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.
Orbital Resonance is for three computer-controlled pianos (Disklaviers, for instance), retuned to include 33 pitches to the octave, the result of eight harmonic series' on the first eight odd-numbered harmonics of Eb. Although the piece 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.
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.
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