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Changing the metaphor: ratio models of musical pitch in the work of Harry Partch, Ben Johnston, and James Tenney.

INTRODUCTION

Models or musical pitch devised by theorists and composers have taken a large number of different forms and have fulfilled a wide range of different functions. Most models are designed not merely to provide a description of a pitch "space" but to suggest or embody an explanation of it. All such models are attempts to circumscribe and make manifest the processes by which we form cognitive representations of musical materials. Clearly, the model and the observations that arise from it are linked: observation is done in the ambience of the model; the model is created in the context of an observation strategy. This interaction helps evolve the adequacy of the model and the sensitivity of the observation.

In the case of music that uses microtonal tunings and therefore a relatively unfamiliar set of pitch materials, the nature of the model or representation of the pitch space may significantly affect aspects of compositional thinking. The study of such models is thus not a sterile exercise in abstract theory but a necessary starting point for a detailed understanding of the music itself. In this paper I shall examine the ratio models of pitch in the writings of three American composer-theorists who have worked with microtonal tuning systems based on extended just intonation: Harry Partch (1901-1974), Ben Johnston (b.1926), and James Tenney (b.1934). In particular, I shall attempt to trace the legacy of ideas inherited by the two younger composers from Partch's book Genesis of a Music,(1) and its transmutation in their own work.

I would not claim that the work of these three composers forms a school of thought, a distinct and coherent tradition. Their work does, however, belong together by virtue of a number of associative linkages, historical and conceptual. Despite the variety of morphologically distinct forms they have taken, and despite differences in terms of aesthetics and semantics, their pitch models still share a measure of systemic identity as descriptions of the same extended pitch space. The composers are here grouped together not for exclusionary purposes, but to highlight the vastly different application each has made of the ratio model of pitch.(2)

In this paper I have tried not merely to paraphrase each of the three composers' thought, which is clearly illuminated in their own writings. Nor do I wish to remove either the need to grapple with those sometimes difficult writings themselves, or the exhilaration in making their meaning clear to oneself. Rather, I have explored in each composer's work the cycles of observation, reflection, reformulation, and have traced some of the points of contact and points of difference between them. I have also attempted to outline how the fruitfulness of the theoretical model for the creative process sheds light on the interaction of theory and compositional practice.

HARRY PARTCH

Partch's theoretical work, from its beginnings in the early 1920s, took root in his dissatisfaction with the intonational and conceptual paradigm of equal temperament. All his mature music is written for instruments of his own design and construction, tuned to a microtonally extended just-intonation scale that derives from, and expands upon, the tuning system of the later ancient Greek modes. His work was not undertaken in the spirit of wishing to replace one exhausted paradigm by another, by a "different set of rigid stipulations";(3) it was the result of what was, for Partch, a searching, creative investigation, and needs to be taken in that light, not as a dogmatic pre- (or pro-) scription, every detail of which is set in amber. In a letter in 1952 he wrote:

I do not always achieve the just intonation which I hold as desirable - the clear choice of consonance or dissonance. Someone has said that ideals are like stars. We can't touch them but we look to them for guidance. I believe in a rational - that is, acoustical - approach to the problems of musical materials, as the only one leading to genuine insight.(4)

Partch's revolutionary work in intonation corresponds to a shifting axis: the creation of a new conceptual milieu, and a language, in which problems of harmony and subtleties of musical pitch relations can be addressed. The first step in that revolution was the very conception of musical pitch relations as ratios - as describable in mathematically precise, quantitative terms. His adoption of the "language of ratios" as the basis of his theoretical work and as the descriptive language for the pitch relationships in his music marks the real breakthrough in his thinking - ironically, by reaffirming a connection between music theory, number, and the natural sciences, the roots of which stretch back to antiquity: in the Western world to Pythagoras, to whom is generally accredited the discovery that proportional lengths of a vibrating string, in small-number-ratio relationships, produce basic musical intervals.

Although sanctioned in ancient Greek thought, the ratio model of pitch remains controversial. Conventional music theory describes the pitch resources of Western music not in numerical terms but by an arbitrary though consistent system of letter names. In the late nineteenth century, ratio terminology had regained currency thanks to the work in acoustics of Helmholtz: the description of intervals in ratio terms was consistent with Helmholtz's basic objective of formulating quantitative rules according to which perceived pitches of sounds are related to physically specifiable properties. Additionally, ratios are an appropriate descriptive language for the pitch relationships of the harmonic series, which Helmholtz discussed at length in his influential book On the Sensations of Tone.(5) Thanks to his encounter with Helmholtz's work, the harmonic series formed a major source of Partch's investigations in the years 1923-28, when his theoretical ideas were crystallizing. In turning to ratio terminology he was less concerned with its relevance to the formal and acoustical underpinnings of the existing system of equal temperament than he was with opening the door to a "new" tuning system - just intonation - for which ratios provided a language. Much of his early theoretical work derives its energy from the exciting sense of exploring a new system of thought incommensurate with that of conventional music-theoretic knowledge.

Helmholtz's shadow is discernible over the pages of Partch's treatise "Exposition of Monophony," which went through five drafts in the years 1928-33: the fifth draft, from June 1933, is the earliest of his extant theoretical writings. In the Foreword to the 1933 draft he writes:

Throughout the history of music there has been a slow and only half-recognized revelation of the universe of tone created by the overtone series. This work is an attempt to found the theory of music definitely on the origin of intervals. It is an exposition of the so-far-accepted in the light of that origin and a disclosure of a further small part of its universe. Understanding of musical structure does certainly devolve upon an understanding of this one source of tonal relationships. Without this, a true knowledge has only chance existence in the arbitrary, traditional, or intuitive.(6)

By "a further small part of its universe" Partch meant his concern to extend the musical palette of consonance and dissonance, as these terms are defined by Helmholtz, by admitting the higher prime number relationships of the overtone series as potential consonances. He acknowledged that a great deal of familiarity was needed with these more complex ratios before they could be heard and reproduced accurately, and that this complexity was hierarchical: the intervals derived from the eleventh partial, which form the limit of his own pitch usage, Partch held to be more complex relationships both physiologically and in terms of historical usage than the intervals derived from the seventh partial, and the septimal intervals in turn as more complex than those derived from the fifth partial, which were (and are) the familiar basis of Western tuning practice.

It was Partch's understanding of ratios and their relationships that gave him the principles that generate his microtonal scale. This generative process, once set in motion, does not result in a "closed" tuning system - like twelve-note equal temperament, which is analogous to a circle on which are marked twelve equally spaced points. Rather, ratio tuning is (by definition) theoretically infinite, and resembles a branching out in several directions at once from a centre - the fundamental, or the ratio 1/1. To use a metaphor that Partch himself does not, we might imagine that each "branch" corresponds to a different prime number, and that along its length is a cycle of multiples of a single interval - the interval formed by the given prime number partial and the fundamental. The branchings, which would theoretically extend forever, have for practical purposes to be chopped off at certain stopping-points - the larger-number ratios at the extremities of the "branches" become, beyond a certain point, ambiguous, because they start to form near-unisons with smaller-number ratios nearer the centre. (This metaphor is somewhat less than exact, because the "branches" grow in parallel, and produce mirror images of themselves as they grow.)

In his early work with ratio tuning Partch experimented with several such stopping-points. Reviews of his work from the early 1930s report the number of tones in his scale as being, at different times, twenty-nine, fifty-five, and thirty-seven tones in the octave, and his early theoretical writings contain descriptions of these and scales of thirty-nine and forty-one tones among others, before he settled on the forty-three-tone scale he analyses in Genesis of a Music. The frequent change in the constitution of his scale, although an interesting aspect of this period of Partch's work, is secondary in importance to the fact that the body of principles underlying the generation of the scales remained consistent.(7)

A better metaphor, and one that Partch frequently used to describe the pitch resources of his system of extended just intonation, is that of a fabric. The word appears in his earliest extant theoretical writings, and it was a usage he retained throughout his life. The metaphor is not merely poetic. The word "fabric," with its connotations of texture, places emphasis on the system's internal coherence - and on the fact that it can grow if woven further - rather than viewing it as a closed structure. The tuning system is seen as an active, expanding fabric of relationships capable of informing, shaping, and ordering the pitch domain: a laying down of the total pitch space.

In his theoretical writings Partch never really abandons, at least in principle, the idea of a scale with a fixed number of degrees. Yet although incipient in his theory, the idea of an expandable source scale - a fabric of tones - is in fact a more accurate description of his music than that of a fixed structure of forty-three tones, of which any given work might use only a subset. His music offers a fully fledged example of the principle: scattered throughout his output we find numerous works that seem, as it were, to lay down their pitch terrain as they unfold, sometimes using only a small piece of the fabric, and occasionally extending that fabric to many more than forty-three tones.(8) As with any real composer, we find that Partch's theory, at least in this respect, lags behind his compositional practice.

These visual metaphors are not simply my own projections. Partch was very much aware of them, and interested in the different possible geometric projections of the pitch resources of his system of Monophony. Some of his reasons for this interest can be documented: some must remain speculative.

* A belief in the use of graphic analysis in explaining this type of musical phenomenon. This is not surprising in view of the still rather arcane nature of the subject for most musicians. In the 1933 "Exposition of Monophony" he explains the source of intervals in relation to the overtone series by use of a graph.

* The links with his activity as instrument builder: the projections or images can translate, directly or less directly, into the layouts of strings or blocks on the instruments.

* Speculative: the feeling that, by scanning the resources of the scale with this in mind, one is identifying patterns that are already there, waiting to be discovered - an appeal to the artist's desire to make sense out of the order of the system s/he has created.

* Speculative: the idea that this type of activity is a process of forming iconic images, arrangements that help to fix each important projection as an event in the mind - an appeal to eidetic memory.

In Genesis of a Music Partch offers two main geometric projections, not merely of the resources of his scale, but meaningfully embodying the relationships inherent within it. (I do not here include the simple image of the scale as a forty-three-step stairway superimposed on a twelve-step stairway, nor the graph of Perpetual Tonal Descent and Ascent, which is drawn to chart the wider field of superparticular ratios from which certain elements are chosen as "secondary ratios" in the formation of his scale.)(9) These are:

* The One-Footed Bride, a graphic representation of the comparative consonance of its constituent intervals (ratios);

* The Tonality Diamond, a projection of the chordal relationships within the primary ratios of the scale.

The One-Footed Bride (reproduced as Example 1), which in a rather embryonic form is present as early as the 1933 "Exposition of Monophony,"(10) manifests a hierarchical disposition of the intervals in Partch's scale according to their comparative consonance. The forty-three scale degrees themselves are presented on the central parallel vertical axes, first ascending and then, once the midpoint of the scale has been reached, descending, with each ratio paired with its "complement," or inversion, in the other part of the scale. The graph shows clearly the symmetrical nature of the scale itself, as well as the straightforward rationale behind Partch's fourfold division of the intervals into the "arbitrary" categories of Intervals of Power, Intervals of Emotion, Intervals of Suspense, and Intervals of Approach. The relationships of each of the scale degrees in turn to the fundamental, 1/1, are rank ordered according to two, potentially contradictory, criteria: the complexities of their prime limits, and their status as "primary" or "secondary" ratios.(11) From the graph it seems clear that the secondary ratios, as a group, are judged the least consonant elements in Partch's scale, quite irrespective of their individual prime limits.

Yet for all the eccentricity of the image, this graphic representation of the comparative consonance of the intervals in his scale still seems insufficiently detailed. For example, no discrimination is made between the comparative consonance of the very different "semitones" 16/15 and 21/20; and it seems odd that the familiar 15/8, the just "major seventh," should be classed as less consonant than the highly complex 11/6, apparently for no other reason than its status in Partch's system as a secondary ratio. These oddities seem less a product of confusions or inconsistencies in his thinking than a result of the very convention of representing complex levels of information in a two-dimensional image. Further refinements or distinctions among the constituent ratios can only be drawn if the diagram were taken into three (or more) dimensions. It is almost as if the convention of two-dimensional representation itself sets limits on the usefulness of the model.

Of the two geometric projections, only the Tonality Diamond became embedded as a figure into the layout of strings or blocks in Partch's instrument designs. In fact, it proved to be enormously fruitful both in that regard and in relation to his compositional thinking. The One-Footed Bride, although a representation of many concepts that bear significantly on his compositional practice, has no relation as an image to any of the instrument layouts. The network of relationships embodied in the Tonality Diamond is one of the central concepts of Genesis of a Music, and it is worth studying the Diamond image itself in some detail before outlining its ramifications in Partch's instrument design and compositional activity. (The eleven-limit Diamond is reproduced as Example 2.)(12)

The Diamond is a mandalic configuration, and is precisely symmetrical around its vertical axis: the complement of a given ratio can be found at the corresponding point in the other half of the Diamond. Mandala, in Sanskrit, means "magic circle," and it is one of the oldest religious symbols. Carl Gustav Jung found that the experience formulated in the mandala pattern was endemic to people who could no longer find the image of God outside themselves: the perimeter of the mandala acted as a form of psychic enclosure, protecting the integration of the personality. Jung writes of a modern mandala that "[t]here is no deity in the mandala, nor is there any submission or reconciliation to a deity. The place of the deity seems to be taken by the wholeness of man."(13) Such wholeness, or psychic integration, is represented in Partch's Diamond by the remarkable coexistence and reconciliation of opposites that the model embodies. The coexistence is of major and minor - of Partch's hexadic Otonalities and Utonalities - in that if one tunes Utonality hexads downward from all six tones of the 1/1-Otonality hexad, all the remaining Otonality hexads appear in the process, without needing any independent generative mechanism.(14) In the Diamond the Utonalities run at right angles to the Otonalities - the Otonalities are given (diagonally) from bottom left to top right, and the Utonalities (diagonally) from bottom right to top left.(15)

Another notable feature of the Diamond image is that it is a pattern that retains its shape as it grows, like spiral forms, or like South Indian kolam chalk patterns. Although the Diamond was consciously used by Partch as a design in his subsequent instrument building activity, it is an image that is not in essence static. Symmetry, as the biologist Gregory Bateson remarked, is a result of the fact of growth: and the Diamond carries within it a memory, a record of its own past growth. We can see this in the simple sense that Partch in his book offers us Diamond figures carried out to different prime number limits (5, 11, and 13: later he designed one, which he never published, to 17). The nature of this growth is in accordance with his explicit theory of our increasing acceptance, through recorded history, of ever-higher prime-number relationships as valid musical material, and points the way to further expansions and refinements of the image and of the musical possibilities it suggests.

It is perhaps not too fanciful to see in the Diamond image and in Partch's other early models of pitch space a latent dimensionality, which seems all the more iconoclastic given the prevailing tendency of the models in the standard literature of the time. Most of the then-current psychoacoustical representations of musical pitch - including those of Helmholtz (1863), Mach (1906), and Koffka (1935)(16) - shared a general tendency to conceive of pitch as a one-dimensional attribute of sound. This was a result of the common sense percept that pitch is a continuum from low to high, bounded only by the perceptual thresholds of the ear. Mach wrote:

A tonal series occurs in something which is an analogue of space, but is a space of one dimension limited in both directions and exhibiting no symmetry ... it more resembles a vertical right line.

This tendency left unsolved the psychoacoustical problem that any representation of pitch as one-dimensional does not account for the phenomenon of octave equivalence: and octaves, moreover, are only one of many musically useful relationships that stand out from this continuum. Revesz (1913)(17) had suggested that pitch be treated instead as a bidimensional attribute, the first dimension representing overall pitch level (what would become known as tone height), and the second defining the position of a tone within the octave (tone chroma): but his work did not appear in English until 1954, by which time the idea had gained currency from other sources.

Against this backdrop the implicit multidimensional aspect of Partch's models stands as a corrective. It would remain for the next generation of theorists to devise such models explicitly. The earliest and still perhaps the most remarkable have been devised by Partch's friend Erv Wilson, whose astonishing reconfiguration of the Tonality Diamond as a set of centered pentagons in a non-Euclidian embedding space is reproduced below as Example 3.(18) This shows that the Diamond image, however inevitable it seems in retrospect, is only one possible geometric projection of the interlocking nature of the primary Otonalities and Utonalities in Partch's scale. In Wilson's reconfiguration the twelve tonalities appear as a set of twelve nested pentagons: those with the apex pointing upward represent Otonalities, those with the apex pointing downward Utonalities. The centre of each pentagon represents the 1 identity of the tonality.(19)

The relationships symbolized in the Tonality Diamond are embedded deep within the substratum of much of Partch's music as the result of two main modes of thought: first, that its form is projected directly in the layouts of several of his instruments, most obviously the Diamond Marimba,(20) and that in working pragmatically with the instruments and composing idiomatically for them he was working directly with the Diamond configuration itself; and second, that the nature of the configuration suggested a particular kind of harmonic language, with which he was then able to work directly, even on an instrument (for example, the Chromelodeon) that had no structural connection with the Diamond layout - a language that stemmed directly from the interlocking nature of the Diamond's constituent hexads.

Partch's harmonic writing often consists of sequences of these hexads - or, to put it a little differently, of tracing different paths through the Diamond. This type of harmonic language can be found in his output as carly as Dark Brother (1942-43), a setting of a passage from Thomas Wolfe, large stretches of which consist of Diamond-derived hexads in a nondirectional sequence. A much later example is Verse 15, for Chromelodeon I and Kithara I, from And on the Seventh Day Petals Fell in Petaluma (1963-66), an example of extended consonance. The music is, in fact, a prolonged passage of what Partch termed "tonality flux": a sequence of chords, each of which resolves onto the next by narrow intervals. This principle often characterizes his writing for the Chromelodeon, whose keyboard layout, with its wide physical spans producing relatively narrow intervals, seems to have influenced him toward the narrow resolving distances in this type of harmonic movement. The chords in the passage reproduced as Example 4(21) are almost all tonalities drawn from the Diamond: the occasional inharmonic tones can be regarded as conventional passing notes (measure 27) or as ornaments (measures 28-29). There is, however, one example of a hexad outside the Diamond, 5/3-Utonality, in the midst of the sequence (measures 11-12, 29). I spoke above of Partch's harmonic writing amounting to tracing different paths through the Diamond: here it is as though the path had led to a hexad that can only be found on a plane parallel to the image on the page. On neither of its appearances does the non-Diamond hexad seem harmonically out of place. This shows that the Diamond does not map out a closed, wholly self-contained harmonic world: it is only a detail of the larger geometry of Partch's scale fabric. The metaphorical correspondence of the Diamond and the mandala, and the idea of the "psychic enclosure" of the perimeter, breaks down at this point.

[Musical Expression Omitted]

The principle of tracing paths through (and beyond) the Diamond becomes even more graphically explicit as Partch's ensemble of instruments developed further. The instruments, with their diverse ranges and layouts of pitches can, on one level, be thought of as embodying different projections of the interrelationships within the scale. Each instrument manifests its own particular sense of the scale's internal geometry. Composition (observation, of a kind) is done in the ambience of these models - these projections - and the adequacy of the models for the fruitfulness of the compositional process helps engender the rotation of the cycle.(22)

The layout of blocks or strings or found objects on Partch's instruments was determined in part by an abstract visual aesthetic, by the patterns created by different projections of the relationships within his tuning system - projections of interval and scale patterns that demanded new geometric forms of their own. These layouts may be complex, like the hexad tunings on the Kitharas, or simple, like the forty-three-tone scale sequence projected on the Chromelodeons, the Boo, and used for some of the Harmonic Canon settings.(23) The study of the layouts themselves is important given the highly idiomatic nature of Partch's instrumental writing. The gestural patterns characteristic of especially the later, percussion-based works - which most often take the form of short reiterated figures involving a particular mallet pattern (on the percussion) or arpeggiation pattern (on the strings) that stays constant as it is applied to different blocks or strings (i.e. different pitches) - result in figures that sometimes overlap with other structural configurations, such as tonalities (as is often the case on the Diamond Marimba or the Kithara), scales (as on the Harmonic Canon or the Boo), tetrachords, harmonic formulae or suchlike, and at other times do not. These gestural patterns are often used together with more abstract musical materials as primary elements in the compositional process.

BEN JOHNSTON

Ben Johnston's work embraces several of Partch's concerns but extends his achievement in several ways: by forging links with European music in ways in which Partch was not interested; by applying Partch's radical ideas on tuning to conventional instruments and in new compositional contexts; by a thorough reassessment of Partch's terminology; and by devising a new notation system and new models of pitch. In the process, Partch's own work begins to appear in a new light.

Partch's ambivalent attitude to European music is often caricatured, notably by those, as Johnston has remarked, who see him only as an unholy patron saint to dropout artists. Although Partch's work implies a rejection of much of the essence and many of the attributes of Western concert tradition, it was not the music itself he detested. Rather, it was its prestige in the eyes of the moneyed minority in America who support the symphonic and operatic worlds, a condition that he regarded as a form of reverential necrophilia that only numerous acts of individual creative anarchism would dissipate. Johnston, by upbringing and inclination, feels far closer to that European heritage and holds no wish for its obsolescence. He has been concerned to affect a reconciliation between Partch's aggressively anarchic creative stance and the expressive worlds of European music, with Partch's radical departures in theory as the catalyst. Johnston began to feel, by the end of the 1970s, that serious music needed Partch "more than he ever needed it. He addressed problems it is ignoring with far more than tentative success, and he diagnosed many of its most serious ills with uncanny accuracy."(24)

Partch was, in the 1920s, largely unfamiliar with the syntactic crisis of contemporary European music. Johnston, however, showed that Partch had formulated a music-theoretic standpoint that implied a striking diagnosis of the immense strain underlying this phase of European music: a strain resulting from the attempt to forge new nontriadic harmonic units and distant reaches of tonal space out of a conceptually exhausted intonational framework that was incapable of realizing them. Partch's own music sidesteps the (wonderful!) syntactic confusions of the European music of circa 1910 and creates an idiom that omits all traces of "serious," Expressionist angst by refusing to push the harmonic language within which it thrived further beyond the breaking point.

Partch's work in tuning provided a radical perspective on early twentieth-century musical developments, but one that it fell to Johnston to formulate. In his writings of the 1960s, Johnston takes the position that Schonberg, "in tacitly accepting as an arbitrary 'given' the twelve-tone equal-tempered scale ... committed music to the task of exhausting the remaining possibilities in a closed pitch system."(25) A different path was suggested by Debussy, "whose harmonic language approximates as well as can be in equal temperament a movement from overtone series to overtone series, with an emphasis upon higher partials":

Schoenberg is an example of a radical thinker motivated strongly by a claustrophobic sense of nearly exhausted resources. Debussy, in sharp contrast, seems motivated by an expansion of harmonic resources and a greatly widened horizon.(26)

Because Schonberg's radicalism has not been eroded by the passing decades, and because the artistic products of fin-de-siecle Paris have come to seem in general more palatable than those of fin-de-siecle Vienna, we forget the revolutionary aspect of many artists' work in Paris in the late nineteenth century. The leprous blue shadows of a Monet painting shocked Parisian gallery-goers in the 1870s, as the radical syntax and form in the work of Mallarme shocked a literary audience; Debussy's work, in its own time, provoked scarcely less controversy than Schonberg's. The aspect Johnston focussed upon was not the "pink Debussy reverie" (as Partch dismissively put it) that so many music lovers esteem, but precisely those technical features of harmony and phrase structure that are so iconoclastic. By the 1980s Johnston had come to see his own work retrospectively as an attempt "to connect Debussy and Partch, to complete the revolution [begun earlier in the century! and connect it with a redefinition of older values."(27)

It was to be nearly ten years after Johnston's six-month "apprenticeship" with Partch on the far northern California coast in 1950-51 before he began to compose music in extended just intonation. Once he had decided to confront head-on the challenge of writing music for conventional instruments using complex microtonal tunings and of re-educating the intonational habits of performers, the transformation in his musical output was rapid. The real breakthrough works were Knocking Piece and Sonata for Microtonal Piano of 1962, followed in 1964 by String Quartet No. 2. The aesthetic of these works posits an alignment of Partch's complex theoretical ideas with the concepts and concerns of "new music" as a whole. In 1963 Johnston wrote:

If contemporary music produces images of tension and anxiety (and worse states) we cannot deny it is holding up a mirror.... A habitual psychological state of high tension such as contemporary life tends to produce is a matter for serious concern. Art can help us by bringing to recognition, analyzing and making intelligible the complex patterns of these tensions.... To extend musical order further into the jungle of ... complexity ... that is perhaps the fundamental aim of contemporary serious music.(28)

The concern with systems of order within complexity is the hallmark of Johnston's work of these years. Indeed, regardless of the changing stylistic orientation of his eclectic output, the underlying aesthetic agenda of all his work has remained how "the extreme complexity of contemporary life [can] be reconciled with the simplifying and clarifying influences of systems of order based upon ratio scales."(29) This concern was parallel to the explorations in the sixties of composers such as Cage and Xenakis, except that Johnston's domain was closer to "the kind of complexity needed to understand the intricate symbiotic interdependence of organic life on earth" than to "the kind which clarifies the statistical behavior of inanimate multitudes."(30)

Johnston found in the work of the Harvard psychophysicist S. S. Stevens the materials from which to establish a formal hierarchy of types of perceptible order. In his 1964 article "Scalar Order as a Compositional Resource" Johnston contended that we listen to music from moment to moment on changing levels of this hierarchy.(31) At the lowest level of perceptible order was the apprehension of distinctions as basic as same and different. At the next, slightly more discerning level, was the apprehension of the distinctions more than and less than; by application of this type of measurement we make such everyday distinctions as louder or softer, higher or lower (with respect to pitch), longer or shorter, and so on. Taking his cue from Stevens, he termed these nominal and ordinal types of measurement. At the higher levels of perceptible order were complex proportional relationships of pitch and rhythm - interval and ratio scales. Johnston argued that our perception of pitch especially was sufficiently acute and discriminatory that we could make much more complex judgements than those in common practice, but that equal temperament - both as a tuning practice and as a language - confounded relationships that were distinct.

Although these ideas found expression in verbal form, they are not irrelevant to a consideration of his music of this period. In the first movement of String Quartet No. 2 the rate of musical events is very rapid: small sound events and structures which pass by quickly function as self-sufficient complex units, and must be so registered. The constitution of these small units, and their succession, is characterized by great variability. Quite often something that strikes us as a distinctive rhythmic figure will have only vaguely memorable pitch contour (measure 1, Example 5.1),(32) or conversely a striking melodic figure will have very complex rhythmic definition (measures 29-31, Example 5.2): this implies a play of interval and ratio types of measurement against more general, "ordinal" measurement of contour. Sometimes Johnston places structural importance on a passage that impresses us mostly by the sudden increase or decrease in textural density or rhythmic activity - the ascent to a high pitch register is associated with an increase of rhythmic activity, the descent to a lower register with a decrease of rhythmic activity and of metric regularity (measures 1-3; measures 4-8; measures 14-17). Moreover, the beginnings and endings of these small units are frequently elided, and the way we register the musical continuity involves a constant flickering of attention between different parameters. Yet equally apparent aurally is the consistency of the musical language, which leads us to suspect that some strict forms of compositional control are being exerted.

[Musical Expression Omitted]

[Musical Expression Omitted]

It may be objected that these observations would apply to almost any twentieth-century music, especially to the idiom of sixties serial music. Yet it remains true that, while not unique to his own music, Johnston's model of the four types of scalar order establishes an aesthetic context in which his ratio scales (of pitch, and sometimes of rhythm) stand at the apex. This aesthetic context has a clear bearing, too, on performance practice: for, notwithstanding the overall difficulty of the music, the greatest challenges are in realizing accurately the tuning and the complex proportional rhythms - the more familiar ensemble problems seem secondary difficulties.

In String Quartet No. 2 the pitch relationships stem from a fifty-three-note scale, the second of the two different fifty-three-note enharmonic scales Johnston describes in "Scalar Order as a Compositional Resource." The first of these scales can be generated by a simplified form of the "branching" type of tuning process I described above: it is a simple three-limit sequence which consists entirely of chains of 3/2s above and below a centre pitch. For the second scale the branching metaphor no longer provides an accurate model: it is a five-limit sequence derived from chains of 3/2s and 5/4s above and below a centre pitch with, in addition, 5/4 relationships above and below some of the scale degrees generated by 3/2s. Although the scale is conceptually simple enough, Johnston does not - in that 1964 article - provide a geometric model of it. Instead, he offers a notation of the scale. This can be seen, in retrospect, as an early manifestation of what would become his most useful model of pitch relationships: far from simply providing Johnston with a way of writing his music down, the very existence of a sophisticated notation system of this kind has itself made possible the composition of his substantial output of music in extended just intonation.

The question of an integrated system of notation had been left unresolved by Partch. His attempts to devise a workable notation for his just intonation resources in the years 1923-33 had led to some potentially successful systems, although all of them inordinately difficult to read and thus to gain familiarity with. Those attempts had given way, in the decade that followed, to the more practical expedient of an elaborate set of tablatures based upon the individualities of each of his instruments. Johnston's system, although designed to be as little different as possible from conventional notation, has one significant difference: it embodies precise frequency ratios, rather than (as in conventional notation) a set of pitch classes, the exact tuning of which is left to unspoken convention.

It might seem that the notation of a fifty-three-note scale would prove a formidable problem, but Johnston solved it by redefining the symbols of conventional staff notation to make them clear and unambiguous as a representation of just intervals. First, the seven notes on the staff without accidentals are assigned precise ratio values - a five-limit just major scale.(33) Second, sharps and fiats are also assigned a precise value: a sharp raises a note by the same interval by which a fiat lowers it, but because that interval - the ratio 25/24 (71 cents) - is not an equal-tempered semitone, enharmonically equivalent notes are kept distinct. He also used double sharps and double fiats. The only "new" notational symbols needed are pluses and minuses, which symbolize respectively raising and lowering a pitch by a syntonic comma, the ratio 81/80.

Johnston uses the pitch resources made possible by this notation system to their fullest advantage in String Quartet No. 2. The first movement is concerned essentially with two interwoven processes. First, each measure contains exactly twelve pitches: not twelve attacks - there are frequent two- or three-part simultaneities - but twelve different pitch classes, which turn out to form permutations of a twelve-note row. Leaving aside for a moment the matter of the spelling of the pitches, and thus the question of enharmonic equivalence, we can say that the strict application of this principle governs much of the pitch content of each measure, and therefore of the whole movement.

However, the second process, at work in conjunction with the first, provides an ingenious twist to the apparently straightforward serial rhetoric just described. It is that the first pitch in each measure, usually on the downbeat, forms in sequence a stepwise progression through the degrees of Johnston's fifty-three-note scale. So, measure 1 begins with C in violins 1 and 2; measure 2 has C+ in violin 2; measure 3 has D[double flat] - in the viola; measure 4 has C[sharp] in violin 1; measure 5 C[sharp]+ in violin 2, and so on (Example 5.1). This process is applied strictly throughout the movement, which appropriately has exactly fifty-four measures, the fifty-fourth marking a return to C.

In a just intonation scale fabric such as the one Johnston employs here, the twelve-note row - in whatever permutation - cannot simply be transposed up by a comma or some other microtonal interval in each measure. Within a just intonation scale, unlike an equal-tempered one, one cannot simply transpose a set of pitches up or down from any given starting pitch and have it come out with exactly the same sequence of intervals as in the original. In a fixed just intonation scale, even an elaborately microtonal one, not every interval can be tuned above (or below) every degree of the scale: that would require an infinite system, which is possible theoretically, but highly problematic in practical terms. So, beginning the row on a different pitch each measure means that the row, as it unfolds within each measure, undergoes a retuning - or, to be more precise, it is subjected to a changing set of tuning relationships. How this is done is specified, in the score, by a network of vertical or diagonal lines between the four parts. The lines establish the tuning relationships for each pitch. In measure 1, for example, the opening C is the note against which the viola's E[flat], is tuned, as a 6/5; the B in the second violin in measure 1 is the note against which the first violin's G[sharp] and the viola's G[natural] are tuned. In one sense these diagonal lines are unnecessary: Johnston's notation system is absolutely precise, and an interval that is written C-E[flat] can only mean the just minor third - no other tuning of the interval is intended. But the lines function for the performer as a visual guide in gauging the tunings in this complex pitch texture at a glance. Analytically, the lines can be thought of as a set of templates that guide us through the different spellings of the row in each bar. Thus, knowing that the first pitch in measure 2 is to be C+, Johnston establishes an audible tuning connection with a note at or near the end of the previous measure's row - in this case, the F in violin 1, which has to become an F+ (F raised by a syntonic comma) to form a 3/2 with the C+. This F+ is in turn tuned to the in the cello (D[flat]/F+ in Johnston's system is a 5/4). It becomes clear that the spelling of each note of the row is determined by this very complex grid of tuning relationships operating within the pitch texture. As such, Johnston may be said to be working not so much with a set of twelve pitch classes, but rather with twelve pitch categories, the precise tuning of each depending on its structural function.

This stepwise progression through the fifty-three-tone scale on the downbeats of the bars is not directly audible. First of all, the pitches are given octave transpositions more or less ad lib.; more importantly, the complex rhythmic and metrical language of this movement and the alternation of time-signatures (5/8 and 8/8) mean that the downbeats have no privileged aural impact. However, the effects of the process are perceptible at a different level when Johnston, later in the movement, presents literal or transformed repeats of some of the material already heard: the predetermined pitch at which they must start, because of the ascending scale progression, causes some internal fine-tuning, and gives the recapitulated material a quite different sonority and a correspondingly different impact than on its first appearance.

In his more recent work Johnston has expanded his notation system to make provision for intervals derived from the higher prime number relationships of the harmonic series - 7, 11, 13, ad infinitum. He has needed to devise only one new symbol for each new prime number (from seven and upwards Johnston calls these symbols chromas). These accidentals are used in combination for the more complex ratios: in his recent music it is not uncommon to find three such symbols applied to one note. A full chart of the symbols and their meanings is given as Example 6. The fact that the notation system has become infinitely expandable makes it an aesthetically appropriate correlate to the tuning system itself.(34)

The notation system helped Johnston realize a move towards an expandable pitch space free from the constraints of a fixed scale, an early manifestation of which we saw in the work of Partch. His music could thus work freely with interval and chord patterns rather than with the possible permutations and combinations of the members of a closed pitch set. A similar process can be seen at this time in the music of Lou Harrison, who explains:

After only a brief study of intervals it becomes clear that there are two ways of composing with them: 1) arranging them into a fixed mode, or gamut, and then composing within that structure. This is Strict Style, and is the vastly predominant world method. However, another way is possible - 2) to freely assemble, or compose with whatever intervals one feels that he needs as he goes along. This is Free Style, and I used this method first in my Simfony in Free Style.(35)

[TABULAR DATA FOR EXAMPLE 6 OMITTED]

It may be noted that the two composers' emphases are different: whereas Harrison posits "strict" and "free" pitch systems as a mutually opposed pair, Johnston's work suggests that all tuning practice (on instruments other than those of fixed pitch) implies an infinite space of which closed systems, such as twelve-note equal temperament, are only static approximations.(36) By the beginning of the 1980s Johnston could write in the program notes for his thirteen-limit String Quartet No. 5 (1979) that "I have no idea as to how many different pitches it used per octave."(37)

There was one other important component in this freedom: Johnston's use of lattice models as representations of seven-limit (and higher) pitch systems. Of all the satisfactory models of ratiometric systems lattices are perhaps the most straightforward geometrically, yet they have the great advantage of being capable, like Johnston's notation system, of great mutability and of infinite expansion. He began working with lattices around 1970, taking his cue from a not-very-deep acquaintance with the writings of Adriaan Fokker.(38)

The metaphor of a just intonation system as a branching out in several directions at once from a centre, which I suggested earlier in this paper, becomes much more exact if, rather than thinking of these lines of ratios as branches on a hypothetical (and mutant) tree, we think of them as comprising a lattice: an array of points in a periodic, multidimensional pattern. Each point on the lattice corresponds to a ratio. The lattice form needed is n-dimensional, each dimension corresponding to a different prime-number partial. Thus a Pythagorean scale - one built exclusively from multiples of 2/1 and 3/2 - would be represented in a lattice of two dimensions only, since the integers of its ratios involve only powers of 2 and 3. The lattice implied by a just diatonic scale, with the ratios 5/4 and 6/5 as the just major and minor third, and 5/3 and 8/5 as the just major and minor sixth, is three-dimensional.(39) Partch's eleven-limit scale would be represented on a five-dimensional lattice, corresponding to the prime factors 2, 3, 5, 7 and 11: all the ratios of 9, which Partch considered an independent "identity" in his theories of chord formation, appear as multiple-number ratios in this five-dimensional space, and do not need to be assigned a dimension of their own. One of the dimensions in each case, that representing 2, is not really necessary in a graphic representation as it only provides octave equivalents of the pitches contained within the other dimensions.

Example 7.1 shows the twenty-two-note scale used by Johnston in String Quartet No. 4 (1973), a seven-limit sequence, and Example 7.2 gives a mapping of the scale onto a three-dimensional lattice (3, 5, 7, omitting the 2 axis). The lattice is not merely a visual representation of the scale, but provides a provisional harmonic analysis of its resources: intervallic and chordal relationships emerge as simple geometric patterns within the lattice space.

[Musical Expression Omitted]

Johnston provides an extensive rationale for his use of lattices, both technical and aesthetic, in his article "Rational Structure in Music," where he explains:

Extrapolating the basic logic from the harmonic practice of traditional Western music is only a first step. A much more challenging and interesting follow-up is the generalization of this logic so that it becomes applicable to unfamiliar pitch materials.(40)

The lattices provided Johnston with the theoretical model he needed to help in the exploration of relationships based on partials higher than the sixth. It also gave him a method of making sophisticated comparisons between triadic systems and larger ones. "I became interested in the morphological analogy between structures, such as scales based upon triadically generated bases and the much more complex ones generated by using higher overtone relationships."(41) Unlike Partch's Tonality Diamond, which offers a projection of the chordal relationships within the primary ratios of his scale but nothing beyond, the lattices provide a total mapping of the pitch space.

The other advantage of lattices was that they demonstrated what Johnston termed "harmonic neighbors." Ratios in near proximity on the lattice are simpler relationships aurally than those further away. For example, in the 2, 3, 5, 7 lattice in Example 7 a just major triad consists of a point in the lattice and one point in a positive direction on the 3- and 5-axes. More complex relationships, such as the septimal minor triad 1/1-7/6-3/2, can be "explained" by examining its geometrically more elaborate configuration on the lattice. This opened up the possibility of specifying closer or more distant harmonic areas: "[p]atterns or 'paths' in the lattice may be used to give a sense of harmonic direction even to passages that lack entirely any conventional tonal or harmonic 'logic.'"(42)

Partch's visual projections of his pitch resources had a direct bearing on his compositional practice, and the same is true, though to a lesser extent, in Johnston's case. Initially, thinking in terms of lattice representations seems to have been useful in clarifying - in the abstract - the tuning relationships within a seven-limit, or higher, space. With practice he could keep the placement of a chord progression or a modulation on the lattice in his mind as he worked. Eventually, his facility with the lattices suggested new compositional applications of a rather complex order: the lattice-derived final movement of String Quartet No. 7 (1984), which stands as a tour de force in his explorations of the relationships within (in this case) a thirteen-limit lattice space, "couldn't have been written without a very careful adherence to ... the lattice and the scales derived from it."(43) The Seventh Quartet, which contains some of the most microtonally elaborate writing in Johnston's output, remains unperformed. In one important sense a full investigation of the work must await its badly overdue premiere: String Quartet No. 7 provides an example of how some of Johnston's compositional designs are less in need of further analysis than they are of more careful and widespread performance.

JAMES TENNEY

Johnston's lattice models have provided the starting point for the more recent multidimensional models of pitch space developed by James Tenney. Tenney had encountered both Johnston and Harry Partch at the University of Illinois during his studies there in 1959-61, and for some six months he was Partch's assistant. As had been the case with Johnston, it would be some time before an interest in tuning - and in particular the relationship between tuning and harmony - manifested itself in Tenney's work.

[M]y early contact with [Partch] - and, more especially, his book - planted a seed that would only begin to sprout over a decade later (in 1972, with Quintext), and has continued to grow. I now see his theoretical work ... not as a complete theory of harmony or harmonic perception (or certainly not as a sufficient one), but as perhaps the most important of a small number of 20th-century contributions toward the development of such a theory.(44)

Partch's theoretical work was not an ex post facto extrapolation from an existing body of compositions, but rather a largely intuitive clearing of the ground in the attempt to give his musical activities a solid theoretical underpinning. In Tenney's case the involvement with tuning manifested itself first in composition and only later in his theoretical work. Quintext: Five Textures for String Quartet and Bass was written in a period when Tenney had wanted to work with very simple musical materials, as though to reinvent the act of composition from scratch: the harmonic series provided a starting point. His theoretical work, in addressing the issue of tuning, inherits a legacy of ideas not only from Partch but from Johnston as well, whose work Tenney readily acknowledges as an important extension of the older composer's ideas.

It may at first seem surprising, then, that Tenney's most definitive presentation to date of his thinking on the matter of tuning is to be found in a 1983 paper with the title "John Cage and the Theory of Harmony."(45) From the outset, Tenney's interest in tuning has been connected to an interest in harmony, and he sees refinements in intonation theory and practice as the essential step in establishing new harmonic relationships of the sort that had been absent in much of the music of European modernism as a consequence of the exhaustion of the tempered scale. In turning his attention to an investigation of tuning Tenney seems to have been concerned to clarify the possible relationship of this newly acquired interest to his own earlier theoretical work, and to his longstanding devotion to the work of Cage. The linchpin is Cage's emphasis on the multidimensional nature of sound, which Tenney relates to his own presentation of the multidimensional model of pitch perception that he calls harmonic space. The rather unusual form of the paper - a first part on Cage's writings on compositional method and aesthetics, and a second part outlining Tenney's model of harmonic space derived from Partch and Johnston, in the course of which (at least on a first encounter) the reader wonders what has happened to Cage - may be illuminated to some extent by the fact that it grew from two quite separate papers, neither of which was completed in its original form. The first was a long and highly technical paper on harmony ("full of unnecessary formulae") at which Tenney worked in the late 1970s, the second a paper planned for Cage's seventieth birthday celebrations in 1982.(46) At some point these two streams of thought reached a kind of confluence, and Tenney came to feel that Cage's work, "while posing the greatest conceivable challenge" to the effort to develop a theory of harmonic perception, "yet contains many fertile seeds for theoretical development."(47) In subsequent statements Tenney has modified this position somewhat, acknowledging more directly that "the work of Harry Partch has become ... an indispensable technical point of departure, just as Cage's work has provided us with an essential aesthetic foundation."(48)

One might go even further and suggest that in the matter of harmony Tenney is perhaps giving Cage credit where none is really due. If Cage's writings were "essential," they were so primarily for Tenney himself in deriving the confidence that an interest in tuning could be made to seem not incompatible with the thrust of Cage's work. Tenney's theory of harmonic perception does not really stand upon Cage's work to the extent that it does on that of Partch and Johnston. In fact by the end of the 1980s things had come full circle, and Cage was led to rethink many of his own ideas on harmony in the encounter with Tenney's music. In a 1990 interview Cage describes listening to a performance of Tenney's Critical Band and realizing that "there was no wall at all, that sound is by its own nature harmonious," that harmony simply meant sounds being together - a rather liberal paraphrase of Tenney's new definition! - and that there could be such a thing as "anarchic harmony."(49)

Insofar as Cage felt that a new definition of harmony was needed he was responding appropriately to one of the main thrusts of Tenney's work. Tenney's contention is that any effort to develop a new theory of harmony must first address a semantic problem: that the term harmony is conventionally bound up with the system of triadic tonality of the eighteenth and nineteenth centuries. A redefinition in more general terms is needed, one in which harmony would come to mean "that aspect of music which involves relations between pitches other than those of sheer direction and distance (up or down, large or small)";(50) so that "what a true theory of harmony would have to be now is a theory of harmonic perception ... one component in a more general theory of musical perception."(51)

Tenney's theoretical work in this domain attempts to make redundant what he has dubbed the "etiquette book" model of earlier harmonic theory, the aspect of it that suggests how to behave in certain situations. He holds instead that a new theory of harmonic perception should aim to be descriptive, which in his terms means it should be quantitative, to the maximum extent possible. In the 1983 paper he offers several other explicit criteria for this type of theory: that it be aesthetically neutral; culturally and stylistically general; and in consequence relevant to music of different periods and cultures. Given the aim of being descriptive, a quantitative theory seems suitably value-free. One might object, however, that the criterion that a theory of harmony be culturally and stylistically general overlooks the fact that the whole idea of precision of pitch and tuning necessary to a quantitative theory is itself a culturally bound notion, irrelevant to the ontology of some musical cultures in the world. Of these criteria it is principally the demand for a quantitative aspect that is satisfied by Tenney's adoption of the ratio model of pitch, and of "the language of ratios," as the basis of his work.

As in his earlier theoretical writings, Tenney's thinking on the subject of harmony is mediated through his interest in Gestalt psychology, but his application is never merely formal. At its root is his hypothesis that our ears interpret pitch relations in the simplest way possible: and in this sense he regards ratio representations of pitch relationships as referential rather than literal.(52) A ratio description of an interval is a reference point to which the intervals we hear - whether just, tempered, or simply out of tune - approximate. In this regard Tenney invokes the mechanism of tolerance, familiar from cognitive psychology, which holds that - within a certain tolerance range, or range of frequencies - some slight mistuning is possible without altering the harmonic identity of an interval. One of the corollaries of this, which will be explored in more detail later, is that Tenney feels no compunction about using tempered systems in his recent music. Although the ratio model implies an extended just intonation system, he has in specific compositional contexts accepted tempered systems, as approximations, as viable ways of working. His use of the ratio model of pitch differs in this rather crucial respect from that of Partch and Johnston.(53)

In Tenney's work a shift is apparent from the aural metaphors that characterized his earlier writings, especially on form,(54) to the visual metaphor of harmonic space that is the basis of his recent writings on the relation of tuning and harmony. It seems endemic of pitch as a perceptual domain that visual metaphors (and mathematical ones, like "space" itself) have proved to be so abundant and so fruitful. By harmonic space Tenney means to imply "a kind of abstract, perceptual space which is in some ways analogous to physical space,"(55) which, following Johnston, he has modelled as a lattice of discrete points. The space of the lattice is thus not a Euclidian one, but rather has a "city-block" metric. Extending the visual metaphor, Tenney speaks of the metrical and topological properties of harmonic space. He feels that harmonic space "is not symmetrical. It clearly has an up and down."(56)

Tenney's model of harmonic space, as he acknowledges, is a direct development from Johnston's lattice models, but by rethinking some details he has provided us with a wholly new perspective. One, as we have seen, is that although the harmonic space model is ratiometric, a tempered system may be mapped onto it by relating its pitches to the simplest ratios within their tolerance range.(57) Another is that although the model is a theoretically infinite n-dimensional lattice - like Johnston, Tenney too delimits the number of dimensions in any given model by the number of different prime factors required to specify the frequency ratios of a given set of pitches - the tolerance principle puts a significant constraint on the unlimited proliferation of points in harmonic space, and of dimensions in the lattice: beyond a certain point, large-number ratios furthest along the axes from the centre point, Tenney contends, become redundant, because the ear interprets them as mistuned versions of small-number ratios.

This latter point provides an instance of two composers' use of closely related models suggesting radical differences in compositional application. Much of Johnston's recent music has made unrestrained use of the higher primes as ways of deriving scale sequences or of tuning chords. Scale sequences using intervals analogous to those in the harmonic series up to the thirty-first partial, for example, are found in his String Quartet No. 9 (1987), and chords - Otonalities and Utonalities, to use Partch's terms - including relationships analogous to those up to the nineteenth partial are found in his jazz arrangements for string quartet. By contrast, within Tenney's harmonic space model the tolerance concept provides an in-built check on the unlimited use of higher primes: the optimum tuning of a chord is that ratio configuration that forms the most compact arrangement of its intervals within the lattice, which in practice often means staying within a five- or seven-limit.

Example 8 shows three chord forms from Johnston's Revised Standards, arrangements for string quartet of songs by Gershwin, Rodgers, and Kern. The first chord, a Utonality on A with the 1, 3, 5, 7, 9, 11, 13, and 15 identities, would be described in jazz terms as a D minor thirteenth chord, with sharpened seventh and a bluesy clash of B?[flat], and B[sharp] (respectively the 15 and 7 identities: the identities of a Utonality work downward from the fundamental). The second chord makes harmonic use of the interval 17/16, which Johnston holds as the "smoothest" tuning of the dominant minor ninth (here the chord is on E, with the 1, 3, 5, 7, and 17 identities). The third chord makes harmonic use of the interval 19/16, the "smoothest" tuning of the dominant augmented ninth; here the root is A, with the 1, 3, 7, 17, and 19 identities (the augmented ninth would in conventional notation be given as B#, but in Johnston's notation system the nineteenth partial is more conveniently seen as a lowering of the twentieth).

[Musical Expression Omitted]

Tenney's criterion of compactness would suggest different mappings of these jazz chords onto the harmonic space model. The most substantial difference is that Utonalities, as used by Partch and Johnston, do not form part of Tenney's harmonic vocabulary: he regards them more as a paper concept than as an acoustical reality, disagreeing with Partch (and others) on the origin of the minor triad in the undertone series.(58) The problem with the Utonality configuration, as Tenney sees it, lies in the ambiguity surrounding the root. In the first chord in Example 8, for instance, the 1/1 is A, even though the ear hears the root quite clearly as D. The configuration of a Utonality in harmonic space provides a direct analogy to this: tracing a Utonality on (i.e. under) any given point on the lattice involves moving one point in a negative direction on each axis, resulting in a complete topological reversal.

Perhaps Tenney's most crucial development of Johnston's lattice models is his concept of harmonic distance, a measure of the relative proximity of two (or more) pitches in the lattice. This represents a more formal statement of the idea of "harmonic neighbors" that we saw in Johnston's theoretical writings of the 1970s. Harmonic distance can be defined between any two points on the lattice as proportional to the sum of the distances traversed on the shortest path connecting them. In mathematical terms, harmonic distance is

proportional to the logarithm of the product of the two terms. This measure of harmonic distance thus shows how far one would have to go in the lattice to get from one point - say the 1/1 or reference pitch - to the point representing the interval, without cutting across the "empty spaces" in the lattice.(59)

The basic formula for harmonic distance is Hd(a/b) = k log(ab), where a/b is the frequency ratio representing the interval in its maximally reduced form, and k determines the unit of measurement (generally "octaves," so k = 1). Thus the harmonic distance of the interval 11/9 is log(11) + log(9). The harmonic distance idea is a necessary complement to the notion of compactness, a quality that can be made explicit "by speaking of the sum of harmonic distances [between] ... various points" in harmonic space.(60)

Tenney's concept of compactness in harmonic space is an interesting refinement of Partch's ideas of consonance and dissonance. Indeed, one of the original motivations for developing the harmonic distance concept was a desire to resolve the consonance and dissonance problem.(61) Tenney states explicitly that "tones represented by proximate points in harmonic space tend to be heard as being in a consonant relation to each other, while tones represented by more widely spaced points are heard as mutually dissonant."(62) This is not intended as a new definition but as an explanation of the correlation between the concept of consonance and dissonance and that of harmonic distance.

Partch's presentation of consonance and dissonance is in accord to a considerable extent with Helmholtz's definition of these terms. In the early "Exposition of Monophony" Partch includes a diagram entitled "The Progress of Consonance," which phenomenon is clearly (if implicitly) equated with the progressive acceptance of intervals derived from the higher prime-number partials as musical materials.(63) In Genesis of a Music he states unequivocally:

The scale of musical intervals begins with absolute consonance (1 to 1), and gradually progresses into an infinitude of dissonance, the consonance of the intervals decreasing as the odd numbers of their ratios increase.(64)

By this definition, consonance is a comparative matter. The main difference between this view and the definition found in the writings of most eighteenth and nineteenth century music theorists, as Tenney has pointed out, is that Partch's theory - like that of Helmholtz - has little to do with the musically functional sense of the terms consonance and dissonance in any given passage of music. Helmholtz's "entitive referents," writes Tenney, "are generally dyads or other simultaneous aggregates isolated from any musical context": his theory has "nothing to do with ... motion, resolution, or chordal connections of any kind. It refers merely to the perceptual character of individual chords."(65) Tenney compares this with the concept of consonance and dissonance found in Rameau and his successors, to which the notion of resolution is central. In his Harmonie-lehre (1911), Schonberg wrote:

the expressions "consonance" and "dissonance," which signify an antithesis, are false. It all simply depends on the growing ability of the analyzing ear to familiarize itself with the remote overtones, thereby expanding the conception of what is euphonious, suitable for art, so that it embraces the whole natural phenomenon.(66)

Elsewhere Schonberg states his position that "what distinguishes dissonances from consonances is not a greater or lesser degree of beauty, but a greater or lesser degree of comprehensibility."(67) Tenney has hoped by his own work to show that it is now possible, thanks to our increasing familiarity with the "remote overtones" and their placement in harmonic space, to move beyond Schonberg's position that "continued evolution of the theory of harmony is not to be expected at present."(68)

In a lecture at Darmstadt in July 1990 Tenney remarked that his work with the harmonic space lattices has led him to conceive of music as "activity in harmonic space." Something of the nature of that activity becomes clearer if we explore the workings of two recent, and very different, compositions: Study No. 1, ("Modesty"), from Book 1 of Changes: Sixty-Four Studies for Six Harps (1985), and Critical Band, for variable ensemble and tape delay (1988).(69)

Changes is different from all the music so far considered in this paper in that the work is not intended to be performed in an extended just tuning. Rather, the harps are tuned in equal temperament but a sixth of a semitone apart from each other, providing a total gamut of seventy-two equal-tempered pitches in the octave. Early in the precomposition process Tenney worked out several of the possible mappings of this seventy-two-note scale in harmonic space, assigning to each interval in the scale one or more possible ratio equivalents. These ratio values were the referents in determining the harmonic procedures of the work, the tempered intervals providing close approximations to them.(70)

In determining the harmonic space lattice for Changes Tenney decided to assume an eleven-limit for the modes (although less complex modal configurations are often in evidence), a seven-limit for root progressions, and to make the closing cadential movement in each Study move in a negative direction along the three-axis, i.e. by descending fifths. Each study is assigned an overall "tonic" pitch class, but as most of the note-to-note details of Changes were determined by stochastic processes the similarity to conventional harmonic practice ends there.

Not all of the individual studies actually use all seventy-two pitches in the octave. Study No. 1 uses fifty-eight different pitches, but in a total duration of 1[minute]55[seconds] the microtonal complexity of the music is still considerable. The study is monophonic, although its "line" consists of a rather free mixture of single notes and two- and three-note aggregates. The tonic of this study is E?? (three-sixths of a semitone below E), although its importance really lies on an organizational level below the perceptual surface, where it is only evident twice, near the beginning and at the end.

The Study manifests different kinds of movement within its harmonic space. In any given phrase for each harp (or, to use a term Tenney coined in his early treatise META+HODOS and which remains relevant to his compositional thinking, in each clang) the pitches used move solely along a three-axis, usually by adjacent ratios, or by skipping one or (at most) two on the axis. In more conventional terms, this means that the pitches in each phrase for each harp consist of stacked fifths (or fourths): the lines actually move by fifths or fourths much of the time, which has the effect of making large stretches of the individual parts virtually sight-readable. However, these intervals are not always directly perceptible, as the continual weaving of the monophonic "line" among the six instruments means that a much richer fabric of interval patterns is built up.

Example 9.1, showing the final five measures of the Study, provides an illustration of this. The pitches used form a compact arrangement in harmonic space, as can be seen in Example 9.2. Working backwards from the final "tonic" the pitches assemble themselves into a modal configuration. Thus, the final pitch, E??, is 1/1; the preceding B?? is 3/2; and the F[sharp]?? 9/8. (Relative to this the C[sharp]?? at the end of measure 25 is 27/16.) The pitches of the Harp 6 part in this passage are thus confined to three-limit relationships. The D??, G??, and A?? in Harp 1 act as a highly diatonic filling in of this closing mode: they can be interpreted respectively as 15/8, 5/4 and "tritone" (ambiguously 7/5 or 45/32). These pitches imply a three-dimensional (five-limit) harmonic space (if the "tritone" is taken as 45/32). The pitches in the Harp 2 line, D?? and A??, 7/4 and 21/16 respectively, imply seven-limit relationships. In the final two measures the "activated" points in harmonic space grow fewer as the cadence approaches - as though the music were closing in on its final tonic.(71) The different possible mappings of the seventy-two-note scale yield, in practice, several useful ambiguities about the precise harmonic identity of specific intervals, such as that of the A?? in the closing measure, but here its effect on the cadence is unsettling in either interpretation because of the "tritone" relationship it forms with the tonic.

Throughout Tenney's output an equally pervasive model of pitch relationships is provided simply by the harmonic series itself. Several pieces - including Spectral CANON for CONLON Nancarrow (1974), Saxony (1978), and Voices (1983-84) - use a set of pitches analogous to the first twenty-four partials of the harmonic series (or, in Saxony, the first thirty-two). Harmonic-series-derived pitch sets are also found in the music of Johnston, notably as the basis of piano tunings (Suite for Microtonal Piano, Twelve Partials for flute and piano, and others), but play no significant role in the music of Partch.

What is interesting about these pieces - other than the intrinsic interest of the music itself - is Tenney's attitude to the nature of their harmony. He has written, in an interesting reversal of Partch's premise, that "it is the nature of harmonic perception in the auditory system which 'explains' the unique perceptual character of the harmonic series, not ... the other way around. The harmonic series is not so much a causal factor in harmonic perception as it is a physical manifestation of a principle which is also manifested (though somewhat differently) in harmonic perception."(72) The pitch set of a work like Voices forms a compact (though nine-dimensional!) configuration in harmonic space.

[Musical Expression Omitted]

A more recent work, Critical Band, uses a set of fifteen different pitch classes that can be derived by the ancient Greek concept of harmonic mean.(73) The harmonic mean, a principle associated with both Pythagoras and Archytas, is the division of an interval into proportionally equal parts - analogous to the division of a string length into halves - which does not result in an aurally equal division. Critical Band grows out of the conventional ensemble tuning process, and begins with a long, sustained A-440, from which other pitches begin to "fan" out geometrically, both above and below. (A graphic representation of the work is given as Example 10.) The ascending sequence of intervals, twelve in all, begins (following the initial 1/1) with the pitches 129/128 and 65/64, which establishes the principle: 129/128 is the harmonic mean of 65/64 (the proportions of the two pitches to the fundamental are as 128:129:130). Then, 65/64 is "reinterpreted" as the harmonic mean of a wider interval, 33/32 (the proportions are 64:65:66). Next, 33/32 is reinterpreted as the harmonic mcan of 17/16 (32:33:34); 17/16 as the harmonic mcan of 9/8 (16:17:18); 9/8 as the harmonic mean of 5/4 (8:9:10). At this point the harmonic mean derivation stops, or rather merges into a harmonic series-related set of proportions for the remaining pitches (10:11:12:13:14:15:16). For the descending sequence Tenney uses a similar principle: in this case the sequence of proportions is 128:127:(126)63:(62)31:(30)15:(14)7:6:5:4:2. Perceptually, the form of the work results from the gradual transition from the dense clustering of pitches near the beginning - the total pitch-range of which does not exceed the limits of the "critical bandwidth" of the opening A-440 (a "major second" above and below) - to the larger intervals that, as the work proceeds, increasingly emerge from the texture as harmonic relationships. Because of the use of very high primes in some of the small intervals near the beginning (e.g. 127), the fifteen pitch classes would assume a very elaborate configuration in harmonic space, but that configuration becomes markedly (and progressively) simpler as the music proceeds. In Critical Band Tenney's conception of music as "activity in harmonic space" is seen to embrace one of the most ancient ratio models of pitch.

CONCLUSION

The musics of Harry Partch, Ben Johnston, and James Tenney have, ultimately, more points of difference than of contact: important differences of style, aesthetic, and technique, even though their pitch vocabularies may be identical, and different from those of the great majority of their contemporaries. Far from forming a coherent and distinct musical tradition linked by a clear linear progression in historical terms, I feel their work is best viewed as a set of concentric circles tracing very different paths around a core of closely related, and quite individual, ideas. The ratio models of pitch found in their theoretical writings have not merely the two functions I mentioned at the outset of this paper - the description and the explanation of an unfamiliar pitch vocabulary - but a third, perhaps the most important: discovery. The present exerts an influence on our perception of the past, and the work of Johnston and Tenney enables us to reassess the nature and significance of Partch's theoretical legacy in the light of its metamorphosis in the present. More than this, I feel their work offers an example of a peculiarly American form of influence: that of a legacy of ideas, the provision of a decisive stimulus in terms of optimism and creative energy without implying a necessary identification with the overall artistic stance. It now seems beyond question that Partch's legacy will prove to be one of the crucial facets of compositional activity in the new century.

ACKNOWLEDGMENTS

I am grateful to Ben Johnston and to James Tenney for their helpful comments on an earlier draft of this paper, and for their permission to quote from previously unpublished writings and correspondence; to Danlee Mitchell, president of the Harry Partch Foundation, for permission to quote from unpublished materials by Harry Partch; to Erv Wilson, for allowing me to reprint from Xenharmonikon his reconfiguration of Partch's Tonality Diamond; and to Larry Polansky, for a careful reading of the paper and for many helpful suggestions.

NOTES

1. First edition, Madison: University of Wisconsin Press, 1949; second edition, enlarged, New York: Da Capo Press, 1974. All references to the book in this paper are to the second edition.

2. Two other important American figures who have acknowledged the centrality of Partch's work to their own use of just intonation, Lou Harrison and LaMonte Young, are not considered in this paper. Their theoretical work seems to me to owe at least as much to other sources, notably non-Western ones, as it does to Partch himself, whereas the writings of Johnston and Tenney confront Partch's thought more directly.

3. Harry Partch, "A Quarter Saw Section of Motivations and Intonation." An extract from this lecture is included in Partch, Bitter Music: Collected Journals, Essays, Introductions, and Librettos, edited with an Introduction by Thomas McGeary (Urbana and Chicago: University of Illinois Press, 1991), 197.

4. Harry Partch to Peter Yates, 25 September 1952. Peter Yates Archive, Mandeville Department of Special Collections, University of California, San Diego.

5. Hermann von Helmholtz, On the Sensations of Tone as a Physiological Basis for the Theory of Music. Second English Edition, translated by Alexander J. Ellis (London: Longmans, Green, and Co., 1885). Originally published as Die Lehre von den Tonempfindungen als physiologische Grundlage fur die Theorie der Musik (Braunschweig: Friedrich Vieweg und Sohn, 1863).

6. Harry Partch, "Exposition of Monophony" (1933 draft, unpublished typescript): manuscript now in the possession of Jonathan Glasier, San Diego, California. "Monophony" was the name Partch gave to his system of extended just intonation.

7. This aspect of Partch's early work is explored in detail in my Doctoral thesis "Harry Partch: the Early Vocal Works 1930-33" (Queen's University, Belfast, 1992).

8. A good example of this can be found in the early works for voice and Adapted Viola: see the analyses in my paper "On Harry Partch's Seventeen Lyrics by Li Po," Perspectives of New Music 30, no. 2 (Summer 1992): 22-58.

9. Genesis of a Music, 134 and 128 respectively. The concept of Perpetual Tonal Descent and Ascent, which Partch discusses in Genesis of a Music, 128-31, but which does not appear in the 1933 Exposition of Monophony, was used by Partch in his work on the design of a Ratio Keyboard in Visalia in the summer of 1932, for a purpose similar to its later use in Genesis of a Music: that of listing all the "successive-number" (or superparticular) ratios for purposes of deciding which should be included in the wide "gaps" at either end of his scale.

10. "Exposition," 34. In the "Exposition" the graph is drawn with respect to a thirty-seven-tone scale, the theoretical norm for Partch's work at that time (June 1933).

11. Secondary ratios are multiple-number ratios included by Partch in his scale to fill the wider gaps in the sequence of the twenty-nine primary ratios of the eleven limit. See Genesis of a Music 127-32.

12. The Diamond does not figure as an image in the 1933 "Exposition of Monophony," though all the relationships that it embodies, and the idea of the interlocking hexads, are made explicit in that early text.

13. Carl Gustav Jung, Psychology and Religion (New Haven: Yale University Press; London: H. Milford, Oxford University Press, 1938), paragraph 139.

14. An Otonality is that set of pitches generated by the numerical factors (which Partch calls identities) 1, 3, 5, 7, 9, and 11 in the numerator over a numerical constant (which he calls the numerary nexus) in the denominator. Conversely, a Utonality is the inversion of an Otonality, a set of pitches with a numerical constant in the numerator over the numerical factors 1, 3, 5, 7, 9, and 11 in the denominator.

15. The concept of the coexistence of major and minor was, of course, not original to Partch, and was noted by Zarlino as early as the sixteenth century. In the 1933 "Exposition of Monophony" Partch credits the discovery to Zarlino: in Genesis of a Music he is less direct, placing the acknowledgment not in the context of the discussion of the Diamond but mentioning it almost in passing in Part Four, on the history of intonation. This may not be subterfuge: it is more likely that he was unwilling to distract from the theoretical discussion in Part Two by introducing a snippet of history.

16. Hermann von Helmholtz, On the Sensations of Tone. Ernst Mach, Erkenntnis und Irrtum: Skizzen zur Psychologie der Forschung, 2d ed. (Leipzig: J. A. Barth, 1906). Kurt Koffka, Principles of Gestalt Psychology (New York: Harcourt, Brace and Company, 1935).

17. Geza Revesz, Introduction to the Psychology of Music, trans. G. I. C. de Courcy (Norman: University of Oklahoma Press, 1954).

18. Reproduced from Xenharmonikon 3 (Spring 1975): cover. Wilson's work on higher-dimensional and non-Euclidian pitch spaces, and his numerous inventive designs for microtonal marimba and keyboard layouts, remain virtually unknown to those who read only the mainstream music journals: almost all his work is published in the journal Xenharmonikon, which he cofounded with John Chalmers in 1974, and which is currently available from Frog Peak Music, Box 1052, Lebanon, NH 03766.

19. It is significant that Wilson essentially reinvented the Diamond image for himself, thinking of it as a network of harmonic and sub-harmonic cross-sets, and only later realized that the relationships that resulted were exactly the same as those of Partch's Diamond. Erv Wilson, conversation with the author, August 1993.

20. In the Diamond Marimba the arrangement of identities in each hexad is different from that in the Tonality Diamond: compare Genesis of a Music, 159 and 261.

21. In this transcription the Chromelodeon I part is given in approximate conventional notation, with cents deviations (relative to the tempered pitches) written above the noteheads. The exact ratios are given at the top of the staves. (The noteheads of the right-hand dyads are printed obliquely purely for ease of reading.) The Kithara I part is given in the same way, with ratio values added below the staff. The score of Petals was published in Source 1, no. 2 (July 1967), and the original recording is available on the compact disc The Music of Harry Partch, CRI CD 7000.

22. An example of this is the marimba that Partch built in 1965, the Quadrangularis Reversum, which reconceives the Diamond Marimba model in a kind of mirror inversion. Partch arrived at the Quadrangularis Reversum pattern intuitively, and the precise nature of its relationship to the Diamond Marimba layout was only later explained to him in detail by Erv Wilson. (The layouts of the two instruments are given in Genesis of a Music, 261 and 269.) The downward arpeggio patterns made possible by the tiered rows of blocks on the Diamond Marimba - which can be heard, for example, in the river motif in Daphne of the Dunes - are complemented by the upward arpeggios characteristic of the Quadrangularis Reversum. The two are played off to great effect in the Exordium from Delusion of the Fury.

23. Harmonic Canon I is tuned this way in "The Wind" and "The Street" from the Eleven Intrusions, and in Ring Around the Moon, and Harmonic Canon III (right canon) is tuned this way in The Dreamer That Remains.

24. Ben Johnston, "Beyond Harry Partch," Perspectives of New Music 22, nos. 1 and 2 (1983-84): 223-32.

25. Ben Johnston, "Three Attacks on a Problem," in 2nd Annual Proceedings of the American Society of University Composers (New York: Department of Music, Columbia University, 1967).

26. Ben Johnston, "A.S.U.C. Keynote Address," Perspectives of New Music 26, no. 1 (Winter 1988): 236-42.

27. Ben Johnston, "A.S.U.C. Keynote Address."

28. Ben Johnston, in the program booklet for the 1963 Festival of Contemporary Arts, University of Illinois at Urbana-Champaign.

29. Ben Johnston, "Extended Just Intonation: A Position Paper," Perspectives of New Music 25, nos. 1 and 2 (1987): 517-19.

30. Ben Johnston, "On Bridge-Building." Unpublished manuscript, circa 1977, courtesy of the author.

31. Perspectives of New Music 2, no. 2 (Spring-Summer 1964): 56-76.

32. Johnston's String Quartet No. 2 is published by Smith Publications, 2617 Gwynndale Ave., Baltimore, MD 21207. A recording of the work by the Composers Quartet was released on Nonesuch H-71224.

33. The sequence (starting from C) is: 1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1.

34. One aspect of Johnston's notation system that has caused some controversy is that its basic scale already represents a hybrid system (2, 3, 5). Rudolf Rasch (personal communication, July 1990) has suggested a modification of the system such that the C major scale on the staff is a Pythagorean (three-limit) scale. This necessitates some additional redefinition of the comma and chroma symbols. See also Paul Rapoport, "Just Inton(ot)ation," in 1/1: The Journal of the Just Intonation Network 7, no. 1 (September 1991), 1, 12-14.

35. Lou Harrison, Lou Harrison's Music Primer (New York: C.F. Peters Corporation, 1971), 6.

36. This point is discussed further in Larry Polansky, "Paratactical Tuning: An Agenda for the Use of Computers in Experimental Intonation," Computer Music Journal 11, no. 1 (1987): 61-68.

37. Quoted in Heidi von Gunden, The Music of Ben Johnston (Metuchen, N.J.: Scarecrow Press, 1986), 159-60.

38. Ben Johnston interviewed by the author, Rocky Mount, April 1991.

39. A twelve-note "chromatic" scale can be represented in ratio terms as a two-dimensional (3,5) projection plane within the three-dimensional (2,3,5) space needed to map the scale. (Octave equivalents would appear on an axis at right angles to the other two.) This configuration appears in Alexander Ellis's additions to Helmholtz's book, 457-64, termed the "harmonic duodene."

40. Ben Johnston, "Rational Structure in Music," in American Society of University Composers Proceedings, 1976-77, 102-118; reprinted in 1/1: The Journal of the Just Intonation Network 2, nos. 3-4.

41. Ben Johnston, "Extended Just Intonation: A Position Paper."

42. "Rational Structure in Music," 11.

43. Ben Johnston interviewed by the author, Rocky Mount, April 1991.

44. James Tenney to the author, 22 August 1990.

45. James Tenney, "John Cage and the Theory of Harmony," Soundings 13: The Music of James Tenney (Santa Fe, New Mexico: Soundings Press, 1984), 55-83; also published in Musicworks 27 (Toronto, 1984), 13-17. All page references to Tenney's article in the present paper refer to its publication in the Soundings volume, which also contains Larry Polansky's monograph "The Early Works of James Tenney."

46. James Tenney in conversation with the author, Toronto, April 1991.

47. "John Cage and the Theory of Harmony," 58.

48. James Tenney, "Reflections after Bridge" (1984): statement in the published score of Bridge (1982-84) for two pianos eight hands in a microtonal tuning system. (Baltimore, MD: Smith Publications, 1984.)

49. John Cage interviewed by Max Nyttelr, Swiss National Radio, 1990, quoted in Eric de Visscher, "John Cage and the Idea of Harmony," Musicworks 52 (Spring 1992), 50-56.

50. Tenney, "Reflections after Bridge."

51. Tenney, "John Cage and the Theory of Harmony," 57.

52. Brian Belet, "An Interview with James Tenney," Perspectives of New Music 25, nos. 1 and 2: 459-66.

53. In his comments to the author on an early draft of this paper, Ben Johnston has pointed out that Tenney's motivations "are in many ways like mine, but his pragmatism about allowing temperament when differences lie [within] a certain threshold is actually a more profound difference from Partch and me than it appears. I don't want to guess what Harry's comment would have been, but my own I will give you: the symbolism of the aim to be perfectly, mathematically accurate in terms of proportionality is more important than the obvious fact that human frailty prevents any performance from attaining to it. I believe that even one's notation, even one's performance practices should symbolize what one is trying to communicate. My own art always concerns the harmonization of relationships, which I also believe is the fundamental interior and exterior religious aim." Ben Johnston to the author, 24 January 1994.

54. James Tenney, META+HODOS: A Phenomenology of 20th-Century Musical Materials and an Approach to the Study of Form (1961; revised edition, Oakland, CA: Frog Peak Music, 1988).

55. James Tenney, Darmstadt lecture 1990: unpublished transcript, courtesy of James Tenney.

56. Belet, already cited, 462.

57. The resulting lattice then becomes periodic in form, repeating itself endlessly in all directions.

58. James Tenney in conversation with the author, Toronto, April 1991.

59. Darmstadt lecture, 1990.

60. Brian Belet, already cited, 462.

61. James Tenney in conversation with the author, Toronto, April 1991.

62. "John Cage and the Theory of Harmony," 77-78.

63. This diagram appears in a revised form as "Chronology of the Recognition of Intervals" in Genesis of a Music, 92.

64. Genesis of a Music, 87.

65. James Tenney, A History of "Consonance" and "Dissonance" (New York: Excelsior Music Publishing Company, 1988), 88-89.

66. Schonberg, Theory of Harmony, trans. Roy E. Carter (London: Faber and Faber, 1978), 21.

67. Schonberg, Style and Idea (New York: St. Martin's Press, 1975), 216.

68. Schonberg, Theory of Harmony, 389.

69. Changes is published by Smith Publications, Baltimore, and Critical Band is distributed by Frog Peak Music: a recording of Critical Band by the ensemble Relache is available on their compact disc On Edge, Mode Records mode 22.

70. Because of the use of a tempered system the harmonic distance formula used in Changes is different from the one given above: for this and more on the work see Tenney, "About Changes: Sixty-Four Studies for Six Harps," in Perspectives of New Music 25, nos. 1 and 2: 64-87.

71. The II-V-I root movement at the very end is analogous, Tenney says, to the conventional practice of Bach (and others!), but I would suggest an equally close resemblance to the Studies for Player Piano by Conlon Nancarrow, of which Tenney was an early champion, many of which end with a V-I progression.

72. Tenney, "John Cage and the Theory of Harmony," 78-79.

73. Under the name Harmonical Proportion this is discussed by Partch in Genesis of a Music, 70-71, 89, 103-105.
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