TY - JOUR
AU - Giacomo, Fiore,
AB - Abstract Composers in the American experimentalist tradition, such as Harry Partch (1902–1976) and Lou Harrison (1917–2003), have turned to the guitar as an accessible and adaptable springboard to pursue alternative tuning systems through retuning, refretting, or a combination of the two. This article examines the tunings employed by composer James Tenney (1934–2006) in his four works for guitars: Harmonium II (1976, rev. 2003), Septet (1981), Water on the mountain … Fire in heaven (1985), and Spectrum 4 (1995). Remarkably, each of these pieces uses complex tunings without the requirement of custom-made or otherwise modified instruments. As is typical for Tenney’s compositions, these works reflect recurring theoretical concerns during his career; each piece is thus analyzed through the application of appropriate elements from the composer’s own theoretical framework. The American composer James Tenney (1934–2006) explored extended just- intonation tunings and complex harmonic constructs in a substantial portion of his music. However, unlike other notable tuning specialists, such as Harry Partch, Lou Harrison, and Ben Johnston, Tenney eschewed both customized instruments and arcane approaches to notation. In particular, Tenney’s four pieces for guitars, composed between 1976 and 1995, illustrate both his practical approach to instrumental writing and his uncompromising theoretical and compositional rigor. From a historical standpoint, his creative tuning solutions established an important precedent for composers of following generations, who would continue to expand the repertoire of extended just- intonation music for standard guitars.1 I consider four pieces: Harmonium II (1976, rev. 2005) for two guitars; Septet (1981) for six electric guitars and electric bass; Water on the mountain … Fire in heaven (1985) for six electric guitars; and Spectrum 4 (1995) for violin, alto recorder, piano, bass clarinet, trombone, vibraphone, guitar, and string bass. Tunings for these pieces are markedly different. Septet and Spectrum 4 are in just intonation, realized on the guitar utilizing two different methods; Water on the mountain … Fire in heaven approximates just intonation by means of a 72-tone equal-tempered system obtained by tuning each of the six guitars a sixth of a semitone apart; finally, the most recent version of Harmonium II uses a hybrid tuning, with each of the two guitars in equal temperament, but pitched approximately 30 cents (hereafter: ¢) away from one another.2 The common and crucial element among these pieces is that Tenney takes advantage of the fact that despite having a fixed number of frets in equal temperament, the guitar is easy to retune. Through extensive retuning of the open strings and careful fingering specifications, the performance of precisely tuned pitches and intervals becomes no more difficult than conventional playing, thus circumventing the exacting requirements—enhanced aural skills and performance techniques—necessary for realizing complex tunings on flexible-pitch instruments. Tenney’s simple adaptations allow him to employ the guitar in harmonic contexts of unprecedented complexity and richness. In the following pages, after identifying the theoretical and aesthetic framework common to Tenney’s works for guitars, I will present a detailed analysis of each piece to highlight both the elegance and efficiency of the composer’s extended tuning practices. A THEORETICAL DIGEST Throughout his life, Tenney was equally active as a composer and theorist. Starting in 1959 he wrote and published dozens of essays on a variety of theoretical subjects including formal analysis, algorithmic composition, harmony and intonation, and musical perception and psychoacoustics.3 Some of these texts have not always been broadly disseminated. For example, Tenney’s most influential work, the Gestalt psychology-based analytical manual Meta+Hodos, was circulated among fellow composers and students in typewritten form for more than twenty-five years until its publication by Frog Peak Music in 1986.4 The theories and concepts that Tenney expressed in his writings provide an invaluable interpretive basis for the analysis of his compositions, especially considering that, with few exceptions, he rarely described the workings of his pieces in detail.5 Tenney’s recurring theoretical interests in form, harmony, and perception reflect artistic activities from key periods in his life: his early years as a solo pianist and a champion of the music of American “ultramodernists” such as Ives and Ruggles (whose musical languages would inspire several of Tenney’s own algorithmic procedures); his roles as a founding member of the Tone Roads performance collective and as a performer in the ensembles of Steve Reich, Philip Glass, and especially Harry Partch (who introduced Tenney to rational tunings during his graduate studies at the University of Illinois); and his tenure as a researcher at the Bell Telephone Laboratories in Murray Hill, New Jersey (1961–64), where he delved into the realms of musical cognition and computer music.6 Tenney’s preoccupation with acoustics and perception led him to compose some of the first (and perhaps still foremost) examples of so-called spectralist compositions in North America.7 These interests also informed his writings on form and analysis (Meta+Hodos), history (A History of “Consonance” and “Dissonance”), and harmony.8 Crucial to many of Tenney’s harmonic constructions is the concept of “tolerance range,” defined as the narrow range within which the ear perceives two different pitches or intervals as equivalent. There are several kinds of tolerance ranges, such as the more or less objective 5¢ range within which the human ear cannot easily distinguish melodic intervals, and wider ranges that depend on the interaction of acoustical factors (spectral complexity and transients, for instance), cultural conventions, and musical context.9 For Tenney, a tolerance range not only allows for a degree of intonational flexibility, but also serves to establish a practical limit—one based on psychoacoustics—for the development of his harmonic theories. Tenney invokes a tolerance range with precisely such a function in the second part of his essay “John Cage and the Theory of Harmony”: Since our perception of pitch intervals involves some degree of approximation, these frequency ratios must be understood to represent pitches within a certain tolerance range—i.e., a range of relative frequencies within which some slight mistuning is possible without altering the harmonic identity of an interval. … Whether all such intervals among a given set of pitches are in fact distinguishable depends, of course, on the tolerance range, and it is this which prevents an unlimited proliferation of “dimensions” in harmonic space. That is, at some level of scale-complexity, intervals whose frequency ratios involve a higher-order prime factor will be indistinguishable from similar intervals characterized by simpler frequency ratios, and the prime factors in these simpler ratios will define the dimensionality of harmonic space in the most general sense.10 Example 1. View largeDownload slide A portion of (3, 5, 7) harmonic space Example 1. View largeDownload slide A portion of (3, 5, 7) harmonic space Tenney adapted Cage’s definition of a “total sound-space,” in which any possible sonic event can be denoted by the values of five determinants—“frequency or pitch, amplitude or loudness, overtone structure or timbre, duration, and morphology (how the sound begins, goes on, and dies away)”11—to his own definition of “harmonic space,” which is defined by axes corresponding to the prime factors of the frequency ratios between tones.12Example 1 illustrates a portion of the three-dimensional harmonic space defined by axes corresponding to the prime numbers 3, 5, and 7.13 In just-intonation systems, pitches are expressed as whole-numbered ratios of values ranging between 1 (unison) and 2 (octave). Mirroring the acoustic properties of the harmonic series, the presence of higher prime number factors corresponds to an increase in harmonic complexity: 2 generates octaves, 3 perfect fifths, 5 pure major thirds, 7 so-called harmonic sevenths, and so on (these numbers also reflect the proportions between the vibrating frequencies of two pitches in an interval). Given an n-dimensional harmonic space, in which the prime limit is determined by the highest prime factor between frequency ratios, any pitch that falls within the prime limit defined by the space can be localized as a point, its position identified through a set of coordinates. Tenney measured the harmonic distance (HD) between points in this space with a “city block” metric: the distance between points in harmonic space is accordingly the sum of the distances along each axis (Tenney 1983, 208). The HD formula measures the shortest distance between any two points in an n-dimensional harmonic space by the formula: HD(a:b) = log(ab), in which a and b are the numerator and denominator of a frequency ratio. In other words, it computes the complexity of the prime factors involved in the definition of any interval.14 Distance on higher-prime axes is thus weighed more heavily than the same number of steps along a lower-prime axis; one step along the 3-axis weighs less and is therefore considered less distant than one step along the 5-axis. Using Example 1 as a reference, the point marked 3:2 lies one block to the right of the fundamental 1:1 (a perfect fifth away); moving upwards results in the pitch 15:8, a just major seventh away from the fundamental (and a just major third away from 3:2). Moving left by one block from 15:8 yields 5:4, which lies, as expected one major third above our original fundamental. The HD values for the frequencies (1:1, 3:2, 15:8, and 5:4) are 1, 2.59, 6.91, and 4.32, respectively. Note that 5:4 is “further” harmonically from the fundamental than 3:2, despite the lattice’s square graphical appearance. The ratio of 15:8, as expected, has the greatest HD value of the set. Tenney expressed the impetus for the definition of harmonic space as the need for a harmonic theory that is descriptive (or “aesthetically neutral”), general, and quantitative (i.e., objectively measurable)—in contrast to most historical harmonic theories, which tended to be prescriptive, specialized, and anything but measurable.15 The concept of harmonic space plays a fundamental role in much of Tenney’s music from the 1980s and onward, providing the backdrop on which complex harmonic itineraries are traced. Tenney’s formula for measuring harmonic distance (HD), which he and others have used extensively as both a compositional and analytical tool, is predicated on the concept of harmonic space. Other compositional and theoretical constructs, such as the “Crystal Growth” algorithm detailed in a 1998 article, rely similarly on Tenney’s previously established concepts of harmonic distance and harmonic space.16 Another relevant concept emerged in the early 1980s, when Tenney returned to algorithmic composition after a hiatus of several years. One of the procedures he adopted models the “dissonant counterpoint” of early twentieth-century American music. First described in Henry Cowell’s notebooks from 1915, dissonant counterpoint was developed further by Cowell and Charles Seeger in the 1920s and 30s, and found use in their own music as well as the music of several of their contemporaries, including Johanna Beyer, Ruth Crawford Seeger, Lou Harrison, and—crucially for Tenney—Carl Ruggles.17 Tenney’s algorithm, in fact, evolved from an earlier statistical study of Ruggles’s melodic style, in which the younger composer mapped the statistical recurrence of pitches in his predecessor’s work over a period of several years.18 The transformation of these analytical findings into a compositional tool, which had been foreshadowed in Tenney’s Bell Labs papers, matured in the 1980s to become his favored method of producing randomness and ensuring variability, whether in terms of pitch succession, instrumentation, register, or other parameters. Tenney used this “dissonant counterpoint” algorithm in the composition of works such as the Spectrum series (1995–2001), Seegersong #1 and #2 (1999), and the late string quartet Arbor Vitae (2006); he also used it as a sort of “random number generator” in several of his pieces since the early 1980s, including Changes (1985) and Water on the mountain … Fire in heaven (1985).19 The common aesthetic trend that unifies these theoretical preoccupations is an interest in the phenomenology of sound, which in turn stems from an “experimental” attitude associated with the more widely used scientific meaning of the term. As a composer, Tenney was uninterested in musical representations of narrative, drama, or subjectivity; rather, he focused on the “neutral” exploration of sonic phenomena, the sounding out of physical and acoustical concepts, and the investigation and realization of musical processes. His recurring compositional challenge lies in solving the problem at hand as elegantly and concisely as possible, much like the way mathematicians approach the proving of a theorem. Although Tenney’s music obviously occurs in time, much of it takes place outside of time—the works as they exist in scores and recordings are just excerpts from processes that can be thought of as having no real beginning or end. In that guise, his work ultimately refers to the physics of sound as it happens and is perceived in the world around us, reaching infinitely upward like each successive order of the harmonic series.20 His invitation to the audience is simply to listen, to follow his acoustic explorations, and to marvel. THE GUITAR PIECES OF JAMES TENNEY: TUNING OVERVIEW Tenney’s four pieces for guitar—Harmonium II, Septet, Water on the mountain … Fire in heaven, and Spectrum 4—span the mature years of his compositional career, reflecting both his renewed interest in the harmonic series as structural material (Harmonium II, Septet) and his return to algorithmic compositional methods. Throughout these works, Tenney uses the extensive retuning of the open guitar strings to produce a range of rationally tuned pitches. In Spectrum 4, following the example of the string quintet Spectra for Harry Partch (1972), Tenney tunes each string to a different partial of the piece’s fundamental. The performer produces a wide range of precisely tuned pitches by playing either a natural harmonic on the open string or the open string itself. For example, by tuning a string to the seventh partial of a given fundamental and touching (not fretting) the string at the fifth node, the performer can produce the thirty-fifth partial of the fundamental. This method does not require any familiarity with just ratios on the performer’s part—once tuned, the performer can play the instrument as if reading from tablature. Conversely, in the Septet Tenney detunes one or two strings of each guitar by an amount corresponding to the deviation from 12TET of a given partial, measured in cents. The actual partials of a given fundamental are then performed by stopping the strings at the appropriate frets on the equal-tempered fingerboard. For instance, the C♯ needed to sound a 5:4 third above the fundamental A is played at the second fret on the second string, previously “detuned” from an equal-tempered B by the necessary 14¢. Water on the mountain … Fire in heaven uses the most extensive departure from conventional tuning. The six guitars are tuned one-sixth of a semitone (∼16.67¢) apart from one another, but treated together as a kind of hyper-instrument in 72TET—a tuning system that provides a close approximation of many important eleven-limit intervals without being wedded to any particular fundamental. Tenney had already experimented with using equal temperaments to approximate just intonation in the 1970s, most notably with Glissande (1972), a piece for strings and tape delay in 84TET.21 HARMONIUM II Harmonium II was written in 1976 as the second installment in a series of seven pieces that explore the harmonic series through sequences of modulating chords. Foreshadowing his future approach to writing for the instrument, Tenney scored Harmonium II for two guitars, which play in hocket throughout and effectively merge into a single hyper-instrument. The original version was written in equal temperament and premiered by Larry Polansky and Claudio Valentini in Toronto on March 17, 1978. After two additional performances (including a version for guitar and harp, performed by Polansky and Claudia Scaletti at the University of Illinois, Urbana-Champaign on December 13, 1979), an unconvinced Tenney withdrew the piece, reworking its harmonic material into a version for three harps tuned ∼14¢ apart, which became Harmonium III.22 Almost three decades later, however, Tenney revisited the shelved composition, sending an edited version of the score to microtonal guitarist and curator John Schneider; the piece was performed at the Microfest 2003 concert series in Los Angeles by guitarists Eric Benzant-Feldra and Michael Kudirka. For this revision, Tenney introduced a tuning approach similar to the one he had used in Harmonium III, pitching one of the guitars lower than the other by the distance between the seventh harmonic and the equal-tempered minor seventh (approximately 31¢).23 Throughout the piece, the distribution of pitches between the two guitars is strictly regimented. Most obviously, pitches corresponding to lower primes (1, 3, and 5) appear in Guitar II (which is tuned normally), whereas higher primes (7, 11, and 17) are played by Guitar I, therefore sounding ∼31¢ flat. This use of tempered tuning is exemplary of Tenney’s utilitarian approach to intonation. In addition to providing a much better-tuned seventh harmonic and closer approximations of higher ratios than 12TET overall (harmonics 7, 11, and 17 are mistuned by ±0, +18, and −26, as opposed to +31, +49, and −5, respectively), the tuning also creates a finer microtonal fabric—one in which the pitches are less immediately familiar—for the piece’s short-distance modulations.24 Over the course of this short piece (approximately five minutes long), Tenney takes the listener across twelve tonal centers, running counterclockwise along the circle of fifths from B to G♭. In the first half of the piece, each successive chord introduces a new dimension in harmonic space as the prime limit for each sonority increases sequentially. For example, the opening B triad (three notes) is followed by an E7 (four notes), an A11 (five notes), and an incomplete D17 (six notes, as the thirteenth is omitted).25 This last sonority can be interpreted two ways. It is both a subset of the octatonic mode made up of the first eight odd primes of the overtone series—an aggregate Tenney had explored extensively in works such as Clang, Chorales for Orchestra, and many others—and a “harmonic series version” of the so-called Petrushka chord (two superimposed major triads a tritone apart), which Tenney would have known intimately from its appearance in the “Emerson” movement of Ives’s Concord Sonata.26 The process with which Tenney moves from one chord to the next is a gradual substitution of closely tuned harmonic pitches. For instance, the D♯ (approximating B5) changes not to E (as would be expected in conventional voice leading), but to the detuned D of Guitar I (E7); the intervening D natural played by Guitar II makes the transition smoother by splitting the distance between the two pitches. Similarly, the F♯ (B3 or E9) creeps up past G (E19) to G♯ (E5), again making use of the sixth-tones available on the way. Example 2 illustrates this process.27 Example 2. View largeDownload slide Opening of Harmonium II, highlighting the microtonal “creeping” of pitches from one guitar to the other. Published by Smith Publications/Sonic Arts Editions. Used by permission. Example 2. View largeDownload slide Opening of Harmonium II, highlighting the microtonal “creeping” of pitches from one guitar to the other. Published by Smith Publications/Sonic Arts Editions. Used by permission. Example 3. View largeDownload slide Rearrangement of primes 3 and 11 in different chord voicings in Harmonium II. Top staff shows unchanging voices as whole noteheads, and moving ones as filled ones. Bottom staff presents the combined rhythmic activity of the two guitars. Published by Smith Publications/Sonic Arts Editions. Used by permission. Example 3. View largeDownload slide Rearrangement of primes 3 and 11 in different chord voicings in Harmonium II. Top staff shows unchanging voices as whole noteheads, and moving ones as filled ones. Bottom staff presents the combined rhythmic activity of the two guitars. Published by Smith Publications/Sonic Arts Editions. Used by permission. As more primes are introduced, the rhythmic patterns become denser, growing in speed to the quintuplets that introduce the seventeen-limit chords of the second half of the piece, which consists of a series of eight direct—rather than gradual—modulations from D17 to G♭17, with the roots moving by a perfect fourth as established previously. Initially, pitches change to their nearest neighbor, which is usually found in the other guitar. Subsequently, primes 3 and 11 exchange places in the chord spacing via a microtonal “switch” to rearrange the chord voicing, as shown in Example 3. As in other pieces by Tenney, the overall trajectory of the piece is determined by its material; Tenney’s compositional challenge lies in realizing such a trajectory in as economical a way as possible—i.e., to find the quickest and smoothest way to travel across the piece’s harmonic space. The last sonority is presented in an interspersed, closed voicing, with the two major triads played in strict hocket between the two voices. The smaller span of this final arpeggio (a mere augmented sixth, compared to the two-octave-plus range of the previous ones) is in part a result of the instrumentation. As shown in Example 4, the D♭ root of the next-to-last chord has to be raised an octave to fall within the range of the guitars; as the final G♭ is also approached “from below,” the resulting chord is the first to feature intervals smaller than a third, creating a novel and shimmering sonority. Example 4. View largeDownload slide Contraction of chord spacing in the final measures of Harmonium II. Published by Smith Publications/Sonic Arts Editions. Used by permission. Example 4. View largeDownload slide Contraction of chord spacing in the final measures of Harmonium II. Published by Smith Publications/Sonic Arts Editions. Used by permission. Harmonium II foreshadows several of Tenney’s theoretical and aesthetic developments—most importantly, the approximation of harmonic series relationships in an equal-tempered context. Beneath an apparent simplicity lies a piece of considerable musical challenges: the two guitarists must act as one, blending their timbres and carefully staggering their attacks to maintain the illusion of a single instrumental voice. Perhaps because of this combination of an innocent façade and a difficult substance, performances of Harmonium II are still relatively rare. In its new, widely available incarnation, however, the piece offers performers and listeners the opportunity to explore a different kind of harmonic motion, preparing them for more ambitious tonal explorations to come.28 SEPTET Not unlike Harmonium II, Tenney wrote his Septet as a modulation study—although in this case the entire work encompasses a single harmonic change. Due to its concise organization, it is perhaps the simplest of the composer’s works for guitar, despite a microscopically detailed surface. Writing for six electric guitars and electric bass, Tenney distributes material among the ensemble in a way that enables complex polyrhythmic and harmonic relations—thus re-creating textures reminiscent of Spectral CANON for CONLON Nancarrow (1974; for player piano) without resorting to mechanical or technological aids. The composer first announced the piece in a 1982 letter to Larry Polansky, who was then at the Mills College Center for Contemporary Music.29 Eventually, Polansky organized a group of guitarists for the premiere, which took place at the Mills Contemporary Ensemble concert on May 8, 1985, along with additional performances in San Francisco, Berkeley, Santa Cruz, and Marin County.30 A recording of one of the San Francisco performances, benefiting the Just Intonation Network, was released in 1986 by Tellus (an audio cassette magazine that ran from 1983 to 1993) as part of an issue dedicated entirely to new works in just intonation.31 In addition, a studio recording dating from around the time of the premiere appeared on the companion CD to issue number 7 of the Leonardo Music Journal.32 The Septet also figures in the repertory of electric guitar ensembles such as DITHER.33 Formally the piece is divided in five sections: the first three consist entirely of materials derived from the harmonic series on A (up to the eleventh partial), and the remaining two shift to a series based on E. However, this “new” fundamental is harmonically related to the first (by a simple 3:2 ratio), and as such all pitches in the piece are ultimately harmonically related to the original A. The pivot occurs at m. 160, underscored by a tempo modulation from quarter note= 60 to 90 (or 2:3). In addition to these macroscopic considerations, harmonic ratios also govern microscopic rhythmic elements, as pitches generally appear in rhythmic patterns that reflect the harmonic numbers of the pitches themselves; each player is responsible for a single ratio (and its octaves) at a time. In order to perform these pitches accurately, each guitarist must retune some of the strings by the appropriate cent deviation to sound the assigned partial. For instance, Guitar 2 tunes the top two strings 14¢ flat, and takes responsibility for the fifth, tenth, and fifteenth partials throughout the piece; the player can thus fret these strings to produce pure major thirds, major tenths, and major sevenths.34 In a similar vein, Guitars 3 and 6, responsible for the eleventh and seventh harmonics, tune their top strings 49¢ and 31¢ flat, respectively. Guitars 1, 4, and 5 are assigned intervals derived from the second and third harmonic, and thus do not need to retune (deviations of 2–4¢ falling well within the range of tuning variations imparted by the idiosyncrasies of fretted-string performance). Example 5. View largeDownload slide View largeDownload slide Superparticular ratios of sounding articulations in Septet. Published by Smith Publications/Sonic Arts Editions. Used by permission. Example 5. View largeDownload slide View largeDownload slide Superparticular ratios of sounding articulations in Septet. Published by Smith Publications/Sonic Arts Editions. Used by permission. Harmonic relationships are one of two interwoven principles governing the unfolding of the piece. The other element at play in the Septet is canonic in nature, in the loose sense of the term as a “musical rule.” For example, the opening section is a canon in unison; the attacks across the parts combine to produce rhythmic ratios that reflect those of the harmonic series.35 In sections II–V, the “rule” is that the order of introduction (or removal) of successive partials follows harmonic series order. As a result, section II, which begins with all instruments sounding unison A2 in superparticular rhythmic subdivisions, progressively grows into a “Rhythmicana,” a musical form dear to Henry Cowell in which partials are played in rhythmic values that reflect their harmonic number (e.g., the third harmonic is played as a triplet, the fifth as a quintuplet, and so forth—as detailed in Ex. 5).36 At the juncture between sections II and III the seven voices form the chord A11—an “otonal hexad” in Partch’s parlance, and an aggregate that appears frequently in Tenney’s works.37 As noted above, Tenney shifts the tonality from A to E in sections III and IV. This process occurs both horizontally and vertically: in m. 107 the bass begins playing successive harmonics from the A series on the downbeat of each repeated gruppetto. As each harmonic sounds, the guitar part with the corresponding pitch drops it from the texture. In Example 6 we see how the note A2 disappears from Guitar 5’s pattern in m. 112 upon the entrance of the same note in the bass part; similarly, Guitar 3 drops the lower of its two Es (A3) in m. 117. As a result, the chord vanishes from the bottom up, as if passed through a slowly sweeping high-pass filter. By m. 157, only the high E (A12) of Guitar 1 is left; a rhythmic modulation preserves the pulse across the overall tempo change (the sixteenth-note triplets at quarter = 60 morphing into straight sixteenths at quarter = 90), and the bass passes its last pitch to Guitar 3, which begins a descent along the “new” harmonic series of E (A12 being of course the same pitch as E8). In section IV, this process is reversed. As Guitar 3 sounds harmonics seven to one of E (played as natural harmonics on the low E string), the pitches are picked up by the respective player and sounded in the appropriate rhythmic pattern. Tenney allows the completed E7 chord to resonate for mm. 193–98, and then embarks in a final filtering procedure, with higher partials disappearing from the texture in favor of unison low Es, which eventually end the piece in m. 218. Ostensibly, this process could be repeated—or reversed—with a return to A. In a sense, the music acts as a model of the lower octaves of the harmonic series “exploded” over the additional dimension of time, and as such could be interpreted as having no real beginning or end. Example 6. View largeDownload slide View largeDownload slide Harmonic simplification leading to modulation, Septet, mm. 107–57. Published by Smith Publications/Sonic Arts Editions. Used by permission. Example 6. View largeDownload slide View largeDownload slide Harmonic simplification leading to modulation, Septet, mm. 107–57. Published by Smith Publications/Sonic Arts Editions. Used by permission. In addition to the deliberate musical treatment of spectral and harmonic phenomena, a critical element in Tenney’s Septet is the choice to limit each player to a small set of intervals, each available on a specific string. Some of the beauty of the Septet owes to its pedagogical efficiency: rather than requiring the musicians to be intonation experts, Tenney offers them the opportunity to become familiar with the sounds and rhythms of harmonic ratios. The wide availability of guitars and guitarists makes it feasible for many music departments (and professional groups) to gather and rehearse an ensemble for Septet. Making it possible for as many performers as possible to explore new tunings and harmonic systems was a concern close to Tenney’s heart. In an interview with composer Donnacha Dennehy, Tenney speaks of the role of accessible, yet precise tuning notation in a pedagogical and progressive sense: I’m also counting on a kind of an evolutionary process involving hearing, people learning to do this more and more. I think there’s a performing practice situation with people that have worked with me on a piece that has these kinds of requirements. They’re going to end up ten years later being teachers or conductors or performers, and it will spread. It will take time, you know, before it really becomes commonplace in the culture. But that’s all right, because it’s going to happen. I really believe it’s going to happen.38 WATER ON THE MOUNTAIN … FIRE IN HEAVEN Composed in 1985 for six guitars in a microtonal, “quasi-just” tuning, Water on the mountain … Fire in heaven (Water … Fire hereafter) is among Tenney’s most harmonically complex pieces. It shares its algorithmic inception, tuning scheme, and harmonic language with Changes, a set of sixty-four studies for six harps from the same year; in fact, the computer output for Water … Fire bears the subtitle “Study 65.”39 Despite this relationship, Water … Fire and Changes differ in significant ways. From a historical standpoint, the first was transcribed into notation soon after the completion of the algorithm and performed during a reading at York University in October 1985 under conductor James McKay.40 A decade later, new music specialist Seth Josel released a studio recording of the piece on his CRI album Long Distance.41 By contrast, the premiere performance of the complete set of Changes only occurred in November 2017.42 Another difference is structural: Changes is a much larger work as a whole, consisting of dozens of individual pieces lasting between 1’20” and 2’40”. Water … Fire specifies a duration of eight minutes for the single movement. Although it is a relatively short work, Water … Fire has not stimulated the same amount of critical interest as Changes.43 From a practical standpoint, the tuning for Water on the mountain … Fire in heaven is a departure from the approach Tenney had taken in the Septet just a few years before. Rather than employing cent-variations for only one or two strings per guitar, each guitar is pitched one-sixth of a semitone (∼17¢) from the next to create a complete 72TET tuning across the six guitars. This tuning affords the same freedom to modulate as any equal temperament while also offering a harmonic context that can closely approximate many significant intervals of eleven-limit just intonation.44Example 7 shows the deviations of the main 72TET intervals used in Water … Fire from their just counterparts; consistent with Tenney’s terminology, the lowest note of the piece (E − 34¢) equals Pc0 in the 72TET pitch-set.45 Note how the largest deviation, corresponding to the 27:16 Pythagorean major sixth, measures just +6¢. More significantly, the 5:4 major third (+3¢) is a considerable improvement on 12TET’s +14¢. The harmonic seventh (7:4) and undecimal tritone (11:8) deviate by only −2¢ and −1¢, respectively. The algorithmic procedures that govern both Changes and Water … Fire are explained in great detail in an article published by Tenney in Perspectives of New Music, where I refer the reader for the most accurate account of the piece’s workings.46 Those aspects that are most valuable for this analysis, however, will be summarized in the following paragraphs. On the simplest level, Water … Fire entails a series of modulations within a two-voice “polyphonic” framework.47 The length of the piece is specified at the onset of the algorithm as 480 seconds (exactly eight minutes). Over the course of this timespan, the algorithm generates a series of six-note modes, each built on a “root” note chosen from the available 72TET pitch set. These modes provide the available pitch set at any given moment of the piece, and can sound both horizontally and vertically as hocketing melodies, dyads, and aggregates. Over time, the algorithm moves away from the original modal root (in this case, Pc0) according to a complex set of harmonic constraints, and eventually returns “home.” As Tenney explains, the program is only successful if it manages to return to the original root within a certain window of time from the specified end of the piece; in the case of Water … Fire it finally reaches the “dominant” at 429 seconds, at which point it stalls by initiating a cadential sequence to buy itself some time before reaching the “tonic.”48 The duration of the piece is divided into ten segments of similar length; each segment, in turn, contains a number of “clangs” proportional to its duration, creating a self-similar structure akin to the “square root” organization of John Cage’s pieces from the 1930s and 1940s.49 Example 7. View largeDownload slide Subset of 72TET pitches used in Water on the mountain … Fire in heaven, their corresponding just ratios, and their deviations in cents Example 7. View largeDownload slide Subset of 72TET pitches used in Water on the mountain … Fire in heaven, their corresponding just ratios, and their deviations in cents The distance between each successive modal root is calculated stochastically using values originally defined at the beginning of the algorithm; however, after each successive modulation the program tweaks these values slightly to maximize the possibility of a successful overall trajectory.50 These modulations happen as a sort of random walk in a tempered, three-dimensional harmonic space defined by the 3, 5, and 7 prime axes (for simplicity, the octave axis is collapsed and therefore omitted in this discussion). Example 8 contains a list of all possible modulatory root motions, their original probability as specified in the algorithm, and their actual recurrence in the piece. Whereas root movement is confined to three-dimensional space, the internal intervallic construction of the modes happens in four dimensions (i.e., expanding to the 11-axis); it results in a variety of more and less familiar hexads, which cover a wide range of harmonic complexity. The modes include (but are not limited to) aggregates such as Partch’s otonalities, the min7/♭9 chord of jazz harmonies, and triadic combinations from common-practice Western harmony, further diversified by the greater intervallic variety of the microtonal context.51 The algorithmic generation of each new mode is a product of Tenney’s HD function, which the composer tuned specifically to favor closely related pitches in harmonic space. Furthermore, the algorithm aims to maintain some commonalities between adjacent modes, both in terms of pivot pitch classes (Pcs) and of internal intervallic structure. HD values provide quantitative measurements that, plotted over time, display the overall harmonic trajectory of the piece. Example 9 shows the harmonic distance of each new root from Pc0, as well as the overall harmonic complexity of the mode built upon each root. It is apparent that, whereas modal complexity stays relatively constant over the course of the piece, the roots of the modes end up traveling far in harmonic space. The peaks and valleys in root harmonic distance distribution correspond to the algorithm’s frustrated attempts to return to Pc0 ahead of time. Example 8. View largeDownload slide Inventory of root modulation motions in Water on the mountain … Fire in heaven, with algorithmic probability and actual occurrences in the piece Example 8. View largeDownload slide Inventory of root modulation motions in Water on the mountain … Fire in heaven, with algorithmic probability and actual occurrences in the piece Example 9. View largeDownload slide Plotting of HD value of each new root from original tonal center, and of HD value of each new mode, over the duration of Water on the mountain … Fire in heaven Example 9. View largeDownload slide Plotting of HD value of each new root from original tonal center, and of HD value of each new mode, over the duration of Water on the mountain … Fire in heaven The trajectory of root modulations can also be mapped graphically in the tempered equivalent of seven-limit harmonic space, representing pure fifths/fourths (3:2 and 4:3, or 42 and 30 in 72TET) on the horizontal axis; pure major thirds/minor sixths (5:4 and 8:5, or 23 and 49) on the vertical axis; and harmonic sevenths/septimal seconds (7:4 and 8:7, or 58 and 14) on the z-axis. Example 10 displays the first twenty-four modulations of the piece, accounting for 125 seconds out of the algorithm’s allotted 480. Each arrow indicates a single modulation, with curves representing more complex movements. Because this is tempered harmonic space, individual pitches appear in multiple locations in the space; for instance, pitches on the horizontal axis (the axis of perfect fifths/fourths) repeat after twelve moves in either direction, as is the case of Pc62 in Example 10. Furthermore, a high enough number of moves along any axis may result in a pitch that could also be factored from a different set of primes. Three moves “forward” on the septimal axis (each equal to fourteen 72TET steps), for example, will yield a pitch that is a perfect fifth away from the original, whereas in strictly rational tuning the pitches would be close, yet distinct: 14 * 3 = 42 8:7 * 8:7 * 8:7 = 512:343 ≠ 3:2 (1.493 ≠ 1.5) Example 10. View largeDownload slide The first twenty-four modulations in Water on the mountain … Fire in heaven, in 72TET equivalent of (3, 5, 7) harmonic space, starting with Pc0 and ending with Pc13. Dotted lines connect recursive tempered pitches. Example 10. View largeDownload slide The first twenty-four modulations in Water on the mountain … Fire in heaven, in 72TET equivalent of (3, 5, 7) harmonic space, starting with Pc0 and ending with Pc13. Dotted lines connect recursive tempered pitches. Example 10 also displays an instance of the algorithm diverting from the most likely move (a modulation by fourth) to avoid returning to the “home” Pc0 before the end of the piece. As the random walk reaches Pc42 a series of complex motions (in order: 8:5, 7:6, 15:8, 4:3, 5:3, 4:3, and 7:6) results in an escape away from Pc0. The last modal root in the example, Pc13, represents another example of a recursive pitch, as the temperament erases the difference between 245:216 (10:9 * 7:4 * 7:6 = 1.134) and 729:640 (8:5 * [3:2]6=1.139). The harmonic complexity of the work is staggering, especially when considering that each of the “tonicized” roots is in turn surrounded by closely grouped fifths, thirds, sevenths, ninths, and so forth, all tuned within a tolerance of ±3 cents. Example 11 is one possible spatial visualization of all tonal centers explored over the course of the piece. To create a comparable harmonic landscape in just intonation may be impossible without using electronic means. Once again, Tenney’s sophisticated approach to tuning and temperament allows for such a labyrinthine journey to be realized with relative ease.52 Choosing from the pitches available within the current mode, the algorithm composes a clang, the duration of which is also algorithmically determined (average clang duration for the piece is about five seconds); the ending of a clang in either voice initiates a change of mode. Though modality can change “in the middle of a clang” (if the other voice’s independent clang reaches its end), at any given time the algorithm can only choose from the six pitches of the mode that is currently active.53 The algorithm also determines the dynamic level and duration of every pitch in the piece. Pitch classes are assigned to the guitar capable of reproducing them (as each guitarist can only play one sixth of the available gamut), whereas duration and loudness are determined after a parametric calculation involving the density profile for the entire piece. As with Changes, this profile is calculated stochastically using I Ching hexagrams; more precisely, three pairs of lines are used to represent four possible levels of temporal, pitch, and dynamic density, with the possibility of different initial and final states for each category. Any “change” in status is then rendered by a half-cosine interpolation for a smooth transition.54Water on the mountain … Fire in heaven, as reflected by its title, utilizes hexagrams thirty-nine (“Limping,” Mountain+Water) and fourteen (“Great Possessing,” Heaven+Fire) to derive its parametric profile. These hexagrams yield a dynamic profile that varies from quiet to loud, temporal density changing from sparse to full, and pitch range going from medium to high. On a macroscopic level, the result is an increase in activity over the course of the piece, with the ending measures sounding much busier, denser, and louder than the sparse beginning.55 Perhaps the most unusual and opaque element in Water … Fire is its notation. The score is a musical transcription of the algorithmic output—which, as we have seen, specifies the parameters for each pitch in the piece. Each guitar is notated on a double staff, with one staff for each “voice” in the global two-voice polyphony (i.e., taken together, the six upper staves of the six guitar parts comprise one voice, and the six lower staves comprise the other; the result is a complex, score-wide hocket). Pitches are notated in 12TET without enharmonics (only sharps are used), with an “overall” deviation in cents indicated for each guitar at the beginning of the part. Such an arrangement makes the piece easier to read and perform at the cost of obscuring the underlying harmonic structures among pitches and voices. Although ostensibly anyone could calculate the relative size of intervals between the parts, such reverse engineering would remain ambiguous without consulting the original printouts. Conductor James McKay noted that the opacity of the score, with no correspondence between sounding and written notes, posed considerable challenges in rehearsal and performance. Reportedly, Tenney responded that hearing the sounds in one’s head was not important, but their precise realization in practice was.56 Example 11. View largeDownload slide “Random Walk” in 72TET equivalent of (3, 5, 7) harmonic space, accounting for all modulations in Water on the mountain … Fire in heaven Example 11. View largeDownload slide “Random Walk” in 72TET equivalent of (3, 5, 7) harmonic space, accounting for all modulations in Water on the mountain … Fire in heaven Example 12 shows a transcription of the first segment of the piece, mm. 1–15.57 The notation indicates actual sounding pitches in both voices, as well as the changing modes with their respective ratios. Due to the nature of the algorithm and transcription, adjacent modes may sometimes overlap for the duration of a held-over note. These occurrences can sometimes “expand” the mode to include pitches from the previous one, which are expressed as more complex ratios to the new fundamental (these ratios are indicated in parentheses in Ex. 12). However, in light of the speed of the modulations and the relatively quick natural decay of clean electric guitar tones, such overlaps are hard to hear in performance and are shown here only for the sake of theoretical fastidiousness. One element that is communicated clearly by the piece’s scoring is the division between the two voices. Perhaps due to the difficulty in reading across staves (and, to an extent, in fingering and holding all pitches for the indicated durations), the composer initially suggested the performance of each voice individually. Both existing recordings first present each voice in isolation, as separate movements, before mixing them together to form the complete harmonic picture. As noted both in Brian Belet’s analysis and in the liner notes to Josel’s recording, this structural arrangement yields an added element of discovery to the work, as its full harmonic structure is revealed progressively by the superimposition of the two layers.58 For the purposes of live realizations, Tenney’s original notes suggest recording the performance of the first staff, then performing the second staff, and finally performing the second staff against the recording of the first; as an alternative, a shorter version entails the recording of the first staff immediately followed by the performing the second against playback. However, Tenney’s handwritten notes for the York Archives tapes indicate his preference for a particular mix-down of the recordings of the two voices to be used as a tape work. To make matters more confusing, by the time the score was published in 1991, a new set of instructions was included with no mention of separate performances for each staff, but instead with a note giving permission to simplify the writing if the polyphonic writing across staves proved too difficult to play. From an aesthetic point of view, Tenney’s flexibility toward realization should not be too surprising, considering how he advocated, for instance, the alternative performance of his player piano pieces by turning the rolls upside-down.59 Example 12. View largeDownload slide View largeDownload slide Segment 1 from Water on the mountain … Fire in heaven, transcribed to show the two polyphonic layers as well as actual sounding pitches. Published by Smith Publications/Sonic Arts Editions. Used by permission. Example 12. View largeDownload slide View largeDownload slide Segment 1 from Water on the mountain … Fire in heaven, transcribed to show the two polyphonic layers as well as actual sounding pitches. Published by Smith Publications/Sonic Arts Editions. Used by permission. At both micro- and macroscopic levels, Water … Fire offers an unprecedented degree of harmonic complexity. Like Septet, it encapsulates Tenney’s brilliant approach to instrumentation with a practical and accessible score in the service of an ambitious musical palette. As a corollary to Changes, the piece allows us to participate in a longer, continuous modulating trajectory, stretching our harmonic horizons with the use of a mere thirty-six strings. SPECTRUM 4 Tenney’s Spectrum Pieces, composed between 1995 and 2001, is a series of eight chamber works for a variety of ensemble forces, ranging from three musicians and delay system in Spectrum 7 to twenty players of mixed instrumentation in Spectrum 3. All are algorithmically composed, and all explore (as indicated by the title) the materials of the overtone series. Furthermore, all pieces utilize F as the fundamental with the exception of Spectrum 2, which uses B♭. As suggested by musicologist Bob Gilmore, the Spectrum Pieces continue a new trend within Tenney’s harmonic series works, following in the direction indicated by In a Large, Open Space (1993–94).60 In this latter work of indeterminate duration and instrumentation, the gamut of available pitches encompasses the first thirty-two partials of the harmonic series on F. Given the “open” nature of the score, the order of entrance of the various pitches is left entirely to the performers and does not necessarily follow harmonic series order, as had been the case with earlier works. The resulting sonority, full of microtonal clusters and devoid of agogic or other form of temporal emphasis on the fundamental, departs from that of earlier pieces—a deliberate response to accusations of excessive euphonia in his previous music.61 Although in the Spectrum Pieces pitch entrances are strictly determinate, the algorithm governing note order and frequency ensures variability to produce a similarly stochastic result. Example 13. View largeDownload slide Basic gamut for Spectrum 4 Example 13. View largeDownload slide Basic gamut for Spectrum 4 Spectrum 4 is the only work in the series to include guitar. Through the employment of a careful tuning scheme, the instrument is not only completely integrated in the microtonal fabric of the piece, but also expands the gamut of available pitches beyond the limits of the rest of the ensemble. As with his previous guitar pieces, Tenney demonstrates the ability to adapt the guitar to rigorous tuning contexts, treating it as a kind of hybrid instrument in which the tuning is fixed, yet still easy to modify. The complete pitch set for Spectrum 4 can be divided in three smaller sets, corresponding to three sections of the ensemble: violin, alto recorder, piano, bass clarinet, trombone, and string bass form group one; the vibraphone group two, and the guitar group three. Group one plays from a twelve-note harmonic set built on a near-infrasonic F (21.83 Hz), detailed in Example 13. The gamut reflects harmonic series spacing, meaning that harmonics are only available in the octave in which they first appear. In more technical terms, the set falls within the nineteen-limit and avoids cognates, thus enabling the (retuned) piano to play all available pitches.62 Other instruments are required to match each pitch’s intonation, and to minimize beats by playing without vibrato. By contrast, the vibraphone’s only available pitched notes are those within ±5¢ of their just counterparts (i.e., F, F♯, G, A♭, and C); seven different unpitched sounds complete the percussionist’s twelve note set, replacing those pitches that would be too far out of tune. The guitar’s collection is a subset of the master one (as it lacks some of the lower harmonics in the gamut), with the addition of harmonics 27, 33, 35, 39, 45, 55, and 65. These higher partials are produced as natural harmonics of the open strings, which are tuned to lower harmonics of the fundamental (see Ex. 14)—an approach similar to the one Tenney had used in “SPECTRA for Harry Partch” from Quintext (1972) and in Arbor Vitae. The combined use of retuned open strings and natural harmonics allows for the production of precisely tuned pitches without the need for calculating cent deviations from 12TET. From a textural point of view, the guitar’s additional pitches populate the upper octave of the gamut with cognates, as shown in Example 15. This feature increases the harmonic and spectral complexity of the entire piece without compromising its playability. Example 14. View largeDownload slide Guitar tuning with open strings and resulting pitches through harmonic fingerings Example 14. View largeDownload slide Guitar tuning with open strings and resulting pitches through harmonic fingerings As with the other pieces in the series, Spectrum 4 employs a proportional notation in which each system corresponds to thirty seconds, white notes indicate long tones (separated by breath marks), and beamed notes represent clangs, to be played legato. The form of the piece is what Tenney called a “half-swell”—the music starts quietly and sparsely for the first seven minutes, then builds up in dynamics and density until the abrupt ending, which corresponds to the mid-point of the swell.63 As in Water on the mountain … Fire in heaven, the form is dictated by sets of parametric limits for dynamics, textural density, and temporal density—one for the beginning of the piece, and one for the end. These limits are once again interpolated in order to provide a smooth transition over time, and the resulting parameters are incorporated in the algorithm to generate musical information (such as the number of notes in a clang, their dynamic level, and register).64 Note, however, that harmonic complexity does not take part in this process: the opening thirty seconds include all twelve notes of the master set, with the guitar contributing higher partials (including harmonic 65, a tridecimal F♯, which is the highest partial in the entire piece). The perceived increase in complexity over the duration of the piece is rather a function of the increase in temporal density and dynamics. As mentioned above, a statistical feedback algorithm modeling dissonant counterpoint practices regulates pitch selection in the Spectrum series. Each of the eight parts in Spectrum 4 is generated individually—i.e., by running multiple instances of the algorithm in parallel; in addition, the guitar version of the code features some modifications to accommodate the different gamut available to the instrument. The sort of enforced variability resulting from the algorithm is not a simple chance operation, as the probability for the selection of any pitch is tweaked to be much lower after the pitch itself has been chosen (a uniform probability distribution, by comparison, would make the repetition of a pitch just as likely as the selection of a different one). Nevertheless, a certain amount of repetition is still present and can be verified at a glance by following the unpitched, numbered sound-events in the percussion part. Although repetition does occur to a degree, the resulting effect is that of ever-changing variability across the instrumental parts, with no particular emphasis on a given note or section of the gamut. Example 15. View largeDownload slide Upper octave of gamut in Spectrum 4; white noteheads indicate pitches introduced by the guitar Example 15. View largeDownload slide Upper octave of gamut in Spectrum 4; white noteheads indicate pitches introduced by the guitar Because the spacing of the gamut is weighted toward the higher partials, and because each part is generated independently, the lower pitches are overall less likely to occur than higher ones. As a result, the instruments capable of reproducing the lowest harmonics (string bass, trombone, bass clarinet, piano, and, to an extent, the guitar) spend most of their time in their higher registers. This absence of a continuous (or even prominent) fundamental in the bass register further contributes to the nebulous atmosphere of the piece, creating a very different sonority than any of the other pieces discussed above. CONCLUSION Whether composed algorithmically or by hand, the guitar works of James Tenney make it possible for the traditionally equal-tuned instrument to explore extended just intonation. As we have seen, Tenney’s tuning modifications do not require the modification of the fretboard or an alternative notational system. Therein lies a crucial difference between Tenney and some of his predecessors such as Partch, Harrison, or Ivor Darreg65—all of whom adopted more or less complex refretting solutions to suit their particular harmonic needs. In a way, the accessibility of Tenney’s works make them closer in inspiration to the raucous strand of experimentalism that flourished in the Downtown New York scene of the late 1970s and early 1980s, exemplified by the multi-guitar works of Rhys Chatham and Glenn Branca, and their reverberations in the music of No Wave groups like Band of Susans and Sonic Youth. Rather than speculating whether Tenney had heard Branca’s The Ascension (1981) or Chatham’s Guitar Trio (1977) by the time he composed Septet, perhaps we can assume that the massive sound of multiple electric guitars might have been “in the air” in the early 1980s, and that the guitar’s inherent tuning flexibility suggested the possibility of great harmonic rewards.66 By composing such ambitious works for the guitar, Tenney invites us on an exploration of the ratios, rhythms, and colors of the harmonic series, while expanding the harmonic horizons of the instrument’s repertoire to unprecedented and bewildering reaches. This article is a revised portion of my Ph.D. dissertation at the University of California, Santa Cruz, “The Just Intonation Guitar Works of Lou Harrison, James Tenney, and Larry Polansky.” I remain indebted to the patience and guidance of my committee members Leta Miller, Larry Polansky, and my dissertation advisor Amy C. Beal. I would like to thank Anna St. Onge at the York University Clara Thomas Special Collection and Archives, for her invaluable assistance in obtaining copies of Tenney’s correspondence, algorithmic output printouts, and a recording of Water on the mountain … Fire in heaven. Composer Michael Winter offered assistance in the early analytical stages, helping me learn to read Tenney’s algorithmic printouts. Finally, thanks to conductor James McKay and guitarist Seth Josel for answering performance-related questions. Footnotes 1 Examples include Larry Polansky’s for jim, ben, and lou (1995), II−V−I (1997), and Yitgadal (2003), Catherine Lamb’s point/wave (2015), and Michael Winter’s ostinato interrupted (2017). 2 There is one additional Tenney piece featuring guitar: Sneezles (An Encore) (1986, rev. 1995) for soprano, alto recorder/flute, clarinet, tenor-bass trombone, electric guitar, violin, double bass, and vibraphone. This four-page song does not focus on tuning and is therefore extraneous to the foregoing study. 3 For a recently published collection of his writings, see Tenney (2015). 4 Tenney (1964 [1986]). 5 Notable exceptions include Tenney (1996 [2015]) and (1987). 6 For a biographical sketch, see Belet (1990, 2–4). 7 Wannamaker (2008, 91–94). 8 See Tenney (1988). 9 One example is the equal-tempered major third, which is widely accepted in twelve-tone equal temperament (hereafter 12TET) contexts as a viable substitute of the natural major third, despite being significantly sharper (+14¢). 10 Tenney (1983, 22–23). 11 Cage (1961, 9). 12 In this light, Tenney’s concept of harmonic space is an extension of the lattice model for tuning systems developed by Ben Johnston. 13 Tenney (1983, 21ff.). 14 See Tenney (1983, 296) and Chalmers (1993, 60). 15 Tenney (1983, 3–4). 16 The crystal growth algorithm is explained at length in Tenney (2008, 47–56). 17 “Dissonant Counterpoint,” as a set of principles, avoids the repetition of pitches and reverses the common-practice treatments reserved for consonant (octave, major/minor thirds/sixths, perfect fourths/fifths) and dissonant (seconds/sevenths, tritones) intervals. For a history of the method, see Spilker (2011). 18 Tenney (1977, 36–69), and Polansky (1983, 253–55). 19 See Polansky, Barnett, and Winter (2011) and Winter (2008). 20 Dennehy (2008, 86–87). Tenney referred to the harmonic series as “the only thing given to us by nature … except sound itself.” 21 An important precedent for the use of higher-order equal temperaments to approximate harmonic ratios is the work of Ezra Sims (1928–2015), who has written almost exclusively with a quasi-just scale derived from 72TET since the 1940s. On his tuning approach, see Sims (1994). Other notable examples include the music of Ivor Darreg, Easley Blackwood Jr., and Wendy Carlos. 22 James Tenney, letter to Larry Polansky (undated, 1981). 23 In a review of the Microfest performance for the Los Angeles Times, Woodard (2003) mistakenly refers to the guitars being tuned a quarter tone apart. “Microtonal Guitarists Open Ears,” Los Angeles Times, April 29, 2003, accessed October 18, 2017, http://articles.latimes.com/2003/apr/29/entertainment/et-woodard29. 24 Belet (1990, 45) gives another example of Tenney’s pragmatic approach to tuning and temperament with the 1972 piece Glissande, in which Tenney opted for the use of fourteenth-tone divisions (as opposed to using the flexible twelfth-tones of 72TET) after evaluating the suitability of each system for the aesthetic and compositional purposes of that particular piece. 25 Hereafter, a subscript indicates the prime limit of a chordal aggregate, whereas superscripts imply the harmonic of a given fundamental. 26 Block (1996, 56) discusses Ives’s possible borrowing of the sonority from Stravinsky. Tenney, as noted above, had performed the Concord Sonata as a recitalist. 27 Because of the hybrid tuning context, and for the sake of clarity, I am borrowing the double-arrowhead accidental from the Helmholtz-Ellis just-intonation notation to indicate pitches one sixth-tone flat in the examples. 28 Whereas the earliest version of Harmonium II was circulated only in manuscript form, the Smith Publication edition is in print and readily available; the author and Polansky have performed the piece numerous times in recent years, including in retrospectives of Tenney’s guitar music in San Francisco, Santa Cruz, and New York City. 29 James Tenney, letter to Larry Polansky (February 18, 1982), James Tenney Fonds, Series S00329, File 19, Clara Thomas Archive, York University. 30 Larry Polansky, letter to James Tenney (undated, July 1985), Tenney Fonds. 31 Just Intonation (TELLUS #14, 1986). 32 Cocks Crow, Dogs Bark: New Compositional Intentions (1997). 33 Performance at Brooklyn’s Invisible Dog, with Polansky conducting an ensemble featuring DITHER plus additional musicians Nick Didkovsky, Dan Josephson, and Devin Maxwell, accessed October 18, 2017, http://www.youtube.com/watch?v=m-oN7El5VM8. 34 The tuning as indicated in the score makes the octave C♯s starting in m. 83. impractical, if not impossible. Tuning the third string down fourteen cents as well would enable the guitarist to play those measures comfortably without compromising other notes. 35 For an in-depth look at the mathematical implication of this process, see Wannamaker (2012). 36 See for example Cowell’s Rhythmicana (1931, for Rhythmicon and orchestra), as well as the earlier Quartet Euphometric and Quartet Romantic (1915–17). 37 Partch defines “otonalities” as collections of harmonic ratios (1–3–5–7–9–11 …), and “utonalities” as their inversions. See Partch (1949, 72–75 and 88–90). 38 Dennehy (2008, 80). 39 A letter from Tenney to Polansky (January 17, 1985) suggests that the composer had originally planned to write the sixty-four studies that would eventually become Changes for seven guitars in 84TET. 40 A tape recording of the reading is held by the York University archive. Tenney Fonds, 1998-038/032. 41 Long Distance (New York: CRI, 1996). 42 The first sixteen studies were performed by the New Music Concerts Ensemble at the Premiere Dance Theater in Toronto under the direction of Robert Aitkev on December 15, 1985. After the completion of the transcription by Michael Winter and Robert Wannamaker, Nicholas Deyoe conducted the premiere performance of the full set at the Box Theater in Los Angeles on November 11, 2017. A commercial recording is forthcoming. 43 In addition to Tenney (1987) and Belet (1990, 93–129), consider for example Young (1988, 204–5), which discusses Tenney’s use of the 11:9 interval in Changes but not in Water … Fire. 44 Tenney, letter to Larry Polansky (undated, 1984). 45 For the notation of 72TET pitches I borrowed some accidentals from the Helmholtz-Ellis notation. In this context, each arrowhead raises/lowers the equal-tempered pitch by ±16.67 cents. 46 Tenney (1987, 64–87). 47 By “polyphony” Tenney indicates the presence of independent musical layers or “voices,” rather than a reference to contrapuntal styles; note, for example, that either voice can contain aggregates as well as single notes. In this piece, the six guitars are collectively playing the two voices. 48 Tenney (1987, 83 and 87). 49 In Tenney (1964 [1986], 87), the composer defines a clang as “a sound or sound-configuration which is perceived as a primary musical unit or aural gestalt.” Not unlike motives, phrases, and periods in conventional formal analysis, clangs can be grouped hierarchically in sequences, sections, and segments; in Water … Fire Tenney’s program organizes the music in clangs and segments only, forfeiting intermediary hierarchies. See Tenney (1987, 68). 50 Tenney (1987, 83–84). 51 Ibid., 76–81. Note that modal construction in Changes allows for seven-note modes. 52 As one example, Polansky’s B’rey’sheet (1984) achieves comparable harmonic results with the help of an adaptive (“paratactical”) computer algorithm that both “listens” to the performer and generates new harmonic pathways for them to follow. 53 Belet (1990, 102–3) states that the Changes algorithm was designed so that two modes could be active at a given time in certain cases, resulting in a 50% likelihood of clangs from adjacent modes actually overlapping. I have found no such instance in the algorithmic output for Water … Fire. 54 Tenney (1987, 66–67). 55 Tenney’s use of I Ching hexagrams and the choice of “Changes” as the title of the sixty-four studies collection represent a connection to Cage’s use of the same book as a source for chance operations, as in, for example, the composition of the third part of the Concert for Prepared Piano (1950–51) and Music of Changes (1951). 56 James McKay, e-mail communication with the author from December 6, 2012. 57 Root movement for the same excerpt encompasses Pcs 0, 30, 7, 26, 56, 14, 44, 2, 32, 62, and 20 in Exx. 10 and 11; note that the first two modes in the pieces are built on the same root (Pc0). 58 Belet (1990, 118); Long Distance (NWCR697), liner notes. 59 Somewhat puzzling is the fact that Tenney personally gave Josel the “old” notes for the piece in 1996, despite the publication of the revision in 1991. Seth Josel, personal communication with the author from April 20, 2015. 60 Gilmore (2009), 1–2. 61 Ibid. Evidently those same critics were not familiar with the hazy cloudiness of Changes or Water … Fire. 62 In tunings other than 12TET, pitches can often share the same “letter name” while being tuned differently (for instance, a B♭ tuned as a harmonic seventh above C, and another B♭ tuned as consecutive perfect fifths below the same C (the pitches are approximately 31 cents apart). In order to avoid cognates, Tenney notates the eleventh harmonic as a 49¢-flat B♮, rather than a 51¢-sharp B♭. For the sake of consistency, and in accordance with the Helmholtz-Ellis notational standards, I will be referring to the same note as an undecimal B♭ (51¢ sharp). 63 Gilmore (2009). Additional variations of this form can be found in “Having Never Written a Note for Percussion” and the three “Swell Pieces” from Tenney’s Postal Pieces set (1965–71). See Polansky (1983, 193–203). 64 The concept of parametric profiles, i.e., the plotting of a given musical parameter over time, is introduced as an analytical/perceptual model in Tenney (1964 [1986], 33). It is especially telling how Tenney employed the parametric profile as a compositional tool, allowing him to control the resulting form of a piece from a perceptual standpoint. 65 Ivor Darreg (born Kenneth Vincent Gerard O’Hara, 1917–94) was a composer, tuning theorist, and instrument builder. He coined the term “xenharmonic” as a neutral alternative to “microtonal,” and issued twelve installments of The Xenharmonic Bulletin, an occasional newsletter outlining theoretical and practical tuning topics. His writings are available in several issues of Xenharmônikon (distributed by Frog Peak Music, 1976–2006). 66 In correspondence from the timeframe of the Septet premiere, Polansky notes a remote similarity to the works of Branca, despite Tenney’s previous admonition that the sound of the ensemble should be “big, but clean” (emphasis as in original). Tenney, letter to Larry Polansky, 17 January 1985; and Polansky, letter to James Tenney, 5 April 1985. Works Cited Belet Brian. 1990 . “An Examination of the Theories and Compositions of James Tenney, 1982–1985.” Ph.D. diss., University of Illinois at Urbana-Champaign. Block Geoffrey. 1996 . Ives, Concord Sonata: Piano Sonata No. 2 (“Concord, Mass., 1840–1860”). Cambridge : Cambridge University Press . Cage John. 1961 . “Experimental Music.” In Silence: Lectures and Writings. 7 – 13 . Middleton, CT : Wesleyan University Press . Chalmers John. 1993 . Divisions of the Tetrachord. Hanover : Frog Peak Music . Dennehy Donnacha. 2008 . “Interview with James Tenney.” Contemporary Music Review 27 ( 1 ): 79 – 89 . Google Scholar Crossref Search ADS Gilmore Bob. 2009 . Liner Notes to Spectrum Pieces. NWR , 80692 – 2 . Partch Harry. 1949 . Genesis of a Music . New York : Da Capo . Polansky Larry. 1983 . “The Early Music of James Tenney.” In Soundings 13: The Early Music of James Tenney. 119 – 297 . Santa Fe : Soundings Press . Polansky Larry , Barnett Alex , Winter Michael . 2011 . “A Few More Words about James Tenney: Dissonant Counterpoint and Statistical Feedback.” Journal of Mathematics and Music 5 ( 2 ): 63 – 82 . Google Scholar Crossref Search ADS Sims Ezra. 1994 . “Long Enough to Reach the Ground, or How Long Should a Man’s Legs Be?” Perspectives of New Music 32 ( 1 ): 208 – 13 . Google Scholar Crossref Search ADS Spilker John. 2011 . “The Origins of ‘Dissonant Counterpoint’: Henry Cowell’s Unpublished Notebook.” Journal of the Society for American Music 5 ( 4 ): 481 – 533 . Google Scholar Crossref Search ADS Tenney James. 1964 [1986] . META+HODOS and META Meta+Hodos . Hanover : Frog Peak Music . Tenney James . 1977 . “The Chronological Development of Carl Ruggles’ Melodic Style.” Perspectives of New Music 16 ( 1 ): 36 – 69 . Google Scholar Crossref Search ADS Tenney James . 1983 . “John Cage and the Theory of Harmony.” In Soundings 13: The Music of James Tenney. 55 – 83 . Santa Fe : Soundings Press . Tenney James . 1987 . “About Changes: Sixty-Four Studies for Six Harps.” Perspectives of New Music 25 ( 1/2 ): 64 – 87 . Tenney James . 1988 . A History of “Consonance” and “Dissonance.” New York : Excelsior . Tenney James . 1996 [2015] . “About Diapason.” In From Scratch. Ed. Polansky Larry , Wannamaker Robert , Winter Michael , Pratt Lauren . 394 – 96 . Chicago : University of Illinois Press . Tenney James . 2008 . “On Crystal Growth in Harmonic Space.” Contemporary Music Review 27 ( 1 ): 47 – 56 . Google Scholar Crossref Search ADS Tenney James . 2015 . From Scratch. Ed. Polansky Larry , Wannamaker Robert , Winter Michael , Pratt Lauren . Chicago : University of Illinois Press . Tenney James , Polansky Larry . 1981–1992 . Correspondence. Tenney Fonds, F0428. Series 2000–045/001. File 19. Clara Thomas Archive, York University, Toronto. Wannamaker Robert. 2008 . “The Spectral Music of James Tenney.” Contemporary Music Review 27 ( 1 ): 91 – 130 . Google Scholar Crossref Search ADS Wannamaker Robert . 2012 . “Rhythmicon Relationships, Farey Sequences, and James Tenney’s Spectral CANON for CONLON Nancarrow (1974).” Music Theory Spectrum 34 ( 2 ): 48 – 70 . Google Scholar Crossref Search ADS Winter Michael. 2008 . “On James Tenney’s Arbor Vitae for String Quartet.” Contemporary Music Review 27 ( 1 ): 131 – 50 . Google Scholar Crossref Search ADS Woodard Josef. 2003 . “Microtonal Guitarists Open Ears.” Los Angeles Times, 29 April 2003. Young Gayle. 1988 . “The Pitch Organization of Harmonium for James Tenney.” Perspectives of New Music 26 ( 2 ): 204 – 12 . Google Scholar Crossref Search ADS Discography The Barton Workshop . 2009 . Spectrum Pieces. NWR , 80692 – 2 . Josel Seth. 1996 . Long Distance. Composers Recordings, Inc., NWCRI , 697 . Josel Seth . 1998 . Go Guitars . O.O Discs , 36 . Various Artists. 1986 . Just Intonation. TELLUS , 14 . Various Artists . 1997 . Cocks Crow, Dogs Bark: New Compositional Intentions. Companion CD to Leonardo Music Journal 7 . Verdejo Adrian. 2014 . Modern Hearts. Redshift Music , TK429. © The Author(s) 2018. Published by Oxford University Press on behalf of The Society for Music Theory. All rights reserved. For permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)
TI - Tuning Theory and Practice in James Tenney’s Works for Guitar
JF - Music Theory Spectrum
DO - 10.1093/mts/mty022
DA - 2018-11-01
UR - https://www.deepdyve.com/lp/oxford-university-press/tuning-theory-and-practice-in-james-tenney-s-works-for-guitar-2e3AMyGqCv
SP - 338
VL - 40
IS - 2
DP - DeepDyve
ER -