Andromeda Memories Futility Row Orbital Resonance
Pavane for a Dead Planet Star Dance Ride the Cosmos
Dark Forces Signify The Lessing is Miracle Busted Grooves
Rings of Saturn Pulsars Neptunian Night
Spacecat Reverse Gravity Romance Postmoderne
Liquid Mechanisms Galactic Jamboree
Dedicated to all those
musical performers who have
ignored my music and inspired me
to become self-sufficient
Hyperchromatica is a work in seventeen movements, lasting two hours and 34 minutes, for three microtonally tuned Disklaviers (computer-driven pianos). The 13-limit just intonation tuning for the three pianos (Disklaviers), with 33 pitches per octave, is given at the bottom of this page. It would be more accurate, actually, to think of this not as music for three pianos, but for one piano with 243 keys. In addition to the microtonality, most of the pieces also use polytempo structures and other rhythmic difficulties that would make performance by human players impossible. The fusion of microtonality and polytempo that I began in Custer and Sitting Bull (1995-99) reaches a second climax here.
Among the collection of movements a great disparity of styles will be noticed. This is deliberate and necessary. Had I only written abstract, austere pieces like Orbital Resonance and Liquid Mechanisms, some listeners would have said, "Well the tuning is interesting, but few people will be attracted to an idiom so peculiar." Had I written only tonal and melodic pieces like Pavane for a Dead Planet and Dark Forces Signify, some would have said, "Well he's not really doing anything new, just going back to old styles and retuning them exotically." In order to thoroughly explore this elegant tuning in a kind of Gradus ad Parnassum - in order to allow the tuning to make its full argument to an audience - I had to go both forward and backward in history, to show what we could have been doing these last few centuries had we not been limited to twelve pitches, and also to project into the future what music could become: and with the same 33 pitches. I had to create not a unified essay, but an alternate universe. Besides, I was like a kid in a candy store with all these new pitches, able to resurrect chord progressions from the past and make them sound otherworldly by restoring the minuscule discrepancies that equal temperament had swept under the carpet. Using the numbers 1, 3, 5, 7, 9, 11, 13, and 15 in the harmonic series, it was natural to emphasize the familiar 1-3-5 in one piece, and the strange 7-9-11 in another, and each has its revelations in this context. The range of style proves the tuning's versatility and wealth of potential. In all the pieces, though, there is a pleasure taken in tiny intervals of 25 to 50 cents - as voice-leading, as melody, as complexly buzzing sonority. And so the style, in every case, can be described as hyperchromatic.
Let me put it more simply: I'm trying to make microtonality attractive and seductive, not scary as it is to most people and in most microtonal music. A lot of people, mostly composers, want to hear the most weird-ass and transgressive s**t I can throw at them, and I try to gratify that in some movements. But more, I want to suggest (and prove) that we can keep conventional tonality and augment it with higher-overtone relationships. This is my strategy for bringing microtonality into the mainstream, where I am convinced it will eventually end up. The goal here is no less than to reinvent tonality.
Andromeda Memories (2016-17)
The cycle opens with a nostalgic interstellar jazz tune, testing how far the tuning can support jazz harmonies. I imagined it being played by a creature with several hands at one of those bars in a science fiction movie where pathetic beings from many planets are passing through. The theoretical impulse that led to the piece (and most of these pieces began with some realization about the capabilities of the scale I was using, for which I then had to find a poetic expression) is that, since D is the note that appears in the most harmonic series (see tuning chart below), it's the pitch that is most versatile and, specifically, has a normal minor scale on it with a minor seventh chord with ninth and eleventh. So Dm7 is the "hook" in jazz syntax that I start from (and keep returning to) to push the jazz language into more exotic parts of the scale.
Futility Row (2015)
Futility Row may well be the first piece written in the key of E-13-flat minor. That is, since my 1/1 is E-flat, the tonic here is the 65th harmonic (major third of the 13th harmonic), 27 cents sharper than E-flat. I have a penchant for minor keys, and it's difficult to write a minor-key piece in a scale constructed from harmonic series'. I gained a new empathy for Haydn, who, in his minor-key symphonies, always seems to modulate into the major as quickly as possible. Schoenberg remarked that Chopin was lucky because, if he wanted to do something that sounded new, all he had to do was write something in F# major. Well I'm way ahead of Chopin, because not only am I the first to write something in E-13-flat minor as far as I know, I have lots of other exotic keys left to use.
This is a particularly Gannian piece in form and gestural style, with a rhythmic ostinato and several interruptive forays into the keys based on the harmonic series', with kind of a humorous "Western noir" feel to it. I got the idea while humming a song by Mikel Rouse, and so I dedicate it to him, whose music has so often been a means of bringing me back to earth.
Orbital Resonance (2015)
When the New Horizons spacecraft took its historic photos of Pluto in July 2015, there was a lot written about Pluto. I learned, for instance, for the first time, that although the orbits of Pluto and Neptune overlap, they are prevented from colliding by the 2-to-3 ratio in their rotations around the sun; Pluto goes around the sun in 247.94 earth years, and Neptune in 164.8, and 247.94/164.8 equals 1.50449.... This kind of mutually influenced periodicity, as it turns out (how was I an astrologer for thirty years without learning this?), is common among pairs, trios, quadruples of planets, moons, asteroids, and so on, and is called orbital resonance. Three of the moons of Jupiter exhibit rotational ratios of 1:2:4, and there's even an asteroid that has a 5:8 dance going with respect to the earth. This is truly the harmony of the spheres, the surprisingly simple mathematical relations that planets in a rotational system fall into in response to each other's gravity.
This suddenly gave me a whole new way to think about the kind of polytempo structures I'd been writing for 35 years. I'm used to having repeating cycles at different tempos, and it has sometimes been an aesthetic problem for me when the articulation points of those cycles coincide by chance. But the solar system, as it turned out, had already solved that problem for me. Inspired by this new knowledge, I started using much simpler ratios than I had been using (3:4, 5:6:7), but shifting each one a slight amount so that the articulated beats would never coincide. It gave me a new way to create melody from the articulated beats among the different cycles. I immediately started a piece titled Orbital Resonance.
Although Orbital Resonance is fairly continuous within its moment form, its successive panels fall into six sections whose progression makes the derivation of the characteristic rhythm increasingly clear:
1. Articulation of the characteristic rhythm by various pitches in the scale less than a quarter-tone apart, with harmonizations.
2. Articulation of the characteristic rhythm by dyads from different harmonic series'.
3. The characteristic rhythm fused into a single melody, accompanied by chords outlining the rhythmic derivation.
4. Articulation of the characteristic rhythm by widely-spaced sonorities separated by extremely parsimonious voice-leading (expansion of the 1st section and 2nd section ideas).
5. Articulation of the characteristic rhythm divided out among increasingly audible independent melodic ostinatos (expansion of the 3rd section idea).
6. A coda returning to the initial idea, with sparser harmonization.
This is a kind of broken symmetry characteristic of my music: the first section is paralleled with the sixth, the fourth combines the first and second, and the fifth expands on the third. I provide the plan not to suggest that the piece should be heard in a corresponding way, but merely to draw attention to the presence of an internal logic that might not be immediately evident.
I had been waiting for many years to write a piece sufficiently ambitious and elaborate to dedicate to my teacher Ben Johnston, who taught me microtonality. This is it.
Pavane for a Dead Planet (2016)
Someone asked me a question about meantone tuning, I mentioned Orlando Gibbons in response, and suddenly I felt compelled to write a pavane. I had been trying to figure out how to write a piece in minor in a tuning based on the harmonic series, and it occurred to me that I could use the harmonic series scale modally, starting on the third step - creating an eight-step Phrygian mode, so to speak. So the primary scale here, in ratios, is 1/1, 11/10, 6/5, 13/10, 7/5, 3/2, 8/5, 9/5, with a very consonant minor seventh chord of two perfect fifths and two minor thirds. Also, since I'm working in just intonation, I thought it would be nice to write a piece emphasizing the pure, "normal" consonances for once. In this scale, the truly interesting voice leadings result in changes between minor and major chords, so the dominant seventh and then the major seventh are gradually worked in.
One might assume that I chose the title to refer to the earth and the specter of climate change, but actually I was thinking more of the earth-like exoplanets scientists keep finding in other solar systems, and how sad it is that they aren't inhabited (as far as we know).
Star Dance (2015-16)
Star Dance is a conceptually atonal piece (although the key notes that run through the tenor register are backed up by bass notes in the same harmonic series) which draws a counterpoint of melodies from the small intervals in the scale (28 to 50 cents). My idea was that the notes going rather randomly in and out of tune with the bass notes suggested the twinkling of stars, and the steady movement evokes the stability of star positions in the sky.
Ride the Cosmos (2016-17)
Ride the Cosmos is a mercurial scherzo inspired by Ravel's Miroirs (1908), which pianist Sarah Cahill introduced me to. The rhythm is extremely fluid, constantly changing tempo in tiny increments; many of the meters have a 32nd-note denominator, and often prime numerators. For the first time in my life I used an arch form (ABCDEDCBA+coda), though the repeating sections return transposed to different harmonic series. In an equal-tempered scale such transpositions would be trivial, but in this very unequal scale I wanted to hear what variations in harmonic connections would result (so it's really ABCDED'C'B'A'+coda).
Dark Forces Signify (2016)
Dark Forces Signify is a tribute to the Black Lives Matter movement (matter is a difficult verb to find a synonym for). The piece is a simple and dignified chorale that aspires to be prayer, march, and waltz in one. A few weeks after I finished it I read that scientists were now talking about the dark matter in outer space being held together by "dark forces," so I was pleased to have another layer of astronomical significance as well.
The Lessing Is Miracle (2015-16)
The Lessing Is Miracle is conceptually very simple, just melody and accompanying harmony; but the harmony's polyrhythms keep up a scintillating texture, while the melody floats through free time-space without tempo or beat. The piece started as a tribute to the music of the late Julius Eastman (1940-1990), with the repeated notes of his multiple-piano pieces, though the mood took a different turn. The title is an inexplicable phrase found written in capital letters in the middle of one of Julius's scores. I first heard Julius perform in 1974 and '75, and got to know him in the early 1980s, last running into him in 1989. He died mysteriously, and ten months later I got wind of the fact and wrote the first obituary of him.
Busted Grooves (2017)
I started Busted Grooves the day I realized that the 49th harmonic could mediate smoothly between minor and major, that is, between the 48th (3/2) and 50th (25/16) harmonics. Since I only had one 49th harmonic in my scale, the piece had to be in the key of G (5/4), to go between the 6/5 and 5/4 of that key via 49/40. Thus the rare intervals 49/40 and 60/49, which I don't think I had ever used before, appear frequently here. The basic ostinato is 6 quarter-note beats, around which weave ostinatos of different lengths and tempos, the texture interrupted at cadences by formulas in unrelated tempos. When I had rehearsed my piece The Planets with the Relache ensemble they called me "the groove-buster" because I so often throw rhythmic wrenches into my grooves. The Disklaviers here don't mind.
Rings of Saturn (2016-17)
Rings of Saturn uses minimalist processes of repetition, additive process, and phase-shifting, variously combined, in each of its five sections, to create organic forms easy to follow on some level but full of anomalies and variation. Technically speaking, the piece continues (as with the Pavane) my exploration of the harmonic series as a mode, most of its tonalities being based on the third harmonic of their respective harmonic series. This gives a scale of 1/1, 13/12, 7/6, 5/4, 4/3, 3/2, 5/3, 11/6, and a tonality which contains a subdominant (something one otherwise rather comes to miss when working with the harmonic series) and no dominant. The bluesy 7/6-5/4 step colors the melodic feel throughout. Rarely in my life has a piece required so much revision during the process, some sections having been rewritten several times.
Pulsars has very few played notes in it, fewer than 300 in ten minutes, but many of them are very close together (like, three pitches within a half-step), so that most of the music consists of the pulsing beat patterns that arise after the notes are played. In many cases the beats don't fully appear for the first five or seven seconds, so each chord lasts at least fifteen seconds to allow them to come out. The music is not in the notes; it happens unpredictably in the air and in the listener's ear. The kind of distinct beating patterns made possible in just intonation have no counterpart in twelve-tone equal temperament. This piece should be listened to with good headphones or speakers; the rhythmic activity will not at all be fully apparent if consigned to laptop speakers. (Despite the stylistic dissimilarity, this movement is closely related to Star Dance; it uses vertically the same sets of closely-spaced pitches that that movement capitalizes on melodically.)
Neptunian Night (2017)
Neptunian Night is the slow, soft adagio of the piece, and is the third in a trio of my "night" pieces, after Long Night (1981) for three pianos played by humans and Unquiet Night (2004) for conventionally tuned Disklavier. All three feature continuous use of the sustain pedal(s) and a watery texture of simultaneous tempos. Almost all the lines float in tuplets, with three to seven tempos going at once. Neptunian Night is something of a theme and variations, though it would be more accurate to say that there is a series of textures and tonalities in which the theme keeps reappearing. The penultimate section creates an illusion of alternating between major and minor, which is not easy to achieve within the harmonic series; actually, it fluctuates between D7b major and C^ minor (the 56th and 55th harmonics of Eb), their tonics only 31 cents apart, a discrepancy almost impossible to notice in this context. The final section leaves melody behind for a mystically Neptunian cloudiness.
Soon after I started this movement, I had a dream telling me to name it Spacecat. I have nothing more to say.
Reverse Gravity (2016-17)
Reverse Gravity is a gnossienne in which - for the theoretical completeness of the set - relative heights within the overtone series are reversed. That is, the higher harmonics such as 7, 11, and 13 are most often heard in the bass, while the melodies above are in smaller-ratio scales on 1, 3, and 5. As a result, we get to hear scales like 12/11, 13/11, 14/11, 15/11, 16/11, and 14/13, 15/13, 16/13, 18/13, 20/13, and so on. In other words, where the Pavane used the harmonic series as a mode starting on the 5th harmonic, and Rings of Saturn used it as a mode starting on the 3rd, Reverse Gravity uses a number of different modes starting on the 7th, 11th, 13th, and others. The tame conventionality of the genre, Satie-inspired, sets off the weirdness of the universe of intervals we're not used to. Some will find it out-of-tune, but out-of-tuneness that is consistent and systematic I find exotic.
Romance Postmoderne (2012)
Romance Postmoderne is a kind of rewriting of a Romantic genre, very faux-pianistic. It was the earliest of these pieces, written in 2012; three years later, with Orbital Resonance, I intuited how to combine the polytempo with the microtonality and the whole set took off. It's one of two pieces here (with Dark Forces Signify) that could be played live, and it has been, by Aron Kallay and Vicki Ray. It's also my first piece on the harmonic series, and consequently conceptually simple, though it takes more advantage of the higher harmonics than some of the others. Its guiding principle is that articulations of the various harmonic series' move very little in register, and that a feeling of tension and release arises depending on which harmonics are nearer the bass; a harmonic series with, say, the 1st and 3rd harmonics in the bass will sound more stable (less tense) than one with the 11th and 13th in the bass.
Liquid Mechanisms (2016)
Liquid Mechanisms is a moment form, a series of panels in a mural. Every section employs notes, chords, or phrases moving at different tempos and going out of phase, with a nonsynchronicity that I think of as a watery feeling of time. Polytonality is as rampant here as polytempo. Within each panel, I think, is enough repetition of elements to begin to hear each set of complex relationships as a whole.
Galactic Jamboree (2015-16)
Galactic Jamboree is intended as the exciting finale of the whole set, coming back to the tuning's home key of Eb, to which the melodies always return over a pair of overlapping bass ostinatos (at ratios of 11 versus 13 beats).
The tuning employs 33 harmonics of Eb. It contains eight harmonics series', each up to the 15th harmonic, based respectively on the 1st, 3rd, 5th, 7th, 9th, 11th, 13th, and 15th harmonics. The 33-pitch tuning of the three pianos (the same in every octave) is as follows, given first in the number of cents above E-flat, and then as ratios to the E-flat 1/1:
Piano Key Cents Ratio Cents Ratio Cents Ratio Piano 1 2 3 D 1088 15/8 977 225/128 1044 117/64 Db 969 7/4 938 55/32 906 27/16 C 857 105/64 773 25/16 840 13/8 B 738 49/32 755 99/64 729 195/128 Bb 702 3/2 590 45/32 609 91/64 A 551 11/8 551 11/8 481 169/128 Ab 471 21/16 440 165/128 408 81/64 G 386 5/4 320 77/64 342 39/32 Gb 204 9/8 275 75/64 275 75/64 F 155 35/32 192 143/128 192 143/128 E 92 135/128 53 33/32 27 65/64 Eb 0 1/1 1103 121/64 1173 63/32
Note that no string needs to be raised higher than its natural tuning except for the B-flat on piano 1, which is 2 cents sharp (or if one prefers, 2 cents could be subtracted from all quantities).
For electronic realization of the piece, it can prove helpful to reconfigure the tuning as a reference pitch in cycles per second for each piano, and ratios derived from that standard:
Pitch name Piano 1 Piano 2 Piano 3 Tuning pitch: 38.891 cps 36.7641 cps 38.2833 cps D 15/8 225/121 13/7 Db 7/4 20/11 12/7 C 105/64 200/121 104/63 B 49/32 18/11 65/42 Bb 3/2 180/121 13/9 A 11/8 16/11 169/126 Ab 21/16 15/11 9/7 G 5/4 14/11 26/21 F# 9/8 150/121 25/21 F 35/32 13/11 143/126 E 135/128 12/11 65/63 Eb 1/1 1/1 1/1
In the configuration of certain tuning softwares, the following sequences might facilitate getting the required tuning:
38.891 cps = Eb0
1/1, 135/128, 35/32, 9/8, 5/4, 21/16, 11/8, 3/2, 49/32, 105/64, 7/4, 15/8
36.7641485 cps = Eb0
1/1, 12/11, 13/11, 150/121, 14/11, 15/11, 16/11, 180/121, 18/11, 200/121, 20/11, 225/121
38.283333 cps = Eb0
1/1, 65/63, 143/126, 25/21, 26/21, 9/7, 169/126, 13/9, 65/42, 104/63, 12/7, 13/7
Overall, the 33-pitch tuning of the three pianos is as follows. In addition to the pitch list on the left, the pitches are grouped into the eight harmonic series' in the right eight columns:
Pitch name Ratio Cents 1/1 3/2 5/4 7/4 9/8 11/8 13/8 15/8 Eb7+ 63/32 1173 9 7 Db^^ 121/64 1103 11 D 15/8 1088 15 5 3 1 Db13 117/64 1044 13 9 C#+ 225/128 977 15 Db7 7/4 969 7 1 C^ 55/32 938 11 5 C+ 27/16 906 9 3 C7+ 105/64 857 15 7 Cb13 13/8 840 13 1 B 25/16 773 5 Bb^ 99/64 755 11 9 Cb77+ 49/32 738 7 Bb13 195/128 729 15 13 Bb 3/2 702 3 1 Bbb713 91/64 773 13 7 A+ 45/32 590 15 9 5 3 Ab^ 11/8 551 11 1 Abb1313 169/128 481 13 Ab7+ 21/16 471 7 3 G^ 165/128 440 15 11 G+ 81/64 408 9 G 5/4 386 5 1 Gb13 39/32 342 13 3 Gb7^ 77/64 320 11 7 F#+ 75/64 275 15 5 F+ 9/8 204 9 3 1 Fb13^ 143/128 192 13 11 F7+ 35/32 155 7 5 E+ 135/128 92 15 9 Eb^ 33/32 53 11 3 Eb13 65/64 27 13 5 Eb 1/1 0 1
(If you don't have enough experience with just intonation to make sense of this chart, try reading the step-by-step Just Intonation Explained section.) In Johnston's notation, + raises a pitch by 81/80, # raises it by 25/24, b lowers it by 24/25, 7 lowers it by 35/36, ^ raises it by 33/32, 13 raises it by 65/64, and F-A-C, C-E-G, and G-B-D are all perfectly tuned 4:5:6 major triads.
I far prefer working in an unequally spaced scale to an equal temperament (you can go here to read why), and this is a very unequal scale, as the following chart shows:
PS: I could go on. I get the sense that a lot of microtonalists first find what looks like a neat scale and then see what they can do with it. I have never worked that way. I have invariably envisioned what effect I wanted my music to create, and then constructed a scale that would produce that effect. For decades I never used the same scale twice, but as I continued studying I became more and more seduced by the harmonic series as an efficient framework for the kinds of voice-leading and tonality flux I had been using all along. Hyperchromatica represents a long-awaited plateau for me in that regard. I don't know that I'll use this scale again. After an intense two and a half years, I'm itching for something new.
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