers, and sound artists all sharing their works and many are

exploring tunings on midi synthesizers and computer music programs. The world seems

ready, finally, to accept what Ives and Carrillo tried to do. The thrust of this movement

207

has taken place outside of academia, in small communities from Ivor Darreg’s San Diego

microtonal community to AFMM in New York, to the New England area. The outsider

composers are divested from academia and its status quo, where according to Kyle Gann,

art remains stultified (and it does not have to be like this).

Like the polytextual composers of the Ars Nova, this author seeks to combine

various tunings (like combining different languages) framing them in different tempos for

the purpose of featuring each microtonal voice in an audible and contrapuntally

distinguishable way. Also, like the relief painters of the Renaissance, he wants to create a

multidimensional texture that would enhance each separate microtonal voice in a

composition, without uniform movement, such that a type of four-dimensional

perspective could be achieved. By studying the works of Ives, Carter, Nancarrow,

Stockhausen, Ferneyhough, and many others, he has taken the most appropriate qualities

to serve this unique musical adventure.

6.1 ARE THESE MICROTONAL SUBTLETIES PERCEPTIBLE TO THE EAR?

Is polymicrotonality perceptible to the ear, or is this a frivolous endeavor? If one

considers the Middle Eastern maqam, with very small intervals, sung by Arabs every day

in the Azan for their call to prayer at daybreak, and sung for centuries, one cannot ask this

question. In Turkey, the octave is divided into 6 whole tones, each of which is divided

further into 9 commas. Each comma has a name. How does that occur if they cannot be

heard? Llewelyn Lloyd also stated that the ear, though it cannot process pitch at 1/100 of

208

a second, can, given a longer compositional duration, hear any interval.397

Hugh Johnson,

Jr. proved in his thesis Tuning Preferences of a Select Group of Singers with Reference to

Just, Pythagorean, and Equal Temperament (1963) that singers prefer sharper thirds398

,

on the order of a comma, which they all can hear, where a comma is circa 25 cents, or an

1/8th

tone. The singers seemed to prefer the 408 cent Pythagorean third, contrary to what

Ellis and Helmholtz had written. Alexander Ellis and Helmholtz, initially, had proved in

On the Sensations of Tone that singers sang in just intonation, but that was in the late

nineteenth century. Apparently over time things tend to change.

Also, David Whaley’s The Microtonal Capability of the Horn (1975) provides a

rich source of microtonal perception surveys by all ranges and levels of horn players,

where the just noticeable difference index for a microtonal change was consistently found

within 13-16 cents.399

E. Chevé of France invented a Pythagorean extended tuning in 53 tones for

vocalists due to the subtle variations of which singers are capable.400

Nevertheless,

Helmholtz held steadfastly to the notion that singers sang in just intonation, and in a

further breakdown, strings performed in Pythagorean tuning, keyboards, historically in

meantone, and brass, naturally used the overtone series to the 19th

limit. Although these

are generalities, performers are not always conscious of this. This still does not qualify as

bona fide polymicrotonality, because intent is lacking. Lack of skill, also does not qualify

as polymicrotonality. There are many ways that human effort can be completely

misconstrued and misinterpreted. Mistakes are still mistakes. But systematic practice of a

397

Lloyd, 1963, 1978, Intervals, Scales, and Temperaments. p. 151. 398

Hugh Johnson, 1963, Tuning Preferences of a Select Group of Singers with References to Pythagorean,

Just, and Equal Temperaments. p. 56. 399

David Whaley, 1975. The Microtonal Capability of the Horn. p. 9. 400

Ellis-Helmholtz, 1954. p. 426.

209

system of tunings will ultimately lead to fewer mistakes, and move towards more control

over intonation. Hearing down to the cent, thus making the 1200 cent master grid our

ultimate gamut, is a noble goal. When one hears the cent, one hears a phase discrepancy

between the pitches such that the entire harmonic series will pass by, audibly, in a very

complete cycle, lasting several seconds. That is your “beat.” Those beatings one hears in

sharp dissonances are actually the overtone series of that particular timbre playing that

particular interval.401

6.2 FUTURE RESEARCH REGARDING POLYTEMPO AND

POLYMICROTONALITY

This paper is just the beginning of a polymicrotonal category of music. There is

much more to explore. This document is a decent foundation, but it is only a start. All

who are interested in this aesthetic can jump in at any time and continue onward.

There are areas of pure microtonal theory and voice leading to consider. Chords in

various tunings and how these sound complexes work together, will be another area of

consideration. One can use his or her ears, or use an FFT, or some other computational

method to measure dissonances, like Sethares’s dissonance curves. One ought not be

overly dismissive towards harsh dissonances, even though many composers are very

concerned about purity and sweetness of tone. There will be many arguments against this

polymicrotonal approach, but with a new footing in polymicrotonal theory (this author’s

next document), some semblance of order can be established. Currently, there is

401

I created a polymicrotonal synthesizer in MaxMSP, where I discovered that at a one cent interval, I

could hear very clearly the overtone series between these two pitches. I realized that this is what a “beat” is

in slow motion.

210

absolutely nothing wrong with allowing our ears to pave the way: first the music, and

then the theory.

One area of research could possibly be to link rhythm, to pitch, and then to color.

Synesthesia is a condition many composers have had, except all the individual color

schemes fail to match. Rimsky-Korsakov’s did not match Skryabin’s, etc. Nevertheless,

all these phenomena are related by wave frequency. Charles Lucy has used multiplication

and a “modulus 12” approach to map directly pitch to color.402

Lucy had to convert

Angstroms into Hertz, but also, these are still apples and oranges, since one form of

energy is mechanical, while the other is electro-magnetic; one deals with hearing, while

the other deals with sight. People become confused by this. This is not to say, however

that pitch and rhythm are completely unrelated, even though they are. They are,

nevertheless, perceived differently. Can one truly hear a pitch tuning of 22TET and its

relationship to a multiple of 22, e.g. 88BPM? No, but there is still logic inherent in this

design. This concerns teleology and intent of the composer. Perhaps it is conceptual to

the extent that Bach’s obsession with numerology, or Messiaen’s “charm of

impossibilities” also was not audible, but even the concept added logic and continuity to

the deeper layers of his music.

Another area of future research includes the aesthetics and metaphysics of

polytemporality, and its stratification, both in music and how it relates to human

existence, being, and consciousness. Time, temporality, and its meaning in human

consciousness is a rarely covered topic, but has had some discussion, from Jonathon

Kramer. Philosophers from Kant to Heidegger have explored the phenomenon of time,

but it could reside within an area of psycho-musical research, investigating polytemporal

402

Charles Lucy. At: http://www.lucytune.com/new_to_lt/pitch_04.html, accessed 3/5/12.

211

stratification, as described by Maury Yeston (from a musical theoretical perspective),

utilizing some of the phenomenological ideas from Kant to Heidegger; areas could

involve the subject/object split, whether or not time as a construct is completely a priori,

and how human consciousness reconciles time as existence, through being, versus

temporal time in music composition and human consciousness’s ability to occupy

polytemporality, as it exists in music and rhythm.

212

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