Quick viewing(Text Mode)

JI Primer Ed.3

JI Primer Ed.3

THE PPRIRIMMEERR An Introduction to the Theory and Practice of Just Intonation by David B. Doty

Third Edtion, December 2002 ISBN 0-9726810-0-0 ©1993, 1994, 2002 The Just Intonation Network A Project of Other Music, Inc. 535 Stevenson Street San Francisco, CA 94103 Phone: (415) 864-8123 Fax: (415) 864-8726 www.justintonation.net [email protected] iii

Contents The Series and the Subharmonic Series Preface to the Edition (2002)...... iv ...... 28 Preface to the Second Edition (1994)...... iv Converting to a Harmonic or Introduction ...... 1 Subharmonic Series Segment ...... 28 Identities...... 30 What is Just Intonation? ...... 1 Prime Numbers, Primary Intervals, A Little History...... 2 and Prime Limits ...... 30 Antiquity ...... 2 What is a Chord? ...... 30 The Middle Ages and the ...... 2 Interval Names ...... 32 The Common-Practice Period Notation...... 32 and the Rise of Temperament ...... 3 Anomalies...... 33 The End of Common Practice...... 5 and Tetrachordal Scales ...... 33 The Twentieth-Century Just Intonation Revival The Ladder of Primes, Part One:Two, ...... 6 Three, and Five...... 35 The Purpose of this Publication...... 7 One, the Foundation ...... 36 Acoustic and Psychoacoustic Background...... 9 Two, the Empty Matrix ...... 36 The Three Limit ()...... 36 Introduction ...... 9 The Pythagorean () .. 36 Periodic Vibrations ...... 9 Pythagorean Chromatic Scales What is an Interval?...... 10 and the ...... 37 Superposition of Pure Tones...... 11 Chords...... 38 Phase Relationships ...... 12 The Five Limit...... 38 Interference Beats ...... 12 Constructing a Five-Limit ...... 39 The Harmonic Series...... 14 The Problem ...... 40 Difference Tones...... 16 The Five-Limit Lattice...... 42 Periodicity Pitch ...... 17 The Great ...... 44 Complex Tones with Harmonic Partials...... 19 The Harmonic Duodene...... 45 Coincident or Beating ...... 20 Five Limit Chords...... 46 Arthur H. Benade’s “Special Relationships” Condissonant Chords in the Five Limit ...... 47 ...... 22 Defi ciencies of the Five Limit ...... 48 Special Relationships Beyond the .. 23 The Ladder of Primes, Part Two: On the Consonance of Relationships Seven and Beyond...... 51 Involving Higher Harmonics ...... 24 Seven-Limit Intervals...... 51 Basic Definitions, Conventions, and Procedures ...... 25 New Consonances...... 51 Whole Tones ...... 54 Introduction ...... 25 ...... 54 Rules and Conventions...... 25 Microtones ...... 54 Calculations with Ratios...... 25 False Consonances...... 54 Addition ...... 25 Two Dimensional Planes in the Subtraction ...... 25 Seven-Limit Fabric ...... 55 Complementation ...... 26 Seven Limit Chords...... 56 Converting Ratios to Cents ...... 26 Subsets of the Dominant- Dividing Just Intervals ...... 26 and Chords...... 56 Calculating Absolute (Hz) ..... 28 ...... 56

Contents iv

Half-Diminished ...... 57 Electronic Organs ...... 70 “4-6-7” Chord ...... 57 Stringed Instruments with Seven-Limit Subharmonic Chords...... 57 One String per Note ...... 70 Condissonant Chords in the Seven Limit ... 58 Fretted Instruments ...... 71 Fixed Scales in the Seven Limit...... 59 Wind Instruments ...... 73 The Higher Primes: Eleven, Thirteen, Continuous-Pitch Instruments ...... 74 and Beyond...... 59 “Natural Tendencies’’...... 74 Eleven and Thirteen ...... 59 Bowed Strings...... 75 Primary Intervals...... 61 Voices...... 76 Chords and ...... 62 ...... 76 Seventeen and Nineteen...... 64 Midi Synthesizers...... 76 Beyond Nineteen...... 65 Pitch Bend...... 76 Practical Just Intonation with Real Instruments with User-Programmable Instruments ...... 66 Tuning Tables...... 77 Tuning Tables plus Pitch Bend ...... 78 Introduction ...... 66 The Midi Tuning-Dump Specifi cation...... 78 Fixed-Pitch Instruments ...... 67 Some Thoughts on Obtaining Acoustic Keyboards...... 67 Satisfactory Performances...... 79 Piano Problems ...... 67 Reed Organs...... 70 Notes...... 81 Pipe Organs...... 70 Index ...... 85 Electroacoustic Keyboards ...... 70

Preface to the Third Edition (2002) Intonation Primer; any future editions will be in the The third edition of The Just Intonation Primer contains form of multimedia products. There seems to be little no signifi cant changes in content from the previous edi- purpose in merely writing about unfamiliar intervals, tions, though a few minor errors have been corrected. scales, and chords when it is now possible to present The type, however, has been completely reset and all of students with the actual sounds. the art has been recreated from scratch. Why? The fi rst Thanks to Lucy A. Hudson for creating software and second editions of The Primer were produced with that simplified the conversion of the text files from the software that has long since become obsolete, necessitat- previous edition. ing the resetting of the type. As for the art, both my skills and the available tools have greatly improved since the Preface to the Second Edition (1994) production of the previous editions, and my standards, The writing and publication of The Just Intonation as well, have risen considerably. Primer were made possible by a gift from a friend of This is not to say that I regard The Primer, as it now the Just Intonation Network. stands, as needing no improvement — quite the contrary. The material in this publication was excerpted The problem is that once I began making improvements, from a larger work in progress, tentatively entitled A there is no telling where the process would end. This Composer’s Guide to Just Intonation. would be contrary to my purpose for creating this third Thanks to Carola B. Anderson and Dudley Duncan edition: to keep the publication available until I am able for proofreading and commenting on The Just Intona- to complete the work that will supersede it. This is tion Primer. Thanks to Jim Horton for the use of his intended to be the last “print-only” edition of The Just collection of obscure publications.

The Just Intonation Primer introduction

What is Just Intonation? They are what the human auditory system recognizes as What is Just Intonation? Although, like most compos- consonance, if it ever has the opportunity to hear them ers working with this unfamiliar tuning system, I am in a musical context. Although the importance of these frequently asked this troublesome question, I have yet to whole-number ratios is recognized both by musical devise an answer that is suitable for casual conversation. tradition and by modern acoustic and psychoacoustic Technically, Just Intonation is any system of tuning in research, for the last two hundred years Western music which all of the intervals can be represented by whole- has been burdened with a tuning system in which all number ratios, with a strongly implied pref- of the supposed consonances, with the exception of the erence for the simplest ratios compatible with a given octave, deviate significantly from their optimal, integer- musical purpose. Unfortunately this defi nition, while forms. Indeed, some consonant intervals are so concise and accurate, is more likely to result in a glazed compromised in twelve-tone that expression indicative of confusion than in the gleam of they are hardly represented at all. understanding. It is, in short, a defi nition that is perfectly Just Intonation provides a greater variety and clear to the comparative few who have the background superior quality of consonances and concords than to understand it and who could, therefore, formulate it equal temperament, but its resources are by no means for themselves, and perfectly opaque to everyone else, limited to unrelieved consonance. Just Intonation including, unfortunately, most trained musicians. (It is also has the potential to provide more varied and my experience that most musicians are as ignorant of the powerful dissonances than the current system. This details of twelve-tone equal temperament, the predomi- is the case in part because the simple, consonant nant tuning system in Western cultures for the past two intervals can be compounded in a great many ways hundred years, as they are of Just Intonation. If you doubt to yield more complex dissonant intervals and, in this, ask the next dozen musicians you meet to explain part, because, the consonant intervals being truly why there are twelve semitones in a and consonant, the dissonances are rendered that much how to accurately tune those twelve equal semitones.) sharper in contrast. Further, because dissonances in A detailed answer that incorporates all the necessary Just Intonation are the products of concatenations of background on the physics of sound, the physiology and simpler intervals, consonance and dissonance coexist psychology of human hearing, the history of music, and in a rational framework and their mutual relations are the mathematics of tuning systems, far exceeds the limits readily comprehensible. of casual conversation. It could, in fact, fi ll a book. The virtues of Just Intonation and the shortcom- A formal definition of Just Intonation may be ings of equal temperament are not limited to the difficult for the novice to grasp, but the aesthetic affective properties of their respective intervals and experience of just intervals is unmistakable. Although chords. An equally serious problem with twelve-tone it is difficult to describe the special qualities of just equal temperament is that it supplies composers with intervals to those who have never heard them, words an artificially simplified, one-dimensional model of such as clarity, purity, smoothness, and stability come musical relationships. By substituting twelve equally readily to mind. The supposedly consonant intervals spaced fixed tones for a potentially unlimited number and chords of equal temperament, which deviate from of tones, interconnected by a web of subtle and com- simple ratios to varying degrees, sound rough, restless, plex musical relationships, equal temperament not only or muddy in comparison. impoverished the sonic palette of Western music, but The simple-ratio intervals upon which Just Intona- also deprived composers and theorists of the means tion is based are “special relationships” that the human for thinking clearly about tonal relationships, causing auditory system is able to detect and distinguish from them to confuse close relationships with remote ones one another and from a host of more complex stimuli. and consonances with dissonances. Not only does Just

Introduction 2

Intonation provide a vast array of superior new musical Antiquity resources, but, when properly understood, provides the Just Intonation is not a new phenomenon. The basic tools necessary for organizing and manipulating these discovery that the most powerful musical intervals are greatly expanded resources. associated with ratios of whole numbers is lost in antiq- Just Intonation is not a particular scale, nor is it tied uity.3, 4 Perhaps it was fi rst discovered by the priestly to any particular musical style. It is, rather, a set of musicians of Egypt or Mesopotamia in the second or principles which can be applied to a limited number third millennium b.c.e. Some scholars, most notably of musically significant intervals to generate an enor- Ernest G. McClain, regard this discovery as of vital mous variety of scales and chords, or to organize music importance to the development of mathematics and reli- without reference to any fixed scale. The principles of gion in these ancient societies. The semimythical Greek Just Intonation are applicable to any style of tonal or philosopher Pythagoras of Samos (c. 560 – 480 b.c.e.) modal music (or even, if you wish, to atonal styles). is generally credited with introducing whole-number- Just Intonation is not primarily a tool for improving the ratio tunings for the octave, , and perfect consonance of existing musics, although it can, in some fi fth into Greek in the sixth century b.c.e. cases, be used this way. Just Intonation can give rise to In the generations following Pythagoras, many Greek new styles and forms of music which, although truly thinkers devoted a portion of their energies to musical innovative, are, unlike those created by the proponents studies and especially to scale construction and tuning. of the various “avant-garde-isms” of the twentieth These musical philosophers, known collectively as the century, comprehensible to the ear of the listener as harmonists, created a host of different tunings of the well as to the intellects of the composer and analyst. various Greek scales, which they expressed in the form Ultimately, Just Intonation is a method for understand- of whole-number ratios. The discoveries of the Greek ing and navigating through the boundless reaches of the harmonists constitute one of the richest sources of tuning pitch continuum — a method that transcends the musical lore in the world and continue to this day to exercise practices of any particular culture. a signifi cant infl uence on Western musical thought. Just Intonation has depth and breadth. Its fundamen- Although most of the original writings of the harmonists tal principles are relatively simple but its ramifications have been lost, much of their work was summarized by are vast. At present, the musical realm that comprises the second century c.e. Alexandrian, Claudius Ptolemy, Just Intonation remains largely unexplored. A few pio- in his Harmonics. Ptolemy made signifi cant contribu- neering composers and theorists have sketched some tions in his own right to the fi eld of music theory, as well of its most striking features, but the map still contains as to astronomy and geography. many blank spaces where the adventurous composer may search for new musical treasures. The Middle Ages and the Renaissance Since the time of the Greek harmonists, the idea of simple A Little History ratios as the determinants of musical consonance has In light of its numerous virtues, why isn’t Just Intonation never been wholly absent from Western musical thought. currently in general use? Like so many of our peculiar Although much Greek music theory was lost to the West customs, this is largely an accident of history. A detailed with the fall of the Roman Empire, some was retained history of tuning in the West would require a book of and passed on to medieval Europe, primarily through the considerable length in its own right, and is thus far musical writings of the late Roman philosopher Anicius beyond the scope of the current work. No one has, as Manlius Severinus Boethius (c. 480 – 525/6 c.e.). (Greek yet, written a comprehensive study of this subject. Until music theory was also preserved and further developed such becomes available, the reader is advised to consult in the Islamic sphere, but this does not appear to have had ’s , especially Chapter much infl uence on musical developments in the West.) Fifteen, “A Thumbnail Sketch of the History of Intona- Throughout the Middle Ages, Western music was theo- tion,” 1 and J. Murray Barbour’s Tuning and Tempera- retically based on what is called Pythagorean intonation, ment.2 The following short sketch is intended only to a subset of Just Intonation based on ratios composed only describe, in general terms, how musical intonation in of multiples of 2 and 3, which will be described in detail the West achieved its current, peculiar state. in Chapter Three. Pythagorean tuning is characterized by consonant , perfect fourths, and perfect fi fths,

The Just Intonation Primer 3

b‚ased on ratios of the numbers 1, 2, 3, and 4. All other tion. The same can hardly be said for fretted strings or intervals in Pythagorean tuning are dissonant. This keyboard instruments. A player of a lute, , or viol property is consistent with the musical practice of the can make some expressive adjustment of pitch, it is true, middle ages, in which polyphony was based on fourths, but certainly has not the same degree of fl exibility as a fi fths, and octaves, with all other intervals, including singer.5 An organ or harpsichord can produce only those thirds and sixths, being treated as dissonances. tones that its pipes or strings have been tuned to. For In the later Middle Ages and early Renaissance, reasons that will not be explained here, but which will be thirds and sixths were increasingly admitted into poly- made plain in subsequent chapters, a fi xed-pitch instru- phonic music as consonances, and music theory was ment intended to play in perfect Just Intonation in more gradually modified to account for the existence of these than a few closely related keys requires far more than consonant intervals, although it appears to have lagged twelve tones per octave, an arrangement that had already considerably behind musical practice. Eventually, theo- become standard by the fi fteenth century. In fact, some rists were forced to partially abandon the Pythagorean experimental keyboard instruments with far more than framework of the middle ages in order to explain the twelve keys per octave were built during the sixteenth existence of consonant thirds and sixths, because the and seventeenth centuries but, presumably because of most consonant possible thirds and sixths are based on their added cost and complexity, these instruments did ratios involving 5. The association of consonant thirds not become popular and the mainstream of musical and sixths with ratios involving 5 was first mentioned thought and activity adopted a different solution to the by the English monk Walter Odington (c. 1300), but problem of intonation on fretted strings and keyboard it took a long time for this idea to penetrate the main- instruments: that of temperament . stream of musical thought and displace the Pythagorean The basic premise of temperament is that the number intonational doctrines—indeed, it can be argued that of pitches required to play in different keys can be it never wholly succeeded in doing so. In the sixteenth reduced by compromising the tuning of certain tones century, the rediscovery of Greek writings on music, so that they can perform different functions in different especially the writings of Ptolemy, gave consider- keys, whereas in Just Intonation a slightly different pitch able added ammunition to the advocates of consonant would be required to perform each function. In other thirds and sixths based on ratios involving 5. In general, words, temperament compromises the quality of inter- music theorists of the Italian Renaissance came to agree vals and chords in the interest of simplifying instrument with the proposition of the Venetian Gioseffe Zarlino design and construction and playing technique. Many (1517 – 1590) that consonance was the product of ratios different schemes of temperament were proposed in of the integers 1 – 6 (the so-called senario ). The ratios the Renaissance and baroque eras, but, at least where that define the triads were discovered keyboard instruments were concerned, they eventually in the senario and were acclaimed as the most perfect coalesced into a type of tuning known as meantone concords, thereby setting the stage for the development temperament. (According to many writers, equal tem- of chordal, harmonic music in the subsequent “common perament was always the preferred system for lutes and practice” period. viols, because it greatly simplified the placement and spacing of the frets.) aims to The Common-Practice Period and achieve perfect major thirds and acceptable major and the Rise of Temperament minor triads in a group of central keys, at the expense Alas, while Renaissance theorists considered just inter- of slightly flatted fifths in those same central keys and vals the foundation of melody and harmony, there was some bad thirds and triads and one very bad in also a fl y in the proverbial ointment, in the form of the more remote keys. The exemplary variety of meantone growth of independent instrumental music featuring temperament, called quarter-comma meantone, pro- fi xed-pitch fretted and keyboard instruments. The poly- duced, in a twelve-tone realization, eight good major phonic music of the Middle Ages and the Renaissance triads and eight good minor triads, with the remaining was predominantly vocal music and the human voice, four triads of each type being badly mistuned.6 Mean- when properly trained and coupled to a sensitive ear, is tone tunings satisfied the needs of composers for a time, readily capable of the subtle intonational adjustments but as instrumental music became more complex and required to perform sophisticated music in Just Intona- the desire to modulate to more remote keys increased,

Introduction 4

the bad triads became a barrier to progress. As a result, . In a sense, the rise of equal temperament musicians gradually adopted another system, twelve- can be seen as a partial resurgence of the old Pythago- tone equal temperament. rean doctrine, since the Pythagorean tuning also pro- There is some uncertainty as to who deserves the duced good perfect fifths (and fourths), wide major credit or blame for the invention of equal temperament. thirds, and narrow minor thirds. The major advantage It seems to have been the product of many minds work- of equal temperament over meantone is that every key ing along convergent lines over a number of decades, in equal temperament is equally good (or equally bad). if not centuries.7 The French monk and mathematician There is no contrast in consonance between keys, so (1588 – 1648) gave an accurate descrip- all twelve tones can serve equally well as keynotes of tion of equal temperament and instructions for tuning it major or minor scales or as the roots of major or minor on a variety of instruments in his most important work, triads. the Harmonie Universelle (1639), thereby contributing Equal temperament was not adopted because it substantially to its popularization, but the practical sounded better (it didn’t then and it still doesn’t, despite adoption of equal temperament, like its invention, was a two hundred years of cultural conditioning) or because gradual process, occurring at different rates in different composers and theorists were unaware of the possibility countries. Equal temperament seems to have first found of Just Intonation. The adoption of twelve-tone equal favor for keyboard instruments in Germany, where some temperament was strictly a matter of expediency. Equal organs were so tuned as early as the last quarter of the temperament allowed composers to explore increasingly seventeenth century, although it was still a subject of complex chromatic and remote modulations debate there seventy-five years later. Meantone seems without increasing the complexity of instrument design still to have been the predominant system in France or the difficulty of playing techniques. These benefits, in the mid-eighteenth century, and in England mean- as we shall see, were not without costs. tone continued to be the predominant tuning, at least Throughout the baroque and classical eras, while for organs, until the middle of the nineteenth century. music, at least on keyboard instruments, was dominated The commonly held assumption that J.S. Bach was an first by meantone temperament, then by equal tempera- advocate of equal temperament and that he wrote the ment, theorists continued to explain musical consonance twenty-four preludes and fugues of The Well-Tempered as the product of simple, whole-number ratios. Consid- Clavier to demonstrate its virtues is at least debatable. erable advances were made in the scientific understand- The term “ ” was used in the eigh- ing of sound production by musical instruments and teenth century to describe a species of temperament in of the human auditory mechanism during this period. which all keys were usable and in which the principal Ironically, Mersenne, who played such a significant role consonances of the most central keys often retained in the popularization of twelve-tone equal temperament, their just forms. In well temperaments, different keys also first detected and described the presence of the had different characters, depending on their closeness harmonic series in the composite tone of a vibrating to or remoteness from the key on which the tuning was string and in the natural tones of the trumpet. Mersenne centered. This latter characteristic was considered desir- was also the first theorist to attribute consonance to able by many baroque composers and theorists, who ratios involving 7, the next step up the harmonic series believed that different keys had characteristic colors from Zarlino’s senario. Later theorists, most notably and emotional effects. Jean Philippe Rameau (1683 – 1784), appropriated the Twelve-tone equal temperament, unlike meantone, harmonic series as further support from “nature” for mistunes all consonant intervals except the octave. the primacy of whole-number ratios as the source of Also unlike meantone, twelve-tone equal temperament consonance. It apparently did not strike most of the favors perfect fifths over thirds. The equally tempered theorists of the seventeenth and eighteenth centuries as is approximately two cents narrower than problematic that, although they formed the theoretical the just perfect fifth (one cent = 1/100 tempered basis for the whole of contemporary harmonic practice, or 1/1200 octave), whereas the equally tempered major simple-ratio intervals were gradually being purged from third is approximately fourteen cents wider than the musical practice in favor of tempered approximations. just , and the equally tempered minor third In the nineteenth century, a vigorous attack on equal is approximately sixteen cents narrower than the just temperament was mounted by Hermann von Helmholtz

The Just Intonation Primer 5

(1821 – 1894), surgeon, physicist, and physiologist, and This is essentially where Western composers found father of modern scientific acoustics and psychoacous- themselves at the beginning of the twentieth century. tics. Helmholtz considerably advanced scientific under- Everything that could be done with the equally tem- standing of the production and perception of musical pered scale and the principles of tonal harmony had sound, and proposed the first truly scientific theory of been tried, and the system was breaking down. This consonance and dissonance. He was a strong advocate situation led many composers to the erroneous conclu- of Just Intonation and deplored the effect that equal sion that consonance, , and even pitch had been temperament had on musical practice, particularly with exhausted as organizing principles. What was really regard to singing. Contemporary with Helmholtz’s stud- exhausted was merely the very limited resources of the ies there was a good deal of interest in the invention of tempered scale. By substituting twelve equally spaced experimental keyboards for Just Intonation (primarily tones for a vast universe of subtle intervallic relation- organs or harmoniums), particularly in Great Britain. ships, the composers and theorists of the eighteenth Among those engaged in this activity, the most notable and nineteenth centuries effectively painted Western were General Perronet Thompson, Colin Brown, and music into a corner from which it has not, as yet, extri- R.H.M. Bosanquet. Unfortunately, the proposals of cated itself. Twentieth century composers have tried in Helmholtz and the other intonational reformers of the vain to invent or discover new organizing principles as nineteenth century appear to have had no detectable powerful as the common-practice tonal system. Instead, effect on contemporary musical practice, although they have created a variety of essentially arbitrary sys- Helmholtz’s work, in particular, was to have a signifi- tems, which, although they may seem reasonable in cant influence on musicians of subsequent generations. the minds of their creators, fail to take into account Nineteenth century composers were still enamored the capabilities and limitations of the human auditory of the facility for modulation and for the building of system. These systems have resulted in music that the increasingly complex harmonies that equal tempera- great majority of the population find incomprehensible ment provided, and it was not until these resources and unlistenable. were exhausted that any alternative was seriously Given that equal temperament had only been in gen- considered. eral use for about 150 years at the time, it may seem strange that so few of the composers of the early twen- The End of Common Practice tieth century recognized that the cure for music’s ills Initially, the effect of equal temperament on Western lay, at least in part, in the replacement of its inadequate music was probably benefi cial. Composers obtained tuning system. (Some theorists and composers did, in the ability to modulate freely and to build complex fact, advocate the adoption of new, microtonal tuning chromatic harmonies that had been impossible under systems, but most of these proposals were for micro- the meantone system. As a result, abstract instrumental tonal equal temperaments, such as quarter tones, third music fl ourished as never before, yielding what is gener- tones, sixth tones, eighth tones, or the like, which merely ally considered the “golden age” of Western music. Like divided the existing twelve-tone scale into smaller, arbi- a plant stimulated by chemical fertilizers and growth trary intervals, and which made no improvement in the hormones, music based on equal temperament grew tuning of Western music’s most fundamental intervals.) rapidly and luxuriously for a short period — then col- However, despite its fairly recent origin, equal tempera- lapsed. If equal temperament played a prominent role ment had already become quite deeply entrenched in in stimulating the growth of harmonic music in the Western musical thought and practice. There were common-practice era, it played an equally large part in several reasons for this. One was the industrial revo- its rapid demise as a vital compositional style. Twelve- lution. The nineteenth century saw the redesign and tone equal temperament is a limited and closed system. standardization of many instruments, particularly the Once you have modulated around the so-called circle of orchestral woodwinds and brass. Strictly speaking, only fi fths, through its twelve major and twelve minor keys, fixed-pitch instruments (the piano, organ, harp, tuned and once you have stacked up every combination of percussion, and fretted strings) require temperament, tones that can reasonably be considered a chord, there is the others being sufficiently flexible as to adjust pitch nowhere left to go in search of new resources. as musical context requires. Nevertheless, brass and woodwind instruments were also standardized to play

Introduction 6

a chromatic scale such that the “centers” of their pitches tuning for a culturally predominant scale, such as a corresponded as closely as possible to the pitches of major, minor, or chromatic scale, the approach of twen- twelve-tone equal temperament. Another reason for the tieth century composers and theorists working with Just persistence of equal temperament was the repertory of Intonation, as exemplified by Partch, has been quite dif- the common-practice period. The previous 150 years ferent. The goal of these artists has been, in most cases, had witnessed the development of the orchestra as we to discover or create tunings that best served their own know it, along with its repertory, and the concert system particular musical goals, whether for a single composi- that supported it. It had also seen the evolution of the tion or for a lifetime’s work, rather than one that could piano, the preeminent equally tempered instrument, as serve the needs of the culture as a whole. the predominant instrument for both solo performance From when he began work in the mid-1920s until and accompaniment, and as the most important tool in the mid-1950s, Partch was the only composer in the musical education. The orchestra, the piano, and their United States doing significant work in Just Intonation. players, trained to perform the works of eighteenth and In the 1950s, Partch was joined by (b. nineteenth century composers, were the resources that 1917) and (b. 1926). Harrison first learned turn-of-the-twentieth-century composers had to use if about Just Intonation from Partch’s Genesis of a Music. they wished to have their music performed. And all of He composed his first major work in Just Intonation, these resources were dedicated to music that assumed Four Strict Songs for Eight Baritones and Orchestra, equal temperament. It was little wonder, then, that in 1954.8 Although, unlike Partch, he does not work few composers were willing to challenge this massive exclusively in Just Intonation, Harrison has written a establishment in order to work in some new, untested large body of work in various just tunings. He is prob- tuning system. ably best known for the creation, in conjunction with his companion, the late William Colvig, of a number of The Twentieth-Century justly tuned American gamelan (Indonesian-style tuned Just Intonation Revival percussion ensembles) and for the body of music he has Although most composers were suffi ciently intimidated composed for this medium, but he has also composed by the weight of eighteenth and nineteenth century just music for a great variety of instrumental and vocal musical practice, fortunately a few were not. The fi rst ensembles, often mixing elements from European and twentieth century composer to make a serious com- Asian musical traditions. Through his teaching at San mitment to Just Intonation and the person primarily Jose State University and Mills College in California responsible for the revival of Just Intonation as a viable and his extensive lecturing, he has introduced many musical resource was Harry Partch (1901 – 1974), the younger composers to Just Intonation. iconoclastic American composer, theorist, instrument Ben Johnston discovered the possibility of Just builder, dramatist, and musical polemicist. When Partch Intonation early in life, when he attended a lecture began his compositional career, no one, to the best of my on Helmholtz at age eleven. Later, he, like Harrison, knowledge, was making music in Just Intonation. Begin- discovered Partch’s Genesis of a Music. Johnston con- ning with tentative experiments in the mid-1920s and tacted Partch and for a six month period in 1950 was his continuing over a span of fi fty years, Partch developed student and apprentice in the remote California coastal a system of Just Intonation with forty-three tones to the town of Gualala. Johnston began composing seriously octave, built a large ensemble of predominantly stringed in Just Intonation in 1959. Unlike Partch and Harrison, and percussion instruments to play in this tuning system, Johnston’s work in Just Intonation employs mainly composed and staged six major musical theater pieces, Western musical forms and instrumental combina- along with numerous lesser works, and produced and tions. His earlier work, through the early 1970s, gener- distributed his own records. In 1947, Partch published ally combines extended microtonal Just Intonation with the fi rst edition of his Genesis of a Music, an account serial techniques. His later work tends to be simpler and of his musical theories, instruments, and compositions more tonal, but still uses at least occasionally. that became the bible for subsequent generations of Just Johnston’s works include eight string quartets in Just Intonation composers. Intonation and numerous vocal and chamber ensemble Whereas in previous centuries the goal of most into- pieces. He is also the inventor of a system of notation for national theorists was to find the ideal or most practical extended Just Intonation that is used in this primer.

The Just Intonation Primer 7

In the 1960s and 1970s, interest in Just Intonation priate software. Hence, a conventional keyboard can be continued to slowly increase. (b. 1935) used to play a virtually unlimited number of different began working with Just Intonation in the early 1960s pitches. This capability has, for all intents and purposes, in the context of his instrumental/vocal performance eliminated the condition that first brought temperament group, The Theater of Eternal Music. In this ensemble, into being: the necessity of restricting the number of Young developed the practice of performing long, static pitches used in music to the number of keys available compositions based on selected tones from the harmonic on an affordable, playable keyboard. series, played on various combinations of amplified Among the many composers currently doing sig- instruments and voices. In 1964, Young began work on nificant work in Just Intonation are William C. Alves, his semi-improvisational, justly tuned piano composi- Lydia Ayers, Jon Catler, David Canright, Dean Drum- tion, The Well-Tuned Piano, which can be from five to mond, Cris Forster, Glenn Frantz, Ellen Fullman, seven hours in duration and which continues to evolve Kraig Grady, Michael Harrison, Ralph David Hill, at the time of this writing. Young is also known for The David Hykes, Douglas Leedy, Norbert Oldani, Larry Dream House, a living environment in which a number Polansky, Robert Rich, Daniel Schmidt, Carter Scholz, of electronically generated, harmonically related tones James Tenney, and Erling Wold. The variety of musical are sustained over a period of months or years. styles represented by this group is extremely diverse, (b. 1935), who was a member of Young’s and the use of Just Intonation may be the only feature Theater of Eternal Music at various times in the early they all share. Although more than half work primarily 1960s, is known primarily as a keyboard composer/ or exclusively with electronic media, they also include improviser. He is perhaps best known as the composer exponents of Partch’s tradition of acoustic instrument of the early minimalist piece In C (1964), which is not building (Drummond and Grady), Lou Harrison’s explicitly a Just Intonation piece, although it has some- American gamelan movement (Schmidt), Young’s and times been performed this way. In the 1970s, Riley Riley’s improvisational keyboard styles (M. Harrison), performed extensively on a modified electronic organ a harmonic singer (Hykes), and even a justly tuned rock tuned in Just Intonation and accompanied by tape delays. guitarist (Catler). More recently, he has been performing his work on justly tuned piano and digital synthesizers, and composing for The Purpose of this Publication other ensembles, especially the string quartet. Although the technical barriers to the composition and In the late 1970s and early 1980s the number of performance of signifi cant music in Just Intonation have composers working with Just Intonation began to been considerably reduced in recent years, barriers of increase significantly, due in part to the development another type remain largely in place, namely the weight of affordable electronic instruments with programmable of custom and the lack of accessible information on prin- tuning capabilities and in part to the coming of age of ciples of Just Intonation. The colleges, universities, and the post – World War II generation of composers. The conservatories continue to teach a curriculum based on achievements of Partch, Harrison, Johnston, Young, music of the common-practice era, in which alternate and Riley made it evident to these younger composers tunings are unlikely to receive more than a passing men- that Just Intonation was a valuable resource for com- tion. With the exception of the fortunate few who fi nd posers of diverse styles and tastes, and the availability themselves in institutions with a microtonal composer or of electronic instruments with programmable tuning theorist on the faculty, students who develop an interest made it possible for the first time for composers to in these matters are unlikely to receive much support experiment with a variety of different tuning systems or encouragement, much less practical instruction, from without having to invent and build novel instruments the academic establishment. Such students, if they per- or to train performers in unusual playing techniques. sist, generally fi nd it necessary to educate themselves, Changing the pitches available on a digital synthesizer and in the process often have to reinvent or rediscover simply means changing the data values in a tuning table principles and structures that are well known to more or switching to a different table. If the instrument and experienced composers. its operating software have been designed to facilitate In an attempt to remedy this situation, in the fall of such changes, either of these functions can be performed 1984, I and my associates in the experimental music virtually instantaneously by a computer running appro- ensemble Other Music, in consultation with a number

Introduction 8

of other West Coast Just Intonation composers and theory and practice of Just Intonation, let alone tuning theorists, founded the Just Intonation Network. The in general or other aspects of composition. Its purpose, Just Intonation Network is a nonprofit group fostering rather, is to provide the reader with the basic information communication among composers, musicians, instru- and skills necessary to read and comprehend intermedi- ment designers, and theorists working with Just Intona- ate and advanced texts such as articles in 1/1 or Harry tion. Its primary goal is to make information about the Partch’s Genesis of a Music, and to prepare the reader theory and practice of composition in Just Intonation to begin independent study and composition. available to all who want or need it. The primary method The Primer is intended for readers with at least an for distributing this information is the network’s jour- elementary knowledge of common-practice Western nal, 1/1, the only current periodical devoted primarily music theory, including the basic terminology of or exclusively to Just Intonation. For the past eighteen intervals, chords, and scales, and the fundamentals of years I have served as editor of this publication. harmony. The reader is not assumed to have any prior A survey of Just Intonation Network members taken knowledge of Just Intonation or of alternative tunings several years ago revealed that more than half were in general, nor is the reader expected to be a mathema- newcomers to the study of Just Intonation who found tician or number theorist. The only math required to a significant portion of the articles in 1 /1 over their understand this book is basic arithmetic, in combina- heads. It was with the goal of assisting these readers tion with some simple procedures explained in Chapter that the Just Intonation Primer was conceived. The Just Three. An inexpensive scientific calculator will prove Intonation Primer, as its title indicates, is not intended useful for comparing the sizes of intervals. to provide a complete or comprehensive course in the

The Just Intonation Primer 85

Index Chromelodeon 70 Flute 19, 74 Circle of fi fths 37 Fokker, Adriaan Clarinet 19 method for representing intervals on the A Classical era 4 lattice 43 Clavichord 67 Formants 19 Added-second chord 57 Clusters. See Tone clusters Forster, Cris 7, 72 Aeolian scale 39 Cochlea 14 Frantz, Glenn 7 Alves, William C. 7 Cocktail-party effect 16 French horn 74 American gamelan 6, 7, 66, 72 Colvig, William 6, 72 French impressionists 63 Anomalies 33. See also great diesis, Combination tones. See Difference tones; Frequency, absolute 11. See also Hertz (Hz) Pythagorean comma, septimal comma, Summation tones calculating 28 , etc. Comma. See Pythagorean comma, septimal of 1/1 36 Ars Antiqua period 38 comma, syntonic comma Frequency ratios 11, 25 – 29. See also Just Ayers, Lydia 7 Comma of Didymus. See Syntonic comma Intervals Common-practice period 6 calculations with 25 – 29 B Common-practice theory 35, 51 addition 25 Complex tones 19 – 20 complementation 26 Bach, J.S. 4 Continuous-pitch instruments 66, 74 – 76 converting to cents 26 Barbershop quartet 35 Critical band 13, 51 division 26 – 28 Barbour, J. Murray 2 subtraction 25 – 26 Baroque era 3, 4 D converting to harmonic or subharmonic series Basilar membrane 14, 18 segment 28 – 29 Bassoon 74 Difference tones 16 – 17, 19 only unambiguous inerval names 32 Beating harmonics. See Harmonics: beating fi rst-order 16 rules and conventions for using 25 Beats. See Harmonics: beating; Interference of condissonant triads 48 superparticular 27 beats of just major triad 46 Fretted instruments 3, 71 frequency (fB) 13 of just minor triad 46 – 47 Fullman, Ellen 7 due to the inharmonicity of piano strings 68 of major-seventh chord 48 Fused tone 13, 14 Benade, Arthur H. of minor-seventh chord 48 experiment to identify “special relationships” higher-order 16 – 17 G 22 – 24 of dominant- 56 Bethea, Rob 76 of dominant-seventh chord 56 Galilei, Vincenzo 35, 51 of subharmonic pentad 58 proposed fretting for the lute 64 Blue notes 35, 55 of tempered major third 17 Genesis of a Music 2, 6, 67, 70 Boethius, Anicius Manlius Severinus 2 Diminished-seventh chord 49 – 50, 64 Grady, Kraig 7, 72 Bosanquet, R.H.M. 5, 70 Diminished major-seventh chord 65 Great diesis 44 – 45, 50 Bowed string instruments 75 Ditone diatonic. See Pythagorean diatonic scale Guitar 71 Branca, Glenn 65 Divisions of the (book) 34 Brass instruments 19, 74 Dominant-minor-ninth chord 50, 64 H Brown, Colin 5, 70 Dominant-ninth chord 31, 55, 56, 58, 62 subsets of 56 – 57 Hair cells 14 C Dominant-seventh chord 31, 35, 48 – 49, 51, Half-diminished-seventh chord 55, 56 – 57 55, 56 Harmonics 15, 16. See also Harmonic series; Canright, David 7, 71 subsets of 56 – 57 Partials: harmonic Carlos, Wendy 73 Drummond, Dean 7, 72 beating 20 – 24 Catler, Jon 7 Duodenarium 46 of an octave (2:1) 20 – 21 Cents 15 Duodene. See Harmonic duodene of a perfect fi fth (3:2) 21 – 22 calculation 26 of a (1:1) 20 defi ned 4 E coincident (matching) 20 – 24 Chalmers, John H. formula for 21 tritriadic scales 64 Electronic organ 70 of just intervals (fi gure) 21 Cheng 71 Ellis, Alexander J. 26, 31, 45, 64, 70 of just major triad 46 Chords 30 – 32. See also Triads, other chord equivalents 44 of just minor triad 46 names Enharmonic spellings 49 Harmonic and subharmonic series (fi gure) 29 condissonant 31 Experimental Musical Instruments (periodical) Harmonic duodene 45 – 46 consonant 31, 46 67 Harmonic series 4, 14 – 16, 18, 19, 28 – 30 dissonant 31 defi ned 14 eleven- and thirteen-limit 62 – 64 F fundamental of 15, 19, 31 fi ve-limit 46 – 48 missing. See Periodicity pitch condissonant 47 – 48 False Consonance 54 – 55 Harmonic spectrum 15, 19 seven-limit 56 – 59 Fender Rhodes Electric Piano 70 cutoff frequency of 19 condissonant 58 – 59 Fixed-Pitch Instruments 67 – 71 of strings 19 subharmonic 57 – 58 Fixed scales 42, 55 Harmonists (ancient Greek theorists) 2 Chromatic scale 45. See also Harmonic seven-limit 59 Harp 70, 72 duodene

Index 86

Harpsichord 67, 72 7:4 (harmonic, subminor, or septimal minor 81:64 (Pythagorean major third, ditone) 36, Harrison, Lou 6, 66, 72, 75 seventh) 35, 51 37, 38, 40, 41 Concerto for Piano with Javanese Gamelan 7:5 (most consonant ) 23, 51, 55, 57 81:80. See Syntonic comma 69 7:6 (subminor or septimal minor third) 23, 128:81 (Pythagorean ) 37, 40 Concerto for Piano with Selected Orchestra 51, 55, 57, 58, 59 128:125. See Great diesis 69, 76 8:1 (triple octave) 24 135:128 (large , large chromatic Four Strict Songs for Eight Baritones and 8:3 24 semitone) 45 Orchestra 6, 76 8:5 (minor sixth) 24, 39, 40, 41, 42, 50, 55 243:128 (Pythagorean ) 40 The Tomb of Charles Ives 76 8:7 (septimal whole tone; supermajor second) 256:243 (Pythagorean limma) 37, 40 Harrison, Michael 7, 69 23, 24, 51, 54, 55, 57 Just Intonation From Ancient Worlds 68 9:4 (major ninth) 24, 37 defi ned 1 – 2 Heim, Bruce 74 9:5 (acute ) 40, 49, 56, 57 discovery 2 Helmholtz, Hermann von 4 – 5, 45, 70 9:7 (supermajor third) 55, 56, 57, 58, 59 eleven- and thirteen-limit 59 – 63 Heptad 62, 63 9:8 (major whole tone) 33, 37, 38, 40, 55, fi ve-limit 35, 38 – 50, 75 Hertz (Hz) 10, 11. See also Periodic vibration: 64, 65 diffi ciencies of 48 – 50 frequency 10:3 24 obtaining satisfactory performances in 79 – 80 Hexad 62, 63 10:7 (septimal tritone) 51, 55 seven-limit 51 – 59 Hill, Ralph David 7, 54 10:9 (minor whole tone) 40, 55 coloristic use of 55 Hohner Clavinet 70 11:1, 11:2, 11:3, 11:4, 11:5 24 seventeen- and ninteen-limit 64 – 65 Hykes, David 7 11:6 62 three-limit 36 – 38 11:8 (primary interval for 11) 61, 62 twentieth-century revival 6 – 7 I 11:9 () 62 Just Intonation Network 8, 78 11:10 62 Identities 12:5 (minor tenth) 24 K as defi ning characteristics of a chord 30, 31 12:7 51 defi ned 30 12:11 62 Keyboard instruments 3, 4, 67 – 70 Idiophones 19, 72 – 73 13:1, 13:2, 13:3, 13:4, 13:5, 13:6 24 Acoustic 67 – 70 Industrial revolution 5 13:8 (primary interval for 13) 61, 62 electroacoustic 70 Interference Beats 12 – 13, 46. See 13:9 62 Koto 71 also Harmonics: beating 13:10 62 Intermodulation products. See Difference tones, 15:8 (diatonic major seventh) 40 L Summation tones 15:13 62 Interval 10 – 11. See also Frequency ratios; Just 15:14 54, 55 Lattice intervals 16:5 24 complementation on 44 conventional names for 10, 32 16:9 (minor seventh) 37, 49 fi ve-limit 42 – 44 defi ned 10 16:11 62 fi gure 43 harmonic vs subharmonic interpretation 28 16:13 62 seven-limit 51 Intonation Systems 71 16:15 (diatonic semitone) 40, 45, 55 fi gure 52, 53 17:16 64 syntonon diatonic (fi gure) 40 J 17:16 (primary interval for 17) 64 transposition on 43 18:11 62 Leedy, Douglas 7 35, 51 18:17 64 Limit of frequency discrimination 13 Johnston, Ben 6, 75 19:16 (primary interval for 19) 64 Lute 71 notation for Just Intonation 33 21:16 (septimal subfourth) 54, 55 ratio scales 64 21:20 54, 55 M remarks on 11 and 13 61 25:16 50 Sonata for Microtonal Piano 70 25:24 (small chromatic semitone) 45, 55 Major-ninth chord 48 Just Intervals 27:16 (Pythagorean ) 37 Major-seventh chord 48 1:1 (unison) 36 27:20 (acute, imperfect, or wolf fourth) 40 Major scale 2:1 (octave) 36, 37, 44 27:25 (great limma) 45 fi ve-limit 3:1 (perfect twelfth) 24 28:25 (intermediate septimal whole tone) 54 constructing 39 – 40 3:2 (perfect fi fth) 36, 37, 38, 39, 40, 42, 45, 28:27 54, 55 Maxwell, Miles 54 46, 51, 54, 55, 57, 59 32:21 54, 55 McClain, Ernest G. 2 4:1 (double octave) 24 32:25 () 45, 50 Mersenne, Marin 4 4:3 (perfect fourth) 33, 36, 37, 38, 40, 42, 32:27 (Pythagorean minor third) 37, 41, 49 Microtones. See also Just Intervals 54, 55 33:32 62 eleven- and thirteen-limit 62 5:1 24 35:32 (small septimal whole tone) 54 seven-limit 54 5:2 (major tenth) 24 36:25 (acute diminished fi fth) 49 Middle Ages 38 5:3 (major sixth) 24, 39 36:35 (septimal or diesis) 54 MIDI 76 – 79 5:4 (major third) 38, 39, 40, 42, 45, 46, 50, 40:27 (grave, imperfect, or “wolf” fi fth) 40, keyboard controller 78 51, 55, 56, 57, 58 41 pitch bend 76 – 77 6:1 24 49:48 (septimal sixth tone) 54 using with tuning tables 78 6:5 (minor third) 24, 39, 45, 49, 50 50:49 (septimal sixth tone) 54 Tuning-Dump Specifi cation 78 – 79 7:1 24 64:63. See Septimal comma (64:63) Minor-major-seventh chord 65 7:2 24, 51 65:64 62 Minor-ninth chord 48 7:3 (septimal minor or subminor tenth) 24, 51 75:64 () 45 Minor-seventh chord 48

The Just Intonation Primer 87

Minor- 64 for seven (7:4) 51 Simple harmonic motion 9. See also Sine wave Missing fundamental. See Periodicity pitch for thirteen (13:8) 61 Simple tone 10 for three (3:2) 36 Sine wave 9, 10 N for two (2:1) 36 Singing. See Voice Prime limit 27, 35 Special relationships 22 – 24 Neutral intervals 62 defi ned 30 absence of among eleven- and thirteen-limit Nondominant-ninth chords 47 Prime numbers 35 dyads 59 Nondominant-seventh chords 47 defi ned 30 beyond the octave 23 – 24 Notation for Just Intonation 32 – 33 Psaltery 70, 72 defi ned 22 Ben Johnston’s 33, 62 Ptolemy, Claudius 2, 3, 39 involving higher harmonics 24 Table 32 Ptolemy’s syntonon diatonic 33, 59. See table 23 Novatone 71 also Major scale: fi ve-limit Sruti box 76 complementation (fi gure) 44 Stone, Tom 71 O relative minor of 41 String players tuning method (fi gure) 39 “natural tendencies” of 74 Oboe 74 Pure tones 10, 15 use of vibrato 75 Odington, Walter 3 superposition of 11 – 12 Subharmonic series 28 – 30 Oldani, Norbert 7 Pythagoras of Samos 2, 36 Subjective pitch. See Periodicity pitch Oliveros, Pauline 70 Pythagorean comma 37 – 38, 44 Subminor-seventh chord 58 1/1 (Journal of the Just Intonation Network) 8 Pythagorean diatonic scale 36 – 37, 40 Summation tones 16, 17 One (foundation of all just tunings) 36 Pythagorean tuning 2 – 3, 4, 36 – 38. See fi rst-order 17 On the (book) 45 also Just Intonation, three-limit Supermajor-seventh chord 59 Orchestra 6 tendency of string players for 75 Supertonic problem 40 – 42 Other Music 7 Synthesizers. See also MIDI Otonalities 29, 47. See also Harmonic series Q with user-programmable tuning tables 7, Overtones. See Harmonics; Harmonic series; 77 – 78 Partials Quartertone scale. See Temperament: twenty- Yamaha DX/TX 77, 79 four-tone equal Syntonic comma 39, 41, 42, 44, 71 P Syntonon diatonic. See Ptolemy’s syntonon R diatonic Partch, Harry 2, 6, 47, 58, 61, 62, 66, 67, 70, 71, 72 Rameau, Jean Philippe 4 T tonality diamond 64 Ratios. See Frequency ratios; Just Intervals Partials 15. See also Harmonics; Harmonic Ratios of eleven Tamboura 76 series table 61 Temperament harmonic 15, 19 – 20 Ratios of thirteen defi ned 3 inharmonic 15, 19 table 62 meantone 3, 35, 42 of a free rectangular bar 72 Recorder 74 twelve-tone equal 4, 11, 33, 35, 37, 42, 49 of a hinged bar 68 Reed instruments 19 benefi ts 5 of piano strings 67 – 69 Reed organ 70 deviation of harmonic series from 15 Periodicity pitch 17 – 19, 20 Renaissance 3 twenty-four-tone equal 62 of eleven- and thirteen-limit dyads 61 Residue tone. See Periodicity pitch well temperament 4, 35 of just major triad 46 Resultant tones. See Difference tones, Tempered major third 17, 23 of just minor triad 46 – 47 Summation tones Tenney, James 7 of subharmonic pentad 58 Rich, Robert 7, 78 Tetrachordal genera 34, 59 of subminor triad 57 Riley, Terry 7, 69, 70, 75 Tetrachordal scales Periodic vibration 9 – 10, 15 In C 7 equal 34, 37, 40 amplitude 10 (of a chord) 31 mixed 34 frequency 10 Rosenthal, David 72 seven-limit 59 graph of 9 Roughness 13, 20. See also Interference Beats (fi gure) 60 period (τ) 10, 16 Tetrachords 33 – 34, 55 Phase Relationship 12 S defi ned 33 Piano 6, 35, 67 – 70 disjunct 34 single-strung 68 Sackbut 76 permutations 34 tuning strategies 69 Sawtooth wave 16 Tetrad. See also Dominant-seventh chord, Pipe organ 70 Saxophone 74 Major-seventh chord, Minor-seventh Pitch 10. See also Periodic vibration: frequency Schmidt, Daniel 7, 72 chord Pitch (periodical) 74 Scholz, Carter 7, 78 subharmonic 58 Place theory 18 Senario 3, 4 Thompson, General Perronet 5, 71 Polansky, Larry 7 Septimal comma (64:63) 54, 65 Tied frets 71 Primary interval Seven Timbre 10, 19 defi ned 30 consonance of 35 Timing theory 18 b for eleven (11:8) 61 7 5 chord 64 Tone clusters 63 m for fi ve (5:4) 38 7 5 chord 65 eleven- and thirteen-limit 63 for ninteen (19:16) 64 Siegel, Jules 77 seven-limit 56 for seventeen (17:16) 64

Index 88

Transposition 11 Tuning Y Triads History 2 – 7 augmented 48, 49 – 50, 65 Tuning and Temperament (book) 2 Young, La Monte 7, 61, 69 condissonant 47 – 48 11 Dream Chords 38, 64, 65 diminished 31, 35, 48, 51, 55, 56 The Dream House 7 subharmonic 58 U 5 V 67 6:38 PM NYC 65 dominant 39 The Four Dreams of China 64, 76 5:7:9 57 Utonalities 29, 47, 58. See also Subharmonic The Lower Map of The Eleven’s Division in 4:6:7 57 series The Romantic Symmetry (over a 60 cycle subharmonic 58 base) in Prime Time from 122 to 119 major 31, 39, 46 V with 144 65 voicings in C (table) 30 The Romantic Symmetry in Prime Time from minor 31, 39, 41, 46 Viol 71 112 to 144 with 119 65 proposed tuning with 19:16 64 – 65 Virtual pitch. See Periodicity pitch The Second Dream of the High Tension Line supertonic 41, 42 Vogt, Walter 71 Step-Down Transformer 64, 76 open 38 Voice 76 The Subsequent Dreams of China 64 Pythagorean 38 The Theater of Eternal Music 7 major 46 W The Well-Tuned Piano 7, 65, 68 minor 46 subdominant 39 The Well-Tempered Clavier 4 Z subminor 31, 55, 57 Widener, Ward 64, 65 supermajor 58 Wilson, Ervin Zarlino, Gioseffe 3 tonic 39 combination product sets 64 Zither 70 76 Wind instruments 73 – 74 Wold, Erling 7

The Just Intonation Primer

Top View