Speeds...

WARNING: MUSIC NERD INFO AHEAD

Ever since I discovered Elliott Carter's music I've been interested in the 'speed' of certain rhythms. What I am talking about is the rate at which a poly-rhythmic pulse moves across the regular 'crotchet' pulse.

For example: if you play dotted crotchets over crotchets you get the rhythm 2 over 3, 2 'hits' over your crotchet pulse, or 2/3. This poly-rhythm moves by at a rate of 0.67. That is, one and a half 'hits' for every 1 crotchet.

What I'm interested in is finding different poly-rhythms that are close in speed. Alternating between these gives the sense of speeding up and slowing down ever-so-slightly, without actually playing rubato. They're not easy to do, and even harder to do over a given structure like a tune, but hey, that's what practice is for.

So I finally got around to figuring out the numbers for most of the poly-rhythms I like using, and organised them from fastest (the rate of 'hits' is high) to slowest (the rate of 'hits' is low).

Above is the poly-rhythm, and below is the rate.

Table 1


7/2 5/2 7/3 7/4 5/3 3/2 7/5 4/3 5/4 7/6 1/1-------------------->

3.5 2.5 2.34 1.75 1.67 1.5 1.4 1.34 1.25 1.17 1



5/6 4/5 7/9 3/4 5/7 2/3 3/5 4/7 5/9 4/9 3/7

0.83 0.8 0.78 0.75 0.71 0.67 0.6 0.57 0.56 0.44 0.43


Obviously rhythm is not finite, and there are plenty of other rhythms I could add here to make this movement much smoother. I have just done it with the poly-rhythms I am most familiar with.

So if you're unsure on how to play these. If we have a poly-rhythm of x/y, x represents the divsion of the beat, and y represents the grouping of that division. So 5/7 is quintuplets grouped in lots of 7:

1 2 3 4 5, 1 2 3 4 5, 1 2 3 4 5, 1 2 3 4 5, 1 2 3 4 5, 1 2 3 4 5, 1 2 3 4 5

The highlighted numbers are you 'hits', the numbers 1-5 represent the 5 quintuplets in the pulse, and the number 1 represents the crotchet pulse.

Obviously in my table of speeds there are some rhythms that are so close the movement between them is almost imperceptible. From 4/9 to 3/7 there is only 0.01 difference, for instance.

Even if this seems un-realistic for your level of playing, you could examine the raltionships of speed from keeping your division of the beat the same, but changing the grouping.

Table 2

4/1 4/2 4/3 4/4 4/5 4/6 4/7 4/8 4/9 ------->etc

4 2 1.34 1 0.8 0.67 0.57 0.5 0.44


This pattern is much obviously much easier to play, as you don't have to change the division of the beat (going from septuplets grouped in 4 to quintuplets grouped in 3 is not very easy). Although I haven't worked it out mathematically, it looks like this way you get an exponential curve when graphing the relationship from one poly-rhythm to the next. This would obviously happen no matter what the division. So in 'Table 2', the rate of acceleration is increasing at an even rate, whereas in 'Table 1', I'm trying to get the acceleration to remain constant. The problem wth that is the number of poly-rhythms I'll need to get that to happen approached infinity.....oh well. I'm content now with my first efforts.
Now, to the practice room!