James Tenney and the Theory of Harmony

In 1979, in the introduction for a planned treatise on harmonic theory, James Tenney penned the following recollections:

Until a few years ago, my own work in composition was such that questions of harmony seemed completely irrelevant to it. Timbre, texture, and formal processes determined by the many musical parameters other than harmonic ones still seemed like unexplored territory, and there was a great deal of excitement generated by this shift of focus away from harmony. Harmonic theory seemed to have reached an impasse sometime in the late nineteenth century, and the innovations of Schoenberg, Ives, Stravinsky, and others in the first two decades of the twentieth century were suddenly “beyond the pale” of any theory of harmony—or so it seemed. I was never really comfortable with this situation, but there was so much to be done—so many other musical possibilities to be explored—that it was easy to postpone questions of harmony in my own music. This situation began to change, however, in about 1970, when I wrote the first of a series of instrumental pieces that were to become more and more involved with specifically harmonic relationships. Then it was no longer the questions that seemed irrelevant but the “answers” offered by the available theories of harmony—both “traditional” and otherwise.¹

A preoccupation—both creative and theoretical—with “specifically harmonic relationships” would occupy much of Tenney’s attention from 1970 until his death in 2006. The harmonic series specifically would appear as a central structural resource beginning in Clang (1972) for orchestra and Quintext, V: Spectra for Harry Partch (1972) for string quintet.² (A general overview of the prevalence, structure, and perceptual correlates of the harmonic series is available in the Postcript to this chapter.) References to the series would appear often in Tenney’s subsequent compositions, such as Harmonium #1 (1976). In the late 1970s, however, alongside formulations directly related to the harmonic series, a second model of harmonic structure began to appear in Tenney’s writings, and then in his music. This involved the depiction of pitches as points within a lattice structure that he dubbed harmonic space. In this model, he associated close harmonic relationship between two or more pitches with their proximity in the lattice. The conceptual underpinnings of Tenney’s harmonic space first appeared modestly in Harmonium #3 (1980), but they would ultimately provide the foundation for such complex large-scale works as Bridge (1984) and Changes: 64 studies for 6 harps (1985).³

Drawing a distinction between his constructions conceived around the harmonic series and those invoking the concept of harmonic space, Tenney remarked that, “obviously there’s a relationship between them, but … the two notions lend themselves in different ways to different kinds of musical ideas.”⁴ Indeed, these distinct paradigms of harmonic structure manifested in musics of notably different character. Tenney’s “harmonic-series pieces” often feature gradual formal processes and a relatively high degree of continuity. These characteristics permit and promote close attention, on the one hand, to aggregate timbre and its gradual variation, and, on the other hand, to constituent tones or partials and their intervallic relationships. Moreover, fluid shifts are possible between these synthetic and analytic modes of listening. An invocation of perceptual duality between macroscopic timbre and internal harmonic relationships is one frequent characteristic of “spectral music.”⁵ In contrast, Tenney’s music conceived around his notion of harmonic space often involves forms that are relatively segmented and changeable, while timbre assumes a role secondary to the perception of relationships between pitches, and an interplay between perceptual synthesis and analysis plays little part. The latter works seem to fit less comfortably under the rubric of spectralism than of “music in non-standard tuning systems.”

Tenney did not simply supplant one approach with another, but continued to invoke each at different times. His archival writings suggest that underlying their overt stylistic differences was a shift of compositional focus between what he conceived as subtly different aspects of “harmonic perception” itself. If multiple modalities do coexist within the facility of harmonic perception, this suggests that any “theory of harmony” (or, perhaps, “theories of harmonies”) will need to be similarly multifaceted in order to describe them.

Tenney traced his first intellectual encounter with the harmonic series and issues concerning intonation to a course in musical acoustics completed in 1956 while he was still an undergraduate student at Bennington College.⁶ This initial exposure to scientific perspectives on sound and perception was greatly extended by graduate coursework at the University of Illinois. Of great importance for his later harmonic theories, while in Illinois he also worked with and performed in the ensemble of composer Harry Partch, who may fairly be described as the primary progenitor of modern extended just intonation in both theory and practice. Tenney subsequently conducted psychoacoustical research at Bell Telephone Laboratories in New Jersey in the early 1960s, and this involved the harmonic series as a matter of course due to the relationship between spectrum and timbre. Despite such early and close engagement with aspects of the series and of intonation, however, it would not manifest explicitly in Tenney’s music until the early 1970s, and his attention would not thereafter turn to attendant theoretical questions until mid-decade.

When, around 1976, Tenney began to formulate his own theory of harmony, it was conceived first as conceptual scaffolding for his own compositional work, and as an alternative to received theories that he found to be unsuited to non-tonal music. In particular, he came to believe that the concept of harmonic function and the imperative resolution of dissonance were culturally and historically specific conventions, not general features of harmonic perception.

There is no harmonic function other than what we choose. That’s a choice, a matter of style, culture, and compositional intention. … The acoustical properties of sound and the physiological, neurological properties of the ear. These are real things that are given, as well as what the brain can do with this.⁷

Tenney recognized, however, that for many musicians and scholars the very meaning of the word “harmony” was associated with Western tonal practice specifically, and he thus concluded that any renovation of harmonic practice and theory would first require clear definition or redefinition of the concept of “harmony” itself. Accordingly, in a landmark essay entitled “John Cage and the Theory of Harmony” (1983), he advanced a generalized definition of harmony, “as that aspect of musical perception that depends on harmonic relations between pitches, i.e., relations other than ‘higher’ or ‘lower’.”⁸ He also articulated a set of scholarly principles to which he believed that a new theory of harmony ought to adhere:

First, it should be descriptive—not pre- (or pro-)scriptive—and thus, aesthetically neutral. That is, it would not presume to tell a composer what should or should not be done but rather what the results might be if a given thing is done.

Second, it should be culturally/stylistically general—as relevant to music of the twentieth (or twenty-first!) century as it is to that of the eighteenth (or thirteenth) century and as pertinent to the music of India or Africa or the Brazilian rainforest as it is to that of Western Europe or North America.

Finally, in order that such a theory might qualify as a “theory” at all in the most pervasive sense in which that word is currently used (outside of music, at least), it should be (whenever and to the maximum extent possible) quantitative. Unless the propositions, deductions, and predictions of the theory are formulated quantitatively, there is no way to verify the theory and thus no basis for comparison with other theoretical systems.⁹

A significant part of Tenney’s research involved the identification of potential acoustic correlates for perceptible harmonic relations and—where possible—quantitative measures of such relations based upon those acoustic correlates. The specific “harmonic relationships” appearing most often among Tenney’s voluminous research notes include octave-equivalence, categorical harmonic identity (between pitch-intervals falling within some sufficiently small “tolerance” range of one another), and a polarity related to the perception of chordal roots and tonics. ¹⁰ Most significantly for present purposes, Tenney formulated a “similarity” relationship (between, for instance, a sound comprising an arbitrary collection of partials and one comprising a complete harmonic series) and a “proximity” relationship that he would come to understand as proximity in harmonic space. These two relationships are different, but both bear upon qualities that are commonly referred to as “consonance” and “dissonance.”

The relationship of the terms “consonant” and “dissonant” to specific percepts is complicated because—despite their widespread usage—these terms possess a complex history from which they have inherited considerable ambiguity. Tenney himself attempted to resolve the terminological equivocacies in a monograph entitled A History of ‘Consonance’ and ‘Dissonance’.¹¹ This detailed semiotic history surveys the changing implications of these terms in Western music theory, identifying five successive semantic paradigms or “consonance/dissonance-concepts” (CDCs). To these Tenney applied the neutral designations CDC-1 through CDC-5 in the chronological order of their appearance in history. In his account, CDC-1 evolved from a classical Pythagorean conception of consonance as “directly tunable” (for example, using certain accepted rational divisions of a monochord) to a pre-polyphonic medieval conception of “relations between pitches,” although it was assumed throughout to apply essentially to melodic dyads rather than simultaneous ones. Tenney dated CDC-2 to the High Middle Ages and concluded that it ascribed “consonance” to the quality of sounding like a single tone, or what psychoacousticians today sometimes call “toneness.”¹² CDC-3, dating from the Renaissance, involved the maintenance of textural clarity in a lower polyphonic voice. CDC-4 represented the functional consonance and dissonance of Western tonal common practice, wherein the consonance or dissonance of an interval is determined in part contextually as its obligation to resolve, and so cannot in general be determined by inherent qualities of that interval in isolation. CDC-5 corresponded to a sensation of what psychoacousticians today refer to as “smoothness/roughness” and is related to the perception of acoustical beating.

Tenney’s CDC-2 and associated concepts provide a theoretical counterpart to his music invoking the harmonic series, while CDC-1 is linked to his conception of relations in harmonic space and his compositions based thereon. Both CDCs will be explored further below. A prefatory account of CDC-5 is also warranted, since it is today probably the consonance/dissonance concept in greatest currency among musicians.

Between sine tones, rates of acoustical beating in the range from about 12 Hz to 300 Hz produce an auditory sensation known as “roughness.”¹³ Roughness is nil when the tones are in unison, reaches a peak at a small interval as their frequencies diverge, and decreases again as the frequency separation increases further. Peak roughness occurs at an interval of slightly less than a semitone for pitches above C4.¹⁴ When intervals are formed by complex tones rather than sine waves, beating between low harmonics that are close but not coincident in frequency may also evoke roughness. The eminent nineteenth-century physicist Hermann von Helmholtz observed that in intervals corresponding to simple frequency ratios the harmonics of two complex tones either coincided precisely or were well-separated in frequency, so that in such cases sensory roughness was relatively low.¹⁵ On the other hand, “dissonant” intervals contained close but non-coincident harmonics that beat, engendering greater roughness. Helmholtz thus hypothesized that “dissonance” corresponded to roughness, and “consonance” to its absence (that is, to “smoothness”). Figure 1 shows illustrative harmonic alignments and interactions for a 3/2 just fifth and a tempered tritone. Gray ellipses indicate pairs of low-order harmonics in the tritone that differ slightly in frequency so as to induce roughness. In the fifth, on the other hand, these same harmonic pairs are either precisely aligned so as to yield no roughness, or are at least better separated so as to induce less roughness. The roughness theory of consonance would thus predict that the fifth was the more consonant interval, in accordance with the judgment of most listeners.

Certain implications of the roughness theory may seem at odds, however, with many musicians’ intuitive sense of what “consonance” and “dissonance” denote. For instance, the rate of beating between two tones is determined by their frequency difference. Due to the nonlinear relationship between frequency and pitch, however, the frequency difference associated with a particular interval grows with registral height. Thus, the degree of roughness of a given interval (and hence its “dissonance” in the Helmholtzian sense) is also dependent on register, other factors being equal. On the other hand, roughness can also vary with intervallic width: the critical bandwidth increases below C4, so that in the treble two sine tones separated by a perfect fourth do not appreciably beat against each other, while in the deep bass they do. Finally, the amplitude of beating—and hence the salience of roughness—can vary with both overall dynamic and the relative amplitudes of interacting partials (and thus with timbre).

Although sensory roughness is perhaps what non-musicians most often mean when they refer to “dissonance,” for musicians its variation with register, intensity, and timbre may rouse a contrary intuition that the dissonance of a particular interval-type should be independent of such factors—an unease that betrays the ambiguity latent in the term “dissonance.” Tenney took the identification of roughness with dissonance in the post-Helmholtzian CDC-5 to concern timbre more than the sort of harmonic relationships with which he was primarily concerned, going so far as to suggest that it be dubbed “timbral consonance and dissonance.”¹⁶

For the composer, certain aspects of Helmholtz’s theory (or its more recent extensions) are quite valuable as tools in the process of orchestration … or, more generally (as in the field of electronic music), in the manipulation and control of timbre, texture, and “sonority.”¹⁷

Roughness/smoothness and audible beating are frequent features of Tenney’s music, but often as phenomena attendant to a more structurally significant manifestation of “consonance/dissonance” in some other sense.

One alternative contemporary theory of consonance and dissonance posits that “consonance” corresponds to the perceptual similarity of a collection of sounds to a single tone, a quality sometimes called “toneness.”¹⁸ A high degree of toneness is associated with a clear auditory image (gestalt) that evokes a clear pitch. The origin of the toneness theory of consonance is usually credited to the nineteenth-century psychologist Carl Stumpf, who in 1898 wrote that:

The combined sound of two tones approximates—now more, now less—the impression of a single tone, and it appears that the more this condition holds, the more consonant is the interval. Even when we perceive and distinguish the tones as two, they nevertheless form a whole in perception, and this whole strikes us as more or less unitary. We find this property with simple tones, just as with those with overtones. That the octave sounds effectively like a unison, even when we can clearly distinguish two tones in it, is always admitted, although it is nothing less than self-evident, but it is a most remarkable fact. This same property becomes weaker, however, even with fifths and fourths, and still weaker with thirds and sixths …¹⁹

The perceived quality of toneness is promoted by a high degree of acoustical “harmonicity”—that is, by similarity between the frequency structure of the sound stimulus and a harmonic series.²⁰ Thus—broadly generalizing—the following list of acoustical signals trends from greater to lesser toneness:

The degree of toneness exhibited even by complete harmonic complexes is subject to registral restrictions, however. In particular, pitch strength weakens significantly for complexes whose fundamental frequencies reside below about 50 Hz (roughly G1).²¹ Of course, for fundamentals below the frequency range of human hearing, pitch and fundamental frequency lose their correspondence entirely. On the other hand, the toneness of harmonic complexes exhibits a maximum for fundamentals near D4, which studies suggest is close to the average pitch of global music as a whole.²² Dyads comprising complex tones possess greater or lesser harmonicity depending on the degree to which the aggregate of their partials resembles a complete harmonic series, and this resemblance decreases for increasingly complex intervals in accordance with Stumpf’s description of their decreasing toneness.

Disentangling the contributions of roughness and toneness to subjective judgments of consonance and dissonance for particular isolated intervals has been complicated by the fact that in most instances these qualities tend to covary: in other words, intervals that are “consonant” (in the experimental psychologist’s sense of “subjectively preferred”) tend locally to both minimize roughness and maximize harmonicity. However, in the past decade new experimental designs have permitted the separation of the two contributions.²³ For instance, electronic synthesis permits the generation of acoustical stimuli that are not found in nature and that vary in smoothness but not harmonicity, or vice versa. Findings have, for instance, indicated that preference for consonance is consistently correlated with preference for harmonicity but not with preference for smoothness.²⁴ As a result of such studies, after decades of scientific dominance by the roughness theory, the toneness theory has lately moved into ascendancy as an explanation of listener preferences.

For the purposes at hand, however, more significant than the statistics of listener preference is the establishment that harmonicity and smoothness make distinct contributions to the perception of tone combinations. If toneness and smoothness have different qualitative referents, then the toneness and roughness theories of consonance are not in competition and any complete account of how tone combinations are perceived will need to address not one but both phenomena.

As an acoustical correlate of toneness, harmonicity provided a crucial formal parameter in Tenney’s compositional practice between 1972 and 1978, and it is relevant in to many of his works thereafter as well. Asked what circumstances led to his use of the harmonic series in composition, he replied:

That’s really hard for me to reconstruct. I wrote two pieces that year, 1972, which used it: a piece for orchestra called Clang and a piece for string quintet called Quintext. In both of them I used the harmonic series in this compositional way. I can’t frankly remember how that came to me—I think it had to do with looking for some new way to integrate a composition. And I’ve always been fascinated by the sheer acoustical and psychoacoustical fact involved there, that the auditory system integrates what, from an acoustical standpoint, is a complex set of frequencies. For one reason or another—and this is an extremely important theoretical question as far as harmony is concerned—the auditory system is able to integrate that complex set into a singular percept. And I think it’s quite possible that just something about that, I began to think of it as a possibility for compositional integration.²⁵

Accordingly, a number of Tenney’s pieces can be understood as involving processes that gradually increase or decrease harmonicity. The late piece Diapason (1996) for chamber orchestra provides a lucid example of a gradual approach to harmonicity.²⁶ At its opening, pitches are restricted to very high harmonics of B♭0, so that all of the low harmonics of this fundamental are absent. The resulting harmonicity is therefore relatively low because the aggregate of sounding partials is far from comprising a complete harmonic series. Gradually, however, the pitch-set of the ensemble descends through the harmonic series of B♭0. The aggregate sound achieves complete harmonicity and strong perceived toneness once the contrabasses and contrabassoons attain that low B♭0, since its complete harmonic series is then supplied by their partials and all partials from the higher instruments also reside within that series (assuming precisely accurate intonation). Various other realizations of such gradual approaches to or departures from harmonicity appear in Tenney’s Three Harmonic Studies, I & II (1974), Symphony (1975), Saxony (1978), Septet (1981), Voice(s) (1982/84), and Critical Band (1988).²⁷

A different musical process involving harmonicity appears in the concluding section of Clang (1972) for orchestra. Therein instrumental pitches are initially drawn from the high odd-numbered harmonics of an infrasonic fundamental. This fundamental is unplayed and unheard, lying far below the frequency regime wherein acoustical harmonicity would evoke perceived toneness. However, in each ensuing section the fundamental of the sounding pitch-set rises by an octave, ultimately traversing the audible register so that toneness is evoked and intensified. Variations of this migration by a fundamental into and through the register wherein harmonicity evokes toneness can also be found in Tenney’s Glissade, V: Stochastic Canonic Variations (1982), Form 2 (1993), the Diaphonic series (1997), and Panacousticon (2005). ²⁸

Yet another strategy for approaching and retreating from harmonicity figures in the first of Tenney’s seven-part Harmonium series (1976–2000). Harmonium #1, for twelve or more sustaining instruments, features repeated note-by-note “modulations” from one harmonic-series subset to another. These subsets constitute prominent “milestone harmonies,” with harmonicity decreased during the transitions between milestones but markedly increasing upon arrival at each new one.

The opening of the score to Harmonium #1 is reproduced in Figure 2 with annotations and transparent overlays. Four “sections” are shown, delimited by double bar lines. Sections, in turn, comprise two-to-five “segments” delimited by single bar lines. Each segment indicates available pitches from which performers freely and independently choose. Pitch deviations from equal temperament (if any) are indicated in cents above the note heads. Each performer executes tones that are four-to-ten seconds in duration, with a symmetrical swell in loudness from pianississimo to the marked dynamic and back again. After finishing a tone, the performer is to pause before playing another (or the same) pitch. Each segment contains at least one open note head, indicating a new pitch that was not available in the preceding segment. Filled note heads, on the other hand, indicate pitches that have been retained. Any performer may initiate the transition to the next segment by introducing a pitch that is new in that segment (i.e., one corresponding to an open note head), although this is to be done such that each section is one to three minutes in duration.

The added annotations in Figure 2 include the section numbers (I–IV), transparent gray rectangles identifying milestone harmonies, and thick gray lines demarcating pitch-classes associated with those harmonies. Each milestone harmony comprises pitches drawn from a single harmonic series whose fundamental changes from section to section, the harmonic numbers appearing being primes 2, 3, 5, 7, 11, and 17, or higher octave equivalents thereof. These milestone harmonies are collected in Figure 3, which shows their constituent harmonic numbers. The pitch of each fundamental is absent, but its pitch-class always appears in the bass of the harmony.

In Sections I–IV the bass descends a succession of tempered perfect fifths, permitting the gradual introduction of higher harmonics in ascending registral order with smooth voice leading. In the first three introductory segments of Section I, pitches corresponding to harmonics 3 and 5 are accumulated above the A in the bass. As Figure 2 shows, in the ensuing segments these pitches are one-by-one replaced by pitches associated with the milestone harmony of the next section, culminating in the appearance of the complete new milestone at the beginning of Segment II. Replacement begins with harmonic number 5, which is lowered to become the seventh harmonic above the new bass. This lowering initially gives the impression of a change from major to minor mode, which in each section provides an aural cue that the process of pitch substitution has begun. Replacement proceeds to harmonic number three, until finally the old bass becomes the third harmonic of the new fundamental (through an effectively negligible intonational adjustment of two cents) as the pitch-class of the new fundamental simultaneously appears in the bass.

This process is repeated until the arrival of Section IV, wherein the accretion of higher harmonics reaches its apogee. The accumulated pitches are reminiscent of higher chordal extensions familiar from tertian tonal practice: chordal sevenths (harmonic 7), sharp-elevenths (harmonic 11), and flat-ninths (harmonic 17). The harmony in Section IV might thus be interpreted as a just-intoned version of a C7♭9♯1 chord, although the registral location of the flat-ninth chord member suggests that it should not be viewed as an “alteration” of a natural ninth associated with harmonic number 9, but that it instead derives from the higher harmonic number 17. In a carefully intoned performance, the aural impression of this harmony evokes the liminal experience of hearing out harmonic partials from within a single complex tone, their pitches entering into various intervallic relationships even as they continually flirt with fusion into a gestalt, an indication of high collective toneness.

The conceptual and experiential crux of Harmonium #1 involves the dramatic qualitative transformations in perceived harmonic sense that occur as milestone harmonies are attained and departed. These perceptual transformations are correlated with changes in the intervallic similarity of the chordal voicing to a harmonic series. For instance, as shown in Figure 2, the milestone harmony of Section III corresponds to harmonics 2, 3, 5, 7, and 11 of a G2 fundamental in their correct registral locations, while the pitch sets of the preceding and following few segments are less simply rationalized. This is illustrated in Figure 4a, which shows each available pitch-set and its notional GCD-pitch or “conceptual fundamental.” This GCD-pitch—shown using a dashed line—is the pitch corresponding to the GCD-frequency, which is the greatest common divisor (GCD) of the set of sounding acoustical frequencies.²⁹ In other words, it is the highest pitch such that among its harmonics would be found all of the acoustically sounding tones (as well as all partials thereof). Thus one measure of the harmonicity of the total aggregate of sounding harmonic partials is supplied by the fraction that they together constitute of a complete harmonic series whose fundamental is this GCD-pitch, under the simplifying assumption that each tone individually includes many harmonics. In Tenney’s parlance, this fraction is the “intersection ratio” of the sounding harmonic series aggregate with a notional harmonic series on its own GCD-pitch.³⁰ This quantity is plotted in Figure 4b.

The unison pitch in the opening segment of Section I is the simplest harmony in the piece, and the only occasion on which the GCD-pitch of the collection is also a member thereof. Being exactly a harmonic series on this fundamental, its intersection ratio is unity, the highest value appearing. The intersection ratios in the next two segments are lower, since the fundamental is no longer part of the available pitch set, but remain relatively high. In Segment 4, however, pitches associated with the next milestone harmony begin to invade the available pitch set, with a resulting plunge in both the GCD-pitch and the intersection ratio. Perceptually, this correlates with a sudden increase of harmonic complexity. The intersection ratio begins to recover at the end of Section I, since in Segment 5 the only pitch of the first milestone harmony that has not been replaced is its bass, which is effectively identical to a pitch in the next milestone harmony. In other words, in Segment 5 the new milestone harmony is complete except for its bass, whose subsequent appearance marks the beginning of Section II and is accompanied by a jump in the intersection ratio to a local maximum, as shown in Figure 4b. This correlates with a dramatic perceptual fusion among the pitches present and a corresponding increase in the apparent toneness of the aggregate. In a 2003 conversation, Tenney described this transformation to me as a, “making of sudden harmonic sense to the pitches you’ve been hearing.” This process of harmonic complexification followed by clarification is repeated at the boundaries of Sections III and IV. Each time, the first appearance of pitches associated with the next milestone harmony precipitates a dramatic disruption of the preceding segment’s harmonicity.

Harmonium #1 explores perceptual fusion and toneness through reference to the harmonic series, an acoustical structure with a uniquely potent capacity to evoke such phenomena. Tenney’s emphasis on the perceptual correlates of the harmonic series over its acoustical properties, however, is paramount to any understanding of its place in his thought and music. He took the nature of perception as the object of his research rather than acoustics per se and, consequently, his notion of harmony did not reduce solely to relationships or percepts derivable from the harmonic series. On the contrary, regarding the resources of just intonation he observed that, “Whoever has taken the position that somehow, because it’s there in the harmonic series, we’re imitating nature when we play with it—well, just immediately, the minor triad throws that theoretical position into disarray.”³¹ Accordingly, while a number of his works explore harmonicity specifically, his theoretical and compositional interests eventually expanded to include a broader notion of harmonic relationship, as described below.

In the late 1970s, Tenney began to explore a set of relationships that he associated with CDC-1.³² He traced that earliest of his five historical Consonance/Dissonance Concepts from ancient Greece through the early Middle Ages, arguing that—unlike later CDCs—it originally applied strictly to successive tones (that is, to melodic intervals). Consequently, unlike the concepts of toneness (CDC-2) and roughness (CDC-5) that prima facie apply only to simultaneous complex tones, Tenney maintained that CDC-1 referred to audible harmonic relationships between successive as well as simultaneous pitches, and between sine tones as well as complex tones. He detected traces of its melodic aspect in Jean-Philippe Rameau’s eighteenth-century rules of root progression and in the concept of “closely related keys” for purposes of modulation, as well as in the early twentieth-century writings of Paul Hindemith and Arnold Schoenberg, the latter casting it as a “relation between tones.”³³

Tenney understood this harmonic relationship as one of harmonic “proximity” between pitches.

The earliest sense of consonance and dissonance—CDC-1—implies that at the octave and perfect fifth, for example, two tones seem much more closely related to each other than at immediately adjacent though smaller intervals (the major seventh and augmented fourth), and this has given rise to numerous attempts to order or “map” pitches in a way that somehow represents these other relations by proximities in a “space” of two or more dimensions while still preserving the relations of pitch-distance. What is implied here is a conception of harmonic space and a measure of the harmonic distance between any two points in that space that is distinct from—but not inconsistent with—the measure of pitch-distance.³⁴

Tenney’s model of harmonic space represented frequency ratios as locations within a multidimensional lattice.³⁵ The appearance of such lattices among Tenney’s compositional plans thus betokens a significant development in his thinking beyond the harmonic-series-based structures of his music in the early 1970s.³⁶

In describing the structure of harmonic space, Tenney wrote that:

For a given set of pitches, the dimensions of this space would correspond to the prime factors required to specify their frequency ratios with respect to a reference pitch. It is a discrete space, not a continuous one, with the line segment connecting any two adjacent points in a graph of the lattice symbolizing a multiplication (or division) of the frequency ratio by the prime number associated with that dimension.³⁷

Tenney’s lattices thus depict the “prime factorizations” of frequency ratios. Any whole-number ratio can be factored in one and only one way as a product of prime numbers:

in which the product extends to include as many prime factors as needed and in which the exponents are integers. For example, the frequency ratio 15/4, such as appears between the fourth and fifteenth harmonics, can be expressed as:

The prime factors on the right of this expression appear with exponents representing their multiplicities.³⁸ Based on this factorization it becomes possible to represent the ratio 15/4 as a unique point in a three-dimensional lattice, as shown in Figure 5. The dimensions of the lattice correspond to the distinct prime factors involved—which, in this example, are 2, 3, and 5—while the coordinates of the point correspond to their exponents—which in this case are (−2, 1, 1). Tenney referred to the three associated axes in the lattice as the “2-axis,” “3-axis,” and “5-axis.” For pictorial purposes, the positive directions along these respective axes are rightwards, upwards, and “outwards” by convention (as in Figure 5). Since coordinates in the lattice correspond to the exponents of these prime factors, the origin at coordinates (0, 0, 0) corresponds to:

A positive step along the figure’s 2-axis corresponds to multiplication by a factor of two or, equivalently, to the pitch interval of an ascending octave that is associated with such a 2/1 frequency ratio. Similarly, a positive step along the 3-axis corresponds to multiplication by a factor of three, or to the ascending just twelfth of a 3/1 ratio, while a positive step along the 5-axis corresponds to multiplication by a factor of five, or to the ascending interval associated with a 5/1 ratio (which is two octaves plus a just major third). This three-dimensional lattice thus permits representation of any ratio involving prime factors no greater than 5 or—in the parlance of composer/theorist Harry Partch—ratios within the “5-limit.”³⁹ In accordance with its coordinates of (−2, 1, 1), and relative to the origin at 1/1, 15/4 is therefore located two negative steps along the 2-axis, one positive step along the 3-axis, and one positive step along the 5-axis, as shown in Figure 5. Similarly, the coordinates of other points in the figure correspond to the exponents in their prime factorizations.

Tenney often invoked octave equivalence to map all ratios differing only by factors of powers of two to the same lattice-point, thereby reducing the dimensionality of the lattice by one. He referred to this operation as “pitch-class projection.”⁴⁰ Each point in the resulting “pitch-class projection space” thus represents a particular “ratio-class” (in other words, the collection of all ratios that differ only by a factor of some power of two) and is customarily labeled with a representative ratio in the range [1, 2), as illustrated in Figure 6. Thus, a step to the right in Figure 6 corresponds to the interval-class of an ascending just fifth (3/2), while a step upwards corresponds to that of an ascending just major third (5/4). As shown, the dimensional reduction permits the depiction, if needed, of an additional prime factor such as 7 using an axis interpreted as perpendicular to the page. Abstract lattices involving additional prime factors are formulable, but cannot be illustrated in three dimensions.

Tenney associated the harmonic simplicity of a pitch set with its “compactness” in harmonic space.⁴¹ For example, if two pitches both reside on the 3-axis, then the less their separation along that axis the greater their harmonic “proximity” (or, in other words, the less the “harmonic distance” between them).⁴² Another important criterion for pitch sets was “connectedness” in harmonic space, wherein each element of a set is adjacent to at least one other element thereof, presumably enabling the ear to range over the entire set via the primitive intervals.⁴³ Various familiar scales and harmonies can be associated with compact connected sets of lattice points in pitch-class projection space. For instance, as shown in Figure 6 a twelve-note just chromatic scale can be associated with a 4×3 configuration of points in the 3,5-plane of pitch-class projection space. This collection contains a major diatonic scale on and above the 3-axis (excluding the Lydian F♯), and a minor diatonic scale on and below it (excluding the Neapolitan D♭). Major and minor triads appear respectively as upward and downward pointing triangular configurations in this same subspace, as shown in the figure. Various seventh chords can be associated with other close-packed subsets, the dominant seventh chord extending into the 7-dimension as illustrated.

In the mid-1980s, Tenney’s harmonic-space model would feature foundationally in some of his largest and most complex compositions. The earliest musical manifestation of this important conceptual development, however, lies almost obscured in the outwardly modest Harmonium #3 (1978).

The score to Harmonium #3 for three harps calls for the instruments to individually be tuned in twelve-tone equal temperament, but Harp I must be tuned 14 cents flat and Harp III 14 cents sharp relative to Harp II. (For analytical convenience, however, I will take Harp III to be the pitch reference with a deviation of 0 cents, with respect to which Harp II is 14 cents flat and Harp I is 28 cents flat.) This intonational scheme affords a highly accurate approximation to the pitch-class of the fifth harmonic, as shown in Figure 7. However, close inspection of the pitch-classes used reveals that not all of them correspond to the best available approximations to harmonic-series pitch-classes. In particular, the pitch-classes that in Harmonium #1 would have corresponded to harmonics 11 and 17 are both played by Harp II, but Harps I and III respectively would have resulted in smaller deviations from the harmonic series (−21 and + 5 cents).

In fact, although Tenney retained the intervallic spacing of the harmonic series as a model for chordal voicing, the “tritone” and “semitone” pitch-class intervals above the bass were not intended to provide approximations to eleventh and seventeenth harmonics. Their intended identities are indicated in the rightmost two columns of Figure 7, which provide alternative interpretations of the intervals appearing in Harmonium #3, expressed as frequency ratio-classes. Instead of the 11/8 and 17/16 ratio-classes that would have been associated with the eleventh and seventeenth harmonics of 1/1, Tenney instead adopted close approximations to the ratios 7/5 (the “tritone” between harmonics 7 and 5) and 21/20. Since 21/20 = 7/5 × 3/2 ÷ 2/1, this “semitone” interval-class is equivalent to a just 3/2 fifth above the adopted 7/5 tritone. The higher prime factors 11 and 17 are therefore eschewed and the greatest prime factor appearing in the ratio-set is 7. Adopting the terminology of Harry Partch, Harmonium #3 thus employs a “7-limit” tuning system. The restriction to ratios involving prime numbers of magnitude no larger than seven revealed a new interest on Tenney’s part in harmonies derivable from collections of relatively simple frequency ratios involving low primes, with more complex relationships (such as 21/20) available as combinations thereof. In 2003, the composer described this development in Harmonium #3 to me as the beginning of his interest in compact sets in “harmonic space,” adding that, “you can get all the complexity you want from 7-limit.”

As voiced in the score, the complete pitch-set could be interpreted as harmonics 10, 15, 25, 35, 56, and 84 of their GCD-frequency, but such an analytic appeal to harmonicity would not reflect certain important aspects of the harmony as perceived. First, the collection as a whole does not fuse strongly both because of the great height of these “harmonics” above the associated GCD-pitch and because—given the register of the music—that GCD-pitch would be infrasonic. Furthermore, this interpretation would fail to capture the relatively simple intervallic connections that are audible between the pitches in the set. These are more clearly reflected in Figure 8, which represents the frequency ratio-classes approximated in Harmonium #3 as points in three-dimensional pitch-class projection space, wherein they occupy a connected and relatively compact region. The 7/5 ratio relative to 1/1 corresponds to the combination of an ascending 7/4 seventh with a descending 5/4 third, 21/20 being arrived at by an additional ascent of a 3/2 fifth.

The elegant geometrical representation of the major triad in pitch-class projection space is visible in Figure 8, outlined by the lattice vertices associated with 1/1, 3/2 and 5/4 ratios. Their geometrical configuration is that of a right triangle with its right angle at lower left. The vertex at 7/5, however, marks the right angle of another such triangle, representing a second just major triad. Thus the hexachord of Harmonium #3 might be regarded as a just-intoned version of Igor Stravinsky’s “Petrushka chord,” a bichord comprising two major triads whose roots are a tritone apart. Unlike in Harmonium #1, here the intonation of each triad is identical and simple, and—most significantly—when voiced as a bichord their combination is readily audible as such in the music. Additionally, the lattice representation captures a consequence of these particular intonations for voice-leading: if the 1/1 “root” descends a 3/2 fifth, the two 3/2 fifths appearing in Figure 8 produce two common tones as 1/1 and 7/5 become 3/2 and 21/20, respectively. In other words, the root and tritone of the first bichord become the chordal fifth and minor ninth in the second one, each appearing as the same pitch class in the same harp, although possibly in a different register.

The rhythmic, voicing, and formal procedures of Harmonium #3 resemble those of Harmonium #1 with a few noteworthy differences. The bass (initially) traverses a similar sequence of tempered fifths, although they now descend from G as shown in Figure 9. Instead of sustained tones, alternating upward and downward arpeggiations serve to delineate the harmonies, entailing some challenging hockets between the instruments. Finally, voice leading is often accomplished using a sequence of minimal pitch adjustments constituting microtonal passing tones between the chord tones of successive milestone harmonies. This is illustrated in Figure 10, in which a dotted line shows the incremental voice leading of a B♭ in Harp I though all available intermediate pitches to a B♮ 114 cents higher in Harp II, thus supplying the 7/5 tritone above a new bass (F) shortly to appear in Harp III (cf. Figure 9). The effect is akin to that of repeatedly tweaking a tuning peg until a desired intonation is finally attained.

Attempts to enumerate the distinct dimensions of harmonic experience result in proliferation. For instance, although they may covary in most musical situations, roughness/smoothness and toneness can be made to vary independently by artificial means. Moreover, although Tenney focused on primitive percepts in search of universals rooted in acoustics and physiology, the prospective facets of harmonic experience quickly multiply if roles for long-term memory and higher cognitive functions are entertained.⁴⁴

In typical listening situations, multiple aspects of harmonic perception may be engaged concurrently, although not all may be equally salient or relevant to the listener and analyst. For Tenney’s Harmonium #1, an analysis of varying harmonicity provides an account of form and process that rationalizes the striking perceptual coherence that accompanies the arrival of milestone harmonies, and that will thus account for the impressions of many listeners. Harmonic distance, however, still has an intelligible meaning in the piece as an index of the more or less complex intervallic relationships between various individual pitches. On the other hand, to the extent that roughness/smoothness is salient at all that quality seems either ancillary or the mark of intonational defect. Many of Tenney’s other “harmonic-series pieces” such as Clang (1972), Quintext, V (1972), Spectral CANON for Conlon Nancarrow (1974), In a Large Open Space (1994), and Diapason (1996), also foreground gradual processes moving toward or away from harmonicity, wherein a constant or infrequently changing GCD-frequency tends to create a stable harmonic orientation even if it is not always sounding.⁴⁵ Such pieces fall into what might retrospectively be described as Tenney’s “spectralist” oeuvre.

On the other hand, the bichordal pitch-set of Harmonium #3 (taken as a whole) has a low harmonicity that does not reflect the perceptibly simple relationships between its members taken pairwise—relationships that are better captured by the criterion of compact connectedness in harmonic space. Harmonicity, however, still seems to contribute to the internal coherence of the two triadic components. Since harmonic space lacks any preferred origin, it lends itself to harmonic progressions or modulations in which the perceived location of “1/1” changes, following which all lattice points are effectively relabeled with ratios relative to the new origin. Harmonic orientation can thus shift relatively rapidly and exhibit definite harmonic direction, as in Changes: Sixty-Four Studies for Six Harps (1985)⁴⁶ or Water on the Mountain … Fire in Heaven (1985).⁴⁷ Alternatively, as in Bridge (1984), pitch activity may cluster around multiple simultaneous points of origin in harmonic space, engendering a form of polymodality in non-standard intonation of which the bichord in Harmonium #3 is a harbinger. ⁴⁸ Such features are less often associated with paradigmatic spectralism, and these works seem more aptly described as music in extended just-intonation systems than as spectral music. In Tenney’s music, at least, the distinction might be viewed in part as a consequence of emphasizing different facets of harmonic perception.

Tenney’s theory of harmony remained incomplete at his death in 2006, and his archives contain voluminous notes on the topic, the bulk of which remain unexcavated. He eventually came to believe that a detailed computational model of neural pitch processing would be a prerequisite to any such theory, and beginning in the 1980s he worked intermittently on such a model. In the scientific community, recent decades have witnessed extensive research regarding the characterization of pitch perception and neurophysiological mechanisms of pitch processing.⁴⁹ It is to be hoped that a better eventual understanding of neural mechanisms may elucidate both received concepts of harmony and the broader domain of perceptual and cognitive harmonic possibilities. On the other hand—as Tenney’s theoretical and compositional work suggests—the diverse, intimate, and exploratory involvement of musicians with tone combinations as phenomena may have a significant contribution to make to such research by illuminating the diverse dimensions and characteristics of harmonic experience for which it must account.

As background information relevant to this chapter, the following postcript offers to interested readers an introduction to the structure and perceptual correlates of the harmonic series.

A “harmonic series” is a sequence of frequency values comprising all whole-number multiples of some lowest value. That lowest value is called the “fundamental frequency” of the set or, informally, the “fundamental.” A collection of sinusoidal acoustical components whose frequencies reside in a harmonic series is called a “harmonic complex,” and the individual components are called “(harmonic) partials” or, informally, “harmonics.” Under acoustical conditions that encompass those typical of speech and pitched music, the components of a harmonic complex strongly tend to perceptually “group” so that they are heard as a unitary “auditory image” or “gestalt.” In this case those components are said to have undergone “harmonic fusion.” Usually the resulting gestalt will possess a pitch called the “residue pitch” associated with the fundamental frequency. An auditory gestalt with a pitch is called a “tone,” so such a fused gestalt is usually called a “(harmonic) complex tone.”

In the normative mode of musical listening, a harmonic complex is perceived as a tone whose pitch corresponds to its fundamental frequency and whose timbre is related to the (possibly evolving) relative amplitudes of its constituent partials. This mode is called “synthetic listening.”⁵⁰ In contrast, with close attention and allowed sufficient time, listeners can instead “hear out” various individual partials within the complex, a mode known as “analytic listening.” In this alternative mode, the pitches of various individual harmonics and the intervals between them become audible. Thus, in the synthetic listening mode the ensemble of harmonic partials contributes through their relative amplitudes to a perception of timbre, while in the analytic mode they contribute through their distinct pitches to a perception of harmony. Analytic listening becomes easier with practice, and is facilitated if the amplitude variations of partials bring certain of them to individual prominence. Partials up to the tenth are usually the most easily discriminable, but considerably higher ones can be heard under suitable circumstances.

Intervals appearing within the harmonic series are said to be just intervals, as distinct from the tempered intervals found on a conventionally tuned piano keyboard, which are produced by dividing the octave into equal intervallic steps. Just intervals are often referenced by the ratio between their fundamental frequenices, which equals the ratio between their harmonic numbers. Thus the fifth appearing between harmonics two and three may be referred to as a “3/2.”

The only just interval that exactly corresponds to a tempered one is the octave. Nonetheless, inspection of Figure 11 reveals that many of the just intervals that appear between harmonics of low number approximate tempered intervals that would be described as “consonant” by Western musicians. On the other hand, less familiar and more “dissonant” intervals involve higher harmonic numbers, although it is possible to find some consonant intervals between higher harmonics as well. More precisely, most listeners describe intervals corresponding to reduced frequency ratios of relatively low height (i.e., ratios in lowest terms whose numerator and denominator are relatively small numbers) as relatively “consonant.”

JTF. James Tenney Fonds (Inventory #F0428). Clara Thomas Archives and Special Collections. York University. Toronto, ON.