mod groups & rhythm

Rhythm, or the duration of notes, is structured a lot like pitch. The common
musical palette for duration is much more limited than than for pitch. But
the ratio of 2/1 is still the fundamental structure, with 3/2 not far behind.

One idea that follows from this - tempered rhythm! Instead of triplets being
exactly 3/2 speed, one could speed them up or slow them down a touch. This
would then permit rhythmic puns.

Rhythm has an extra dimension that pitch doesn't - phase. Two steady
pulses with the same speed can still differ, they can be offset from each
other in time.

Really, pitch and pace - pardon my dread ignorance of proper musical terms -
are not wholly separate. A steadily repeated click sets up a regular
pulse at say 3 beats per second. But speed it up to 24 beats per second and
it's a very deep pitch. Somehow as this pulse/pitch transition is traversed,
the sensitivity to phase or registration in time is lost. Perhaps that loss
of phase is just what the transition is about.

Adding phase makes the mathematical structure messier. Pitch intervals do
seem to form a group. Moving one interval after another is the same as moving
some third interval. With rhythm or pulse, there are two operations. One can
shift the pulse in time, or change the speed.

Shifting a pulse back and forth in time looks like a group structure -
there's a nice inverse. But changing speed is messier. It makes good sense to
double the speed of a pulse. But if I halve the speed, then there are two
ways to do this - I need to skip every other beat, but do I choose to skip
the even or odd beats? So the speed-up operation appears not to have a
well-defined inverse.

Jim

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