A few simple examples of using a simple rhythmic proportion, two-to-one, to generate unexpectedly rich ensemble textures.
In the first, an example that I associate with a brief impromptu lesson I had with Ornette Coleman in the back of a theatre in New Haven. A melody is played and then restated at twice the speed over the same accompaniment, here a simple ground bass. This is especially fun to place with extended melodies, going through the tune twice in double time while the accompaniment stays put (this is an idea I associate with a brief impromptu lesson I had with Ornette Coleman in the back of a theatre in New Haven). The rhythmic point in the middle of such textures can be a useful point of departure for composition, beginning with a point in which consonance or dissonance can be most heightened.
The second is cribbed from South Indian Solkattu (rhythmic solfege) lessons. A single pattern is learned at three speeds, here notated as quarters, eighths, and sixteenths. In this example, the superposition of a three note pattern in each speed over a pattern which repeats at the slowest speed results in a nicely propulsive series of polyrhythms (3:2, 4:3) and then syncopations created by the combinations of the smaller note values.
And this, too: A perfect mensural canon in the proportions 1:2:3, from my string trio, Figure & Ground (1994-95). The proportions are applied to both the pitches and rhythms and correspond precisely to the tunings of the three instruments. Thus the cello plays its material once through while the viola plays the same material an octave higher and twice as fast two times through and the violin plays the same material a twelfth higher and three times as fast three times through. Here, getting the music to work meant focusing first on the seams between the repetitions in the upper instruments, here at the one-half point articulated by the viola, and the one-third and two-thirds points articulated by the violin.