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THE

J UST INTONATIONP RIME RRIME R

by David B. Doty

An Introduction to the Theory and Practice of Just Intonation

Third Edtion, December 2002

ISBN 0-9726810-0-0

1993, 1994, 2002 The Just Intonation Network A Project of Other Music, Inc.

535 Stevenson StreetSan Francisco, CA 94103

Phone: (415) 864-8123Fax: (415) 864-8726

[email protected]


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iv

The Just Intonation Primer

Half-Diminished Seventh Chord ................ 574-6-7 Chord ............................................. 57Seven-Limit Subharmonic Chords.............. 57Condissonant Chords in the Seven Limit ... 58

Fixed Scales in the Seven Limit ........................ 59The Higher Primes: Eleven, Thirteen,and Beyond........................................................ 59

Eleven and Thirteen .................................... 59Primary Intervals........................... .............. 61Chords and Harmony.................................. 62Seventeen and Nineteen.............................. 64Beyond Nineteen........................... .............. 65

Prac t ica l Jus t In tona t ion wi th Rea lIn s t r umen t s . . . . . . . . . .. . . . . . . . .. . . . . . . . . .. . . . . . . . 6 6

Introduction ....................................................... 66Fixed-Pitch Instruments .................................... 67

Acoustic Keyboards.................................... 67Piano Problems ........................................... 67Reed Organs................................................ 70Pipe Organs................................................. 70Electroacoustic Keyboards ......................... 70

Electronic Organs ....................................... 70Stringed Instruments withOne String per Note .................................... 70Fretted Instruments ..................................... 71

Wind Instruments .............................................. 73Continuous-Pitch Instruments ....................... .... 74

Natural Tendencies .................................. 74Bowed Strings............................................. 75Voices........................ ...................... ............ 76Trombones .................................................. 76

Midi Synthesizers.............................................. 76Pitch Bend................................................... 76Instruments with User-ProgrammableTuning Tables............ ...................... ............ 77Tuning Tables plus Pitch Bend ................... 78The Midi Tuning-Dump Speci cation........ 78

Some Thoughts on ObtainingSatisfactory Performances................................. 79No te s . . . . . . . . . . . . . . .. . . . . . . . . .. . . . . . . . .. . . . . . . . . .. . . . 81

Index . . . . . . . . .. . . . . . . . .. . . . . . . . . .. . . . . . . . .. . . . . . . . . .. 85

Preface to the Third Edition (2002)

The third edition of The Just Intonation Primer containsno signi cant changes in content from the previous edi-tions, though a few minor errors have been corrected.The type, however, has been completely reset and all of the art has been recreated from scratch. Why? The rstand second editions of The Primer were produced withsoftware that has long since become obsolete, necessitat-ing the resetting of the type. As for the art, both my skillsand the available tools have greatly improved since theproduction of the previous editions, and my standards,as well, have risen considerably.

This is not to say that I regard The Primer , as it nowstands, as needing no improvement quite the contrary.The problem is that once I began making improvements,there is no telling where the process would end. Thiswould be contrary to my purpose for creating this thirdedition: to keep the publication available until I am ableto complete the work that will supersede it. This isintended to be the last print-only edition of The Just

Intonation Primer ; any future editions will be in theform of multimedia products. There seems to be little purpose in merely writing about unfamiliar intervals,scales, and chords when it is now possible to presentstudents with the actual sounds.

Thanks to Lucy A. Hudson for creating softwarethat simplified the conversion of the text files from the

previous edition.

Preface to the Second Edition (1994)

The writing and publication of The Just IntonationPrimer were made possible by a gift from a friend of the Just Intonation Network.

The material in this publication was excerptedfrom a larger work in progress, tentatively entitled AComposers Guide to Just Intonation .

Thanks to Carola B. Anderson and Dudley Duncanfor proofreading and commenting on The Just Intona-tion Primer . Thanks to Jim Horton for the use of hiscollection of obscure publications.


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Introduction

introduction

What is Just Intonation?

What is Just Intonation? Although, like most compos-ers working with this unfamiliar tuning system, I amfrequently asked this troublesome question, I have yet todevise an answer that is suitable for casual conversation.Technically, Just Intonation is any system of tuning inwhich all of the intervals can be represented by whole-number frequency ratios, with a strongly implied pref-erence for the simplest ratios compatible with a givenmusical purpose . Unfortunately this de nition, whileconcise and accurate, is more likely to result in a glazed

expression indicative of confusion than in the gleam of understanding. It is, in short, a de nition that is perfectlyclear to the comparative few who have the backgroundto understand it and who could, therefore, formulate itfor themselves, and perfectly opaque to everyone else,including, unfortunately, most trained musicians. (It ismy experience that most musicians are as ignorant of thedetails of twelve-tone equal temperament, the predomi-nant tuning system in Western cultures for the past twohundred years, as they are of Just Intonation. If you doubtthis, ask the next dozen musicians you meet to explainwhy there are twelve semitones in a chromatic scale and

how to accurately tune those twelve equal semitones.)A detailed answer that incorporates all the necessarybackground on the physics of sound, the physiology andpsychology of human hearing, the history of music, andthe mathematics of tuning systems, far exceeds the limitsof casual conversation. It could, in fact, ll a book.

A formal definition of Just Intonation may bedifficult for the novice to grasp, but the aestheticexperience of just intervals is unmistakable. Althoughit is difficult to describe the special qualities of justintervals to those who have never heard them, wordssuch as clarity, purity, smoothness, and stability comereadily to mind. The supposedly consonant intervalsand chords of equal temperament, which deviate fromsimple ratios to varying degrees, sound rough, restless,or muddy in comparison.

The simple-ratio intervals upon which Just Intona-tion is based are special relationships that the humanauditory system is able to detect and distinguish fromone another and from a host of more complex stimuli.

They are what the human auditory system recognizes asconsonance, if it ever has the opportunity to hear themin a musical context. Although the importance of thesewhole-number ratios is recognized both by musicaltradition and by modern acoustic and psychoacousticresearch, for the last two hundred years Western musichas been burdened with a tuning system in which allof the supposed consonances, with the exception of theoctave, deviate significantly from their optimal, integer-ratio forms. Indeed, some consonant intervals are socompromised in twelve-tone equal temperament that

they are hardly represented at all.Just Intonation provides a greater variety andsuperior quality of consonances and concords thanequal temperament, but its resources are by no meanslimited to unrelieved consonance. Just Intonationalso has the potential to provide more varied and

powerful dissonances than the current system. Thisis the case in part because the simple, consonantintervals can be compounded in a great many waysto yield more complex dissonant intervals and, in

part, because, the consonant intervals being trulyconsonant, the dissonances are rendered that much

sharper in contrast. Further, because dissonances inJust Intonation are the products of concatenations of simpler intervals, consonance and dissonance coexistin a rational framework and their mutual relations arereadily comprehensible.

The virtues of Just Intonation and the shortcom-ings of equal temperament are not limited to theaffective properties of their respective intervals andchords. An equally serious problem with twelve-toneequal temperament is that it supplies composers withan artificially simplified, one-dimensional model of musical relationships. By substituting twelve equallyspaced fixed tones for a potentially unlimited number of tones, interconnected by a web of subtle and com-

plex musical relationships, equal temperament not onlyimpoverished the sonic palette of Western music, butalso deprived composers and theorists of the meansfor thinking clearly about tonal relationships, causingthem to confuse close relationships with remote onesand consonances with dissonances. Not only does Just


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Intonation provide a vast array of superior new musicalresources, but, when properly understood, provides thetools necessary for organizing and manipulating thesegreatly expanded resources.

Just Intonation is not a particular scale, nor is it tiedto any particular musical style. It is, rather, a set of principles which can be applied to a limited number of musically significant intervals to generate an enor-mous variety of scales and chords, or to organize musicwithout reference to any fixed scale. The principles of Just Intonation are applicable to any style of tonal or modal music (or even, if you wish, to atonal styles).Just Intonation is not primarily a tool for improving theconsonance of existing musics, although it can, in somecases, be used this way. Just Intonation can give rise tonew styles and forms of music which, although trulyinnovative, are, unlike those created by the proponents

of the various avant-garde-isms of the twentiethcentury, comprehensible to the ear of the listener aswell as to the intellects of the composer and analyst.Ultimately, Just Intonation is a method for understand-ing and navigating through the boundless reaches of the pitch continuum a method that transcends the musical practices of any particular culture.

Just Intonation has depth and breadth. Its fundamen-tal principles are relatively simple but its ramificationsare vast. At present, the musical realm that comprisesJust Intonation remains largely unexplored. A few pio-neering composers and theorists have sketched some

of its most striking features, but the map still containsmany blank spaces where the adventurous composer may search for new musical treasures.

A Little History

In light of its numerous virtues, why isn t Just Intonationcurrently in general use? Like so many of our peculiarcustoms, this is largely an accident of history. A detailedhistory of tuning in the West would require a book of considerable length in its own right, and is thus farbeyond the scope of the current work. No one has, asyet, written a comprehensive study of this subject. Untilsuch becomes available, the reader is advised to consultHarry Partch s Genesis of a Music , especially ChapterFifteen, A Thumbnail Sketch of the History of Intona-tion, 1 and J. Murray Barbour s Tuning and Tempera-ment .2 The following short sketch is intended only todescribe, in general terms, how musical intonation inthe West achieved its current, peculiar state.

Antiquity

Just Intonation is not a new phenomenon. The basicdiscovery that the most powerful musical intervals areassociated with ratios of whole numbers is lost in antiq-uity. 3, 4 Perhaps it was rst discovered by the priestlymusicians of Egypt or Mesopotamia in the second orthird millennium b.c.e. Some scholars, most notablyErnest G. McClain, regard this discovery as of vitalimportance to the development of mathematics and reli-gion in these ancient societies. The semimythical Greek philosopher Pythagoras of Samos (c. 560 480 b.c.e. )is generally credited with introducing whole-number-ratio tunings for the octave, perfect fourth, and perfectfth into Greek music theory in the sixth century b.c.e. In the generations following Pythagoras, many Greek thinkers devoted a portion of their energies to musicalstudies and especially to scale construction and tuning.

These musical philosophers, known collectively as theharmonists , created a host of different tunings of thevarious Greek scales, which they expressed in the formof whole-number ratios. The discoveries of the Greek harmonists constitute one of the richest sources of tuninglore in the world and continue to this day to exercisea signi cant in uence on Western musical thought.Although most of the original writings of the harmonistshave been lost, much of their work was summarized bythe second century c.e . Alexandrian, Claudius Ptolemy,in his Harmonics . Ptolemy made signi cant contribu-tions in his own right to the eld of music theory, as well

as to astronomy and geography.

The Middle Ages and the Renaissance

Since the time of the Greek harmonists, the idea of simpleratios as the determinants of musical consonance hasnever been wholly absent from Western musical thought.Although much Greek music theory was lost to the Westwith the fall of the Roman Empire, some was retainedand passed on to medieval Europe, primarily through themusical writings of the late Roman philosopher AniciusManlius Severinus Boethius (c. 480 525/6 c.e. ). (Greek music theory was also preserved and further developedin the Islamic sphere, but this does not appear to have hadmuch in uence on musical developments in the West.)Throughout the Middle Ages, Western music was theo-retically based on what is called Pythagorean intonation,a subset of Just Intonation based on ratios composed onlyof multiples of 2 and 3, which will be described in detailin Chapter Three. Pythagorean tuning is characterizedby consonant octaves, perfect fourths, and perfect fths,


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Introduction

based on ratios of the numbers 1, 2, 3, and 4. All otherintervals in Pythagorean tuning are dissonant. Thisproperty is consistent with the musical practice of themiddle ages, in which polyphony was based on fourths,fths, and octaves, with all other intervals, includingthirds and sixths, being treated as dissonances.

In the later Middle Ages and early Renaissance,thirds and sixths were increasingly admitted into poly- phonic music as consonances, and music theory wasgradually modified to account for the existence of theseconsonant intervals, although it appears to have laggedconsiderably behind musical practice. Eventually, theo-rists were forced to partially abandon the Pythagoreanframework of the middle ages in order to explain theexistence of consonant thirds and sixths, because themost consonant possible thirds and sixths are based onratios involving 5. The association of consonant thirds

and sixths with ratios involving 5 was first mentioned by the English monk Walter Odington (c. 1300), butit took a long time for this idea to penetrate the main-stream of musical thought and displace the Pythagoreanintonational doctrinesindeed, it can be argued thatit never wholly succeeded in doing so. In the sixteenthcentury, the rediscovery of Greek writings on music,especially the writings of Ptolemy, gave consider-able added ammunition to the advocates of consonantthirds and sixths based on ratios involving 5. In general,music theorists of the Italian Renaissance came to agreewith the proposition of the Venetian Gioseffe Zarlino

(1517 1590) that consonance was the product of ratiosof the integers 1 6 (the so-called senario ). The ratiosthat define the major and minor tr iads were discoveredin the senario and were acclaimed as the most perfectconcords, thereby setting the stage for the developmentof chordal, harmonic music in the subsequent common practice period.

The Common-Practice Period andthe Rise of Temperament

Alas, while Renaissance theorists considered just inter-vals the foundation of melody and harmony, there wasalso a y in the proverbial ointment, in the form of thegrowth of independent instrumental music featuringxed-pitch fretted and keyboard instruments. The poly-phonic music of the Middle Ages and the Renaissancewas predominantly vocal music and the human voice,when properly trained and coupled to a sensitive ear, isreadily capable of the subtle intonational adjustmentsrequired to perform sophisticated music in Just Intona-

tion. The same can hardly be said for fretted strings orkeyboard instruments. A player of a lute, guitar, or violcan make some expressive adjustment of pitch, it is true,but certainly has not the same degree of exibility as asinger. 5 An organ or harpsichord can produce only thosetones that its pipes or strings have been tuned to. Forreasons that will not be explained here, but which will bemade plain in subsequent chapters, a xed-pitch instru-ment intended to play in perfect Just Intonation in morethan a few closely related keys requires far more thantwelve tones per octave, an arrangement that had alreadybecome standard by the fteenth century. In fact, someexperimental keyboard instruments with far more thantwelve keys per octave were built during the sixteenthand seventeenth centuries but, presumably because of their added cost and complexity, these instruments didnot become popular and the mainstream of musical

thought and activity adopted a different solution to theproblem of intonation on fretted strings and keyboardinstruments: that of temperament .

The basic premise of temperament is that the number of pitches required to play in different keys can bereduced by compromising the tuning of certain tonesso that they can perform different functions in differentkeys, whereas in Just Intonation a slightly different pitchwould be required to perform each function. In other words, temperament compromises the quality of inter-vals and chords in the interest of simplifying instrumentdesign and construction and playing technique. Many

different schemes of temperament were proposed inthe Renaissance and baroque eras, but, at least wherekeyboard instruments were concerned, they eventuallycoalesced into a type of tuning known as meantonetemperament . (According to many writers, equal temperament was always the preferred system for lutes andviols, because it greatly simplified the placement andspacing of the frets.) Meantone temperament aims toachieve perfect major thirds and acceptable major andminor triads in a group of central keys, at the expenseof slightly flatted fifths in those same central keys andsome bad thirds and triads and one very bad fifth inmore remote keys. The exemplary variety of meantonetemperament, called quarter-comma meantone, pro-duced, in a twelve-tone realization, eight good major triads and eight good minor triads, with the remainingfour triads of each type being badly mistuned. 6 Mean-tone tunings satisfied the needs of composers for a time,

but as instrumental music became more complex andthe desire to modulate to more remote keys increased,


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the bad triads became a barrier to progress. As a result,musicians gradually adopted another system, twelve-tone equal temperament.

There is some uncertainty as to who deserves thecredit or blame for the invention of equal temperament.It seems to have been the product of many minds work-ing along convergent lines over a number of decades,if not centuries. 7 The French monk and mathematicianMarin Mersenne (1588 1648) gave an accurate descrip-tion of equal temperament and instructions for tuning iton a variety of instruments in his most important work,the Harmonie Universelle (1639), thereby contributingsubstantially to its popularization, but the practicaladoption of equal temperament, like its invention, was agradual process, occurring at different rates in differentcountries. Equal temperament seems to have first foundfavor for keyboard instruments in Germany, where some

organs were so tuned as early as the last quarter of theseventeenth century, although it was still a subject of debate there seventy-five years later. Meantone seemsstill to have been the predominant system in Francein the mid-eighteenth century, and in England mean-tone continued to be the predominant tuning, at leastfor organs, until the middle of the nineteenth century.The commonly held assumption that J.S. Bach was anadvocate of equal temperament and that he wrote thetwenty-four preludes and fugues of The Well-Tempered Clavier to demonstrate its virtues is at least debatable.The term well temperament was used in the eigh-

teenth century to describe a species of temperament inwhich all keys were usable and in which the principalconsonances of the most central keys often retainedtheir just forms. In well temperaments, different keyshad different characters, depending on their closenessto or remoteness from the key on which the tuning wascentered. This latter characteristic was considered desir-able by many baroque composers and theorists, who

believed that different keys had characteristic colorsand emotional effects.

Twelve-tone equal temperament, unlike meantone,mistunes all consonant intervals except the octave.Also unlike meantone, twelve-tone equal temperamentfavors perfect fifths over thirds. The equally tempered perfect fifth is approximately two cents narrower thanthe just perfect fifth (one cent = 1/100 tempered semitoneor 1/1200 octave), whereas the equally tempered major third is approximately fourteen cents wider than the

just major third, and the equally tempered minor thirdis approximately sixteen cents narrower than the just

minor third. In a sense, the rise of equal temperamentcan be seen as a partial resurgence of the old Pythago-rean doctrine, since the Pythagorean tuning also pro-duced good perfect fifths (and fourths), wide major thirds, and narrow minor thirds. The major advantageof equal temperament over meantone is that every keyin equal temperament is equally good (or equally bad).There is no contrast in consonance between keys, soall twelve tones can serve equally well as keynotes of major or minor scales or as the roots of major or minor triads.

Equal temperament was not adopted because itsounded better (it didnt then and it still doesnt, despitetwo hundred years of cultural conditioning) or becausecomposers and theorists were unaware of the possibilityof Just Intonation. The adoption of twelve-tone equaltemperament was str ictly a matter of expediency. Equal

temperament allowed composers to explore increasinglycomplex chromatic harmonies and remote modulationswithout increasing the complexity of instrument designor the difficulty of playing techniques. These benefits,as we shall see, were not without costs.

Throughout the baroque and classical eras, whilemusic, at least on keyboard instruments, was dominatedfirst by meantone temperament, then by equal tempera-ment, theorists continued to explain musical consonanceas the product of simple, whole-number ratios. Consid-erable advances were made in the scientific understand-ing of sound production by musical instruments and

of the human auditory mechanism during this period.Ironically, Mersenne, who played such a significant rolein the popularization of twelve-tone equal temperament,also first detected and described the presence of theharmonic series in the composite tone of a vibratingstring and in the natural tones of the trumpet. Mersennewas also the first theorist to attribute consonance toratios involving 7, the next step up the harmonic seriesfrom Zarlinos senario. Later theorists, most notablyJean Philippe Rameau (1683 1784), appropriated theharmonic series as further support from nature for the primacy of whole-number ratios as the source of consonance. It apparently did not strike most of thetheorists of the seventeenth and eighteenth centuries as

problematic that, although they formed the theoretical basis for the whole of contemporary harmonic practice,simple-ratio intervals were gradually being purged frommusical practice in favor of tempered approximations.

In the nineteenth century, a vigorous attack on equaltemperament was mounted by Hermann von Helmholtz


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Introduction

(1821 1894), surgeon, physicist, and physiologist, andfather of modern scientific acoustics and psychoacous-tics. Helmholtz considerably advanced scientific under-standing of the production and perception of musicalsound, and proposed the first truly scientific theory of consonance and dissonance. He was a strong advocateof Just Intonation and deplored the effect that equaltemperament had on musical practice, particularly withregard to singing. Contemporary with Helmholtzs stud-ies there was a good deal of interest in the invention of experimental keyboards for Just Intonation (primarilyorgans or harmoniums), particularly in Great Britain.Among those engaged in this activity, the most notablewere General Perronet Thompson, Colin Brown, andR.H.M. Bosanquet. Unfortunately, the proposals of Helmholtz and the other intonational reformers of thenineteenth century appear to have had no detectable

effect on contemporary musical practice, althoughHelmholtzs work, in particular, was to have a signifi-cant inf luence on musicians of subsequent generations.

Nineteenth century composers were still enamoredof the facility for modulation and for the building of increasingly complex harmonies that equal tempera-ment provided, and it was not until these resourceswere exhausted that any alternative was seriouslyconsidered.

The End of Common Practice

Initially, the effect of equal temperament on Western

music was probably bene cial. Composers obtainedthe ability to modulate freely and to build complexchromatic harmonies that had been impossible underthe meantone system. As a result, abstract instrumentalmusic ourished as never before, yielding what is gener-ally considered the golden age of Western music. Likea plant stimulated by chemical fertilizers and growthhormones, music based on equal temperament grewrapidly and luxuriously for a short period then col-lapsed. If equal temperament played a prominent rolein stimulating the growth of harmonic music in thecommon-practice era, it played an equally large part inits rapid demise as a vital compositional style. Twelve-tone equal temperament is a limited and closed system.Once you have modulated around the so-called circle of fths, through its twelve major and twelve minor keys,and once you have stacked up every combination of tones that can reasonably be considered a chord, there isnowhere left to go in search of new resources.

This is essentially where Western composers foundthemselves at the beginning of the twentieth century.Everything that could be done with the equally tempered scale and the principles of tonal harmony had been tried, and the system was breaking down. Thissituation led many composers to the erroneous conclu-sion that consonance, tonality, and even pitch had beenexhausted as organizing principles. What was reallyexhausted was merely the very limited resources of thetempered scale. By substituting twelve equally spacedtones for a vast universe of subtle intervallic relation-ships, the composers and theorists of the eighteenthand nineteenth centuries effectively painted Westernmusic into a corner from which it has not, as yet, extri-cated itself. Twentieth century composers have tried invain to invent or discover new organizing principles as powerful as the common-practice tonal system. Instead,

they have created a variety of essentially arbitrary sys-tems, which, although they may seem reasonable inthe minds of their creators, fail to take into accountthe capabilities and limitations of the human auditorysystem. These systems have resulted in music that thegreat majority of the population find incomprehensibleand unlistenable.

Given that equal temperament had only been in gen-eral use for about 150 years at the time, it may seemstrange that so few of the composers of the early twen-tieth century recognized that the cure for musics illslay, at least in part, in the replacement of its inadequate

tuning system. (Some theorists and composers did, infact, advocate the adoption of new, microtonal tuningsystems, but most of these proposals were for micro-tonal equal temperaments, such as quarter tones, thirdtones, sixth tones, eighth tones, or the like, which merelydivided the existing twelve-tone scale into smaller, arbi-trary intervals, and which made no improvement in thetuning of Western musics most fundamental intervals.)However, despite its fairly recent origin, equal tempera-ment had already become quite deeply entrenched inWestern musical thought and practice. There wereseveral reasons for this. One was the industrial revo-lution. The nineteenth century saw the redesign andstandardization of many instruments, particularly theorchestral woodwinds and brass. Strictly speaking, onlyfixed-pitch instruments (the piano, organ, harp, tuned percussion, and fretted strings) require temperament,the others being sufficiently flexible as to adjust pitchas musical context requires. Nevertheless, brass andwoodwind instruments were also standardized to play


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a chromatic scale such that the centers of their pitchescorresponded as closely as possible to the pitches of twelve-tone equal temperament. Another reason for the persistence of equal temperament was the repertory of the common-practice period. The previous 150 yearshad witnessed the development of the orchestra as weknow it, along with its repertory, and the concert systemthat supported it. It had also seen the evolution of the

piano, the preeminent equally tempered instrument, asthe predominant instrument for both solo performanceand accompaniment, and as the most important tool inmusical education. The orchestra, the piano, and their

players, trained to perform the works of eighteenth andnineteenth century composers, were the resources thatturn- of-the-twentieth-century composers had to use if they wished to have their music performed. And all of these resources were dedicated to music that assumed

equal temperament. It was little wonder, then, thatfew composers were willing to challenge this massiveestablishment in order to work in some new, untestedtuning system.

The Twentieth-CenturyJust Intonation Revival

Although most composers were suf ciently intimidatedby the weight of eighteenth and nineteenth centurymusical practice, fortunately a few were not. The rsttwentieth century composer to make a serious com-mitment to Just Intonation and the person primarily

responsible for the revival of Just Intonation as a viablemusical resource was Harry Partch (1901 1974), theiconoclastic American composer, theorist, instrumentbuilder, dramatist, and musical polemicist. When Partchbegan his compositional career, no one, to the best of myknowledge, was making music in Just Intonation. Begin-ning with tentative experiments in the mid-1920s andcontinuing over a span of fty years, Partch developeda system of Just Intonation with forty-three tones to theoctave, built a large ensemble of predominantly stringedand percussion instruments to play in this tuning system,composed and staged six major musical theater pieces,along with numerous lesser works, and produced anddistributed his own records. In 1947, Partch publishedthe rst edition of his Genesis of a Music , an accountof his musical theories, instruments, and compositionsthat became the bible for subsequent generations of JustIntonation composers.

Whereas in previous centuries the goal of most into-national theorists was to find the ideal or most practical

tuning for a culturally predominant scale, such as amajor, minor, or chromatic scale, the approach of twen-tieth century composers and theorists working with JustIntonation, as exemplified by Partch, has been quite dif-ferent. The goal of these artists has been, in most cases,to discover or create tunings that best served their own

particular musical goals, whether for a single composi-tion or for a lifetimes work, rather than one that couldserve the needs of the culture as a whole.

From when he began work in the mid-1920s untilthe mid-1950s, Partch was the only composer in theUnited States doing significant work in Just Intonation.In the 1950s, Partch was joined by Lou Harrison (b.1917) and Ben Johnston (b. 1926). Harrison first learnedabout Just Intonation from Partchs Genesis of a Music .He composed his first major work in Just Intonation,Four Strict Songs for Eight Baritones and Orchestra ,

in 1954.8

Although, unlike Partch, he does not work exclusively in Just Intonation, Harrison has written alarge body of work in various just tunings. He is prob-ably best known for the creation, in conjunction withhis companion, the late William Colvig, of a number of

justly tuned American gamelan (Indonesian-style tuned percussion ensembles) and for the body of music he hascomposed for this medium, but he has also composed

just music for a great variety of instrumental and vocalensembles, often mixing elements from European andAsian musical traditions. Through his teaching at SanJose State University and Mills College in California

and his extensive lecturing, he has introduced manyyounger composers to Just Intonation.

Ben Johnston discovered the possibility of JustIntonation early in life, when he attended a lectureon Helmholtz at age eleven. Later, he, like Harrison,discovered Partchs Genesis of a Music . Johnston con-tacted Partch and for a six month period in 1950 was hisstudent and apprentice in the remote California coastaltown of Gualala. Johnston began composing seriouslyin Just Intonation in 1959. Unlike Partch and Harrison,Johnstons work in Just Intonation employs mainlyWestern musical forms and instrumental combina-tions. His earlier work, through the early 1970s, gener-ally combines extended microtonal Just Intonation withserial techniques. His later work tends to be simpler andmore tonal, but still uses serialism at least occasionally.Johnstons works include eight string quartets in JustIntonation and numerous vocal and chamber ensemble pieces. He is also the inventor of a system of notation for extended Just Intonation that is used in this primer.


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Introduction

In the 1960s and 1970s, interest in Just Intonationcontinued to slowly increase. La Monte Young (b. 1935)

began working with Just Intonation in the early 1960sin the context of his instrumental/vocal performancegroup, The Theater of Eternal Music . In this ensemble,Young developed the practice of performing long, staticcompositions based on selected tones from the harmonicseries, played on various combinations of amplifiedinstruments and voices. In 1964, Young began work onhis semi-improvisational, justly tuned piano composi-tion, The Well-Tuned Piano , which can be from five toseven hours in duration and which continues to evolveat the time of this wr iting. Young is also known for The

Dream House , a living environment in which a number of electronically generated, harmonically related tonesare sustained over a period of months or years.

Terry Riley (b. 1935), who was a member of Youngs

Theater of Eternal Music at various times in the early1960s, is known primarily as a keyboard composer/improviser. He is perhaps best known as the composer of the early minimalist piece In C (1964), which is notexplicitly a Just Intonation piece, although it has some-times been performed this way. In the 1970s, Riley performed extensively on a modified electronic organtuned in Just Intonation and accompanied by tape delays.More recently, he has been performing his work on justlytuned piano and digital synthesizers, and composing for other ensembles, especially the string quartet.

In the late 1970s and early 1980s the number of

composers working with Just Intonation began toincrease significantly, due in part to the developmentof affordable electronic instruments with programmabletuning capabilities and in part to the coming of age of the post World War II generation of composers. Theachievements of Partch, Harrison, Johnston, Young,and Riley made it evident to these younger composersthat Just Intonation was a valuable resource for composers of diverse styles and tastes, and the availabilityof electronic instruments with programmable tuningmade it possible for the first time for composers toexperiment with a variety of different tuning systemswithout having to invent and build novel instrumentsor to train performers in unusual playing techniques.Changing the pitches available on a digital synthesizer simply means changing the data values in a tuning tableor switching to a different table. If the instrument andits operating software have been designed to facilitatesuch changes, either of these functions can be performedvirtually instantaneously by a computer running appro-

priate software. Hence, a conventional keyboard can beused to play a virtually unlimited number of different

pitches. This capability has, for all intents and purposes,eliminated the condition that first brought temperamentinto being: the necessity of restricting the number of pitches used in music to the number of keys availableon an affordable, playable keyboard.

Among the many composers currently doing sig-nificant work in Just Intonation are William C. Alves,Lydia Ayers, Jon Catler, David Canright, Dean Drum-mond, Cris Forster, Glenn Frantz, Ellen Fullman,Kraig Grady, Michael Harrison, Ralph David Hill,David Hykes, Douglas Leedy, Norbert Oldani, LarryPolansky, Robert Rich, Daniel Schmidt, Car ter Scholz,James Tenney, and Erling Wold. The variety of musicalstyles represented by this group is extremely diverse,and the use of Just Intonation may be the only feature

they all share. Although more than half work primarilyor exclusively with electronic media, they also includeexponents of Partchs tradition of acoustic instrument

building (Drummond and Grady), Lou HarrisonsAmerican gamelan movement (Schmidt), Youngs andRileys improvisational keyboard styles (M. Harrison),a harmonic singer (Hykes), and even a justly tuned rock guitarist (Catler).

The Purpose of this Publication

Although the technical barriers to the composition andperformance of signi cant music in Just Intonation have

been considerably reduced in recent years, barriers of another type remain largely in place, namely the weightof custom and the lack of accessible information on prin-ciples of Just Intonation. The colleges, universities, andconservatories continue to teach a curriculum based onmusic of the common-practice era, in which alternatetunings are unlikely to receive more than a passing men-tion. With the exception of the fortunate few who ndthemselves in institutions with a microtonal composer ortheorist on the faculty, students who develop an interestin these matters are unlikely to receive much supportor encouragement, much less practical instruction, fromthe academic establishment. Such students, if they per-sist, generally nd it necessary to educate themselves,and in the process often have to reinvent or rediscoverprinciples and structures that are well known to moreexperienced composers.

In an attempt to remedy this situation, in the fall of 1984, I and my associates in the experimental musicensemble Other Music, in consultation with a number


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of other West Coast Just Intonation composers andtheorists, founded the Just Intonation Network. TheJust Intonation Network is a nonprofit group fosteringcommunication among composers, musicians, instru-ment designers, and theorists working with Just Intona-tion. Its primary goal is to make information about thetheory and practice of composition in Just Intonationavailable to all who want or need it. The primary methodfor distributing this information is the networks jour-nal, 1 /1 , the only current periodical devoted primarilyor exclusively to Just Intonation. For the past eighteenyears I have served as editor of this publication.

A survey of Just Intonation Network members takenseveral years ago revealed that more than half werenewcomers to the study of Just Intonation who founda significant portion of the articles in 1 /1 over their heads. It was with the goal of assisting these readers

that the Just Intonation Primer was conceived. The Just Intonation Primer , as its title indicates, is not intendedto provide a complete or comprehensive course in the

theory and practice of Just Intonation, let alone tuningin general or other aspects of composition. Its purpose,rather, is to provide the reader with the basic informationand skills necessary to read and comprehend intermedi-ate and advanced texts such as articles in 1 /1 or HarryPartchs Genesis of a Music , and to prepare the reader to begin independent study and composition.

The Primer is intended for readers with at least anelementary knowledge of common-practice Westernmusic theory, including the basic terminology of intervals, chords, and scales, and the fundamentals of harmony. The reader is not assumed to have any prior knowledge of Just Intonation or of alternative tuningsin general, nor is the reader expected to be a mathema-tician or number theorist. The only math required tounderstand this book is basic arithmetic, in combina-tion with some simple procedures explained in Chapter

Three. An inexpensive scientific calculator will proveuseful for comparing the sizes of intervals.


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Index

Index

A

Added-second chord 57Aeolian scale 39Alves, William C. 7American gamelan 6, 7, 66, 72Anomalies 33. See also great diesis,

Pythagorean comma, septimal comma,syntonic comma, etc.

Ars Antiqua period 38Ayers, Lydia 7

B

Bach, J.S. 4Barbershop quartet 35Barbour, J. Murray 2Baroque era 3, 4Basilar membrane 14, 18Bassoon 74Beating harmonics. See Harmonics: beatingBeats. See Harmonics: beating; Interference

beatsBeat frequency (f B) 13

due to the inharmonicity of piano strings 68Benade, Arthur H.

experiment to identify special relationships 22 24

Bethea, Rob 76Blues 35, 51Blue notes 35, 55Boethius, Anicius Manlius Severinus 2Bosanquet, R.H.M. 5, 70Bowed string instruments 75Branca, Glenn 65Brass instruments 19, 74Brown, Colin 5, 70

C

Canright, David 7, 71Carlos, Wendy 73Catler, Jon 7Cents 15

calculation 26dened 4

Chalmers, John H.tritriadic scales 64

Cheng 71Chords 30 32. See also Triads, other chord

namescondissonant 31

consonant 31, 46dissonant 31eleven- and thirteen-limit 62 64ve-limit 46 48

condissonant 47 48seven-limit 56 59

condissonant 58 59subharmonic 57 58

Chromatic scale 45. See also Harmonicduodene

Chromelodeon 70Circle of fths 37Clarinet 19Classical era 4Clavichord 67Clusters. See Tone clustersCochlea 14Cocktail-party effect 16Colvig, William 6, 72Combination tones. See Difference tones;

Summation tonesComma. See Pythagorean comma, septimal

comma, syntonic commaComma of Didymus. See Syntonic commaCommon-practice period 6Common-practice theory 35, 51Complex tones 19 20Continuous-pitch instruments 66, 74 76Critical band 13, 51

D

Difference tones 16 17, 19rst-order 16

of condissonant triads 48of just major triad 46of just minor triad 46 47of major-seventh chord 48of minor-seventh chord 48

higher-order 16 17of dominant-ninth chord 56of dominant-seventh chord 56of subharmonic pentad 58of tempered major third 17

Diminished-seventh chord 49 50, 64Diminished major-seventh chord 65Ditone diatonic. See Pythagorean diatonic scale

Divisions of the Tetrachord (book) 34Dominant-minor-ninth chord 50, 64Dominant-ninth chord 31, 55, 56, 58, 62

subsets of 56 57Dominant-seventh chord 31, 35, 48 49, 51,

55, 56subsets of 56 57

Drummond, Dean 7, 72Duodenarium 46Duodene. See Harmonic duodene

E

Electronic organ 70Ellis, Alexander J. 26, 31, 45, 64, 70Enharmonic equivalents 44Enharmonic spellings 49

Experimental Musical Instruments (periodical)

67F

False Consonance 54 55Fender Rhodes Electric Piano 70Fixed-Pitch Instruments 67 71Fixed scales 42, 55

seven-limit 59

Flute 19, 74Fokker, Adriaan

method for representing intervals on thelattice 43

Formants 19Forster, Cris 7, 72Frantz, Glenn 7French horn 74French impressionists 63Frequency, absolute 11. See also Hertz (Hz)

calculating 28of 1/1 36

Frequency ratios 11, 25 29. See also JustIntervals

calculations with 25 29addition 25complementation 26converting to cents 26division 26 28subtraction 25 26

converting to harmonic or subharmonic seriessegment 28 29

only unambiguous inerval names 32rules and conventions for using 25superparticular 27

Fretted instruments 3, 71Fullman, Ellen 7Fused tone 13, 14

G

Galilei, Vincenzoproposed fretting for the lute 64

Genesis of a Music 2, 6, 67, 70Grady, Kraig 7, 72Great diesis 44 45, 50Guitar 71

H

Hair cells 14Half-diminished-seventh chord 55, 56 57Harmonics 15, 16. See also Harmonic series;

Partials: harmonicbeating 20 24

of an octave (2:1) 20 21of a perfect fth (3:2) 21 22of a unison (1:1) 20

coincident (matching) 20 24formula for 21of just intervals ( gure) 21of just major triad 46of just minor triad 46

Harmonic and subharmonic series ( gure) 29Harmonic duodene 45 46

Harmonic series 4, 14 16, 18, 19, 28 30dened 14fundamental of 15, 19, 31

missing. See Periodicity pitchHarmonic spectrum 15, 19

cutoff frequency of 19of strings 19

Harmonists (ancient Greek theorists) 2Harp 70, 72


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Harpsichord 67, 72Harrison, Lou 6, 66, 72, 75

Concerto for Piano with Javanese Gamelan 69

Concerto for Piano with Selected Orchestra69, 76

Four Strict Songs for Eight Baritones and Orchestra 6, 76

The Tomb of Charles Ives 76Harrison, Michael 7, 69

From Ancient Worlds 68Heim, Bruce 74Helmholtz, Hermann von 4 5, 45, 70Heptad 62, 63Hertz (Hz) 10, 11. See also Periodic vibration:

frequencyHexad 62, 63Hill, Ralph David 7, 54Hohner Clavinet 70Hykes, David 7

I

Identitiesas de ning characteristics of a chord 30, 31dened 30

Idiophones 19, 72 73Industrial revolution 5Interference Beats 12 13, 46. See

also Harmonics: beatingIntermodulation products. See Difference tones,

Summation tonesInterval 10 11. See also Frequency ratios; Just

intervalsconventional names for 10, 32dened 10harmonic vs subharmonic interpretation 28

Intonation Systems 71

J

Jazz 35, 51Johnston, Ben 6, 75

notation for Just Intonation 33ratio scales 64remarks on 11 and 13 61Sonata for Microtonal Piano 70

Just Intervals1:1 (unison) 362:1 (octave) 36, 37, 443:1 (perfect twelfth) 243:2 (perfect fth) 36, 37, 38, 39, 40, 42, 45,

46, 51, 54, 55, 57, 594:1 (double octave) 244:3 (perfect fourth) 33, 36, 37, 38, 40, 42,

54, 555:1 245:2 (major tenth) 245:3 (major sixth) 24, 395:4 (major third) 38, 39, 40, 42, 45, 46, 50,

51, 55, 56, 57, 586:1 246:5 (minor third) 24, 39, 45, 49, 507:1 247:2 24, 517:3 (septimal minor or subminor tenth) 24, 51

7:4 (harmonic, subminor, or septimal minorseventh) 35, 51

7:5 (most consonant tritone) 23, 51, 55, 577:6 (subminor or septimal minor third) 23,

51, 55, 57, 58, 598:1 (triple octave) 248:3 248:5 (minor sixth) 24, 39, 40, 41, 42, 50, 558:7 (septimal whole tone; supermajor second)

23, 24, 51, 54, 55, 579:4 (major ninth) 24, 379:5 (acute minor seventh) 40, 49, 56, 579:7 (supermajor third) 55, 56, 57, 58, 599:8 (major whole tone) 33, 37, 38, 40, 55,

64, 6510:3 2410:7 (septimal tritone) 51, 5510:9 (minor whole tone) 40, 5511:1, 11:2, 11:3, 11:4, 11:5 2411:6 6211:8 (primary interval for 11) 61, 6211:9 (neutral third) 6211:10 6212:5 (minor tenth) 2412:7 5112:11 6213:1, 13:2, 13:3, 13:4, 13:5, 13:6 2413:8 (primary interval for 13) 61, 6213:9 6213:10 6215:8 (diatonic major seventh) 4015:13 6215:14 54, 5516:5 2416:9 (minor seventh) 37, 4916:11 6216:13 6216:15 (diatonic semitone) 40, 45, 5517:16 6417:16 (primary interval for 17) 6418:11 6218:17 6419:16 (primary interval for 19) 6421:16 (septimal subfourth) 54, 5521:20 54, 5525:16 5025:24 (small chromatic semitone) 45, 5527:16 (Pythagorean major sixth) 3727:20 (acute, imperfect, or wolf fourth) 4027:25 (great limma) 4528:25 (intermediate septimal whole tone) 5428:27 54, 5532:21 54, 5532:25 (diminished fourth) 45, 5032:27 (Pythagorean minor third) 37, 41, 49

33:32 6235:32 (small septimal whole tone) 5436:25 (acute diminished fth) 4936:35 (septimal quarter tone or diesis) 5440:27 (grave, imperfect, or wolf fth) 40,

4149:48 (septimal sixth tone) 5450:49 (septimal sixth tone) 5464:63. See Septimal comma (64:63)65:64 6275:64 (augmented second) 45

81:64 (Pythagorean major third, ditone) 36,37, 38, 40, 41

81:80. See Syntonic comma128:81 (Pythagorean minor sixth) 37, 40128:125. See Great diesis135:128 (large limma, large chromatic

semitone) 45243:128 (Pythagorean major seventh) 40256:243 (Pythagorean limma) 37, 40

Just Intonationdened 1 2discovery 2eleven- and thirteen-limit 59 63ve-limit 35, 38 50, 75

dif ciencies of 48 50obtaining satisfactory performances in 79 80seven-limit 51 59

coloristic use of 55seventeen- and ninteen-limit 64 65three-limit 36 38twentieth-century revival 6 7

Just Intonation Network 8, 78

K

Keyboard instruments 3, 4, 67 70Acoustic 67 70electroacoustic 70

Koto 71

L

Latticecomplementation on 44ve-limit 42 44

gure 43seven-limit 51

gure 52, 53syntonon diatonic ( gure) 40transposition on 43

Leedy, Douglas 7Limit of frequency discrimination 13Lute 71

M

Major-ninth chord 48Major-seventh chord 48Major scale

ve-limitconstructing 39 40

Maxwell, Miles 54McClain, Ernest G. 2Mersenne, Marin 4Microtones. See also Just Intervals

eleven- and thirteen-limit 62seven-limit 54Middle Ages 38MIDI 76 79

keyboard controller 78pitch bend 76 77

using with tuning tables 78Tuning-Dump Speci cation 78 79

Minor-major-seventh chord 65Minor-ninth chord 48Minor-seventh chord 48


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Index

Minor-sixth chord 64Missing fundamental. See Periodicity pitch

N

Neutral intervals 62Nondominant-ninth chords 47Nondominant-seventh chords 47Notation for Just Intonation 32 33

Ben Johnston s 33, 62Table 32

Novatone 71

O

Oboe 74Odington, Walter 3Oldani, Norbert 7Oliveros, Pauline 701/1 (Journal of the Just Intonation Network) 8One (foundation of all just tunings) 36On the Sensations of Tone (book) 45Orchestra 6Other Music 7Otonalities 29, 47. See also Harmonic seriesOvertones. See Harmonics; Harmonic series;

Partials

P

Partch, Harry 2, 6, 47, 58, 61, 62, 66, 67, 70,71, 72

tonality diamond 64Partials 15. See also Harmonics; Harmonic

seriesharmonic 15, 19 20inharmonic 15, 19

of a free rectangular bar 72of a hinged bar 68of piano strings 67 69

Periodicity pitch 17 19, 20of eleven- and thirteen-limit dyads 61of just major triad 46of just minor triad 46 47of subharmonic pentad 58of subminor triad 57

Periodic vibration 9 10, 15amplitude 10frequency 10graph of 9period ( ) 10, 16

Phase Relationship 12Piano 6, 35, 67 70

single-strung 68tuning strategies 69

Pipe organ 70Pitch 10. See also Periodic vibration: frequency Pitch (periodical) 74Place theory 18Polansky, Larry 7Primary interval

dened 30for eleven (11:8) 61for ve (5:4) 38for ninteen (19:16) 64for seventeen (17:16) 64

for seven (7:4) 51for thirteen (13:8) 61for three (3:2) 36for two (2:1) 36

Prime limit 27, 35dened 30

Prime numbers 35dened 30

Psaltery 70, 72Ptolemy, Claudius 2, 3, 39Ptolemy s syntonon diatonic 33, 59. See

also Major scale: ve-limitcomplementation ( gure) 44relative minor of 41tuning method ( gure) 39

Pure tones 10, 15superposition of 11 12

Pythagoras of Samos 2, 36Pythagorean comma 37 38, 44Pythagorean diatonic scale 36 37, 40Pythagorean tuning 2 3, 4, 36 38. See

also Just Intonation, three-limittendency of string players for 75

Q

Quartertone scale. See Temperament: twenty-four-tone equal

R

Rameau, Jean Philippe 4Ratios. See Frequency ratios; Just IntervalsRatios of eleven

table 61Ratios of thirteen

table 62Recorder 74Reed instruments 19Reed organ 70Renaissance 3Residue tone. See Periodicity pitchResultant tones. See Difference tones,

Summation tonesRich, Robert 7, 78Riley, Terry 7, 69, 70, 75

In C 7Root (of a chord) 31Rosenthal, David 72Roughness 13, 20. See also Interference Beats

S

Sackbut 76Sawtooth wave 16

Saxophone 74Schmidt, Daniel 7, 72Scholz, Carter 7, 78Senario 3, 4Septimal comma (64:63) 54, 65Seven

consonance of 357 b5 chord 647 m5 chord 65Siegel, Jules 77

Simple harmonic motion 9. See also Sine waveSimple tone 10Sine wave 9, 10Singing. See VoiceSpecial relationships 22 24

absence of among eleven- and thirteen-limitdyads 59

beyond the octave 23 24dened 22involving higher harmonics 24table 23

Sruti box 76Stone, Tom 71String players

natural tendencies of 74use of vibrato 75

Subharmonic series 28 30Subjective pitch. See Periodicity pitchSubminor-seventh chord 58Summation tones 16, 17

rst-order 17Supermajor-seventh chord 59Supertonic problem 40 42Synthesizers. See also MIDI

with user-programmable tuning tables 7,77 78

Yamaha DX/TX 77, 79Syntonic comma 39, 41, 42, 44, 71Syntonon diatonic. See Ptolemy s syntonon

diatonic

T

Tamboura 76Temperament

dened 3meantone 3, 35, 42twelve-tone equal 4, 11, 33, 35, 37, 42, 49

bene ts 5deviation of harmonic series from 15

twenty-four-tone equal 62well temperament 4, 35

Tempered major third 17, 23Tenney, James 7Tetrachordal genera 34, 59Tetrachordal scales

equal 34, 37, 40mixed 34seven-limit 59

(gure) 60Tetrachords 33 34, 55

dened 33disjunct 34permutations 34

Tetrad. See also Dominant-seventh chord,

Major-seventh chord, Minor-seventhchordsubharmonic 58

Thompson, General Perronet 5, 71Tied frets 71Timbre 10, 19Timing theory 18Tone clusters 63

eleven- and thirteen-limit 63seven-limit 56


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Transposition 11Triads

augmented 48, 49 50, 65condissonant 47 48diminished 31, 35, 48, 51, 55, 56

subharmonic 58dominant 395:7:9 574:6:7 57

subharmonic 58major 31, 39, 46

voicings in C (table) 30minor 31, 39, 41, 46

proposed tuning with 19:16 64 65supertonic 41, 42

open 38Pythagorean 38

major 46minor 46

subdominant 39subminor 31, 55, 57supermajor 58tonic 39

Trombone 76

TuningHistory 2 7

Tuning and Temperament (book) 2Twelfth root of two 11

U

Utonalities 29, 47, 58. See also Subharmonicseries

V

Viol 71Virtual pitch. See Periodicity pitchVogt, Walter 71Voice 76

W

The Well-Tempered Clavier 4Widener, Ward 64, 65Wilson, Ervin

combination product sets 64Wind instruments 73 74Wold, Erling 7

Y

Young, La Monte 7, 61, 69Dream Chords 38, 64, 65The Dream House 75 V 67 6:38 PM NYC 65The Four Dreams of China 64, 76The Lower Map of The Elevens Division in

The Romantic Symmetry (over a 60 cyclebase) in Prime Time from 122 to 119with 144 65

The Romantic Symmetry in Prime Time from112 to 144 with 119 65

The Second Dream of the High Tension LineStep-Down Transformer 64, 76

The Subsequent Dreams of China 64The Theater of Eternal Music 7The Well-Tuned Piano 7, 65, 68

Z

Zarlino, Gioseffe 3Zither 70

primersample

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