For whatever reason, I was never much drawn to the type of nested 3:2 and 4:3 crossrhythms that I described in my last metametrics post, but I did use it once, and could have mentioned it. In 1985, on the verge of turning 30, I was under some psychological pressure to write something ambitous. I decided on a big set of variations for two pianos, somewhat inspired by the two-piano tradition of Busoniâ€™s Fantasia Contrappuntistica, Wallingford Rieggerâ€™s Variations, The Art of Fugue, and such huge solo piano variations as the Diabelli and Brahmsâ€™s Handel. (This was before Larry Polansky wrote his Lonesome Road Variations, which dwarfed mine.) My Iâ€™itoi Variations was a set of eleven variations on the â€œBlack Mountainâ€ song from the Iâ€™itoi cycle of the Papago Indians. I was very involved in collecting, transcribing, and analyzing American Indian music in those days, because I was attracted to its, um, proto-totalist rhythmic character – specifically the way it shifted back and forth among tempos.
Another influence on Iâ€™itoi Variations (and on much of my music) was Beethovenâ€™s Op. 111 Sonata – variations 1 through 6 were stormy and tense like Beethovenâ€™s first movement, variations 7 through 10 were impersonal and calm like the second movement, and no. 11, in an attempted tour de force, went back through the first ten variations in reverse order. Variation 8 was based on a nested 3:2. The idea was simple. (I try to make all my ideas simple). There were two foreground lines, the bass line in piano 2 and the right-hand melody of piano 1, plus a line of quarter-note chords shared by both pianos. The bass was in dotted quarters, the treble in triplet quarters, for a 4:6:9 tempo resultant. I hoped the treble and bass would float in seeming unrelatedness to each other, though subtly mediated by the chords in between. Hereâ€™s a passage from the beginning of the variation:
and another from the end, where the treble and bass lines have doubled in tempo:
The bass line is the inversion of the theme, with the intervals doubled in size (a trick I picked up from Messiaenâ€™s bird song music). I suppose it wonâ€™t much harm my reputation to admit that the changes of key are timed according to a descending sequence of Fibonacci numbers. Everyone who studied Bartokâ€™s music had to try them once. This was back in 1985 when I had not yet heard of Mikel Rouse, Michael Gordon, or Art Jarvinen, and was blissfully unaware that they were using the same ideas.
Hereâ€™s the recording of Variation 8 from Iâ€™itoi Variations, played by the Double Edge duo – Nurit Tilles and Edmund Niemann, who premiered it at Cooper Union in New York circa 1990.