I had a thought today about polyrhythms. First of all, I think of a polyrhythm as an embellishing rhythm over a steady pulse. When playing a 3/4 polyrythm (my terminology for playing 3 beats from one source and 4 beats from another in the same amount of time), I start with a steady beat, like the tempo of a song, and I play 3 hits over 4 of those steady beats.
My idea was to superimpose polyrhythms. What if I played 3 hits over 4 steady beats, then 3 hits from a new source over 4 of the first hits? Things get wacky. I haven't worked it out on paper yet, but it'll be messy.
If I remember my math terms correctly, any polyrhythm must be subdivided into the lowest common multiple of the number of beats from the different sources. A 3/4 polyrhythm must be subdivided into 12 beats: one source plays every third while the other plays every fourth.
If I try to superimpose a polyrhythm in the manner suggested above, it would simplify to 9 beats over 12 beats over 16 beats in a complete loop. That is, one hand plays 9 beats while the other plays 12 beats while a foot taps 16 beats, all in the same amount of time. There's a 3/4 ratio between the hands and between the second hand and the foot. Now the whole thing needs to be subdivided into the lowest common multiple, 144. What a pain in the ass. I'll work on this tonight.