So this is going to veer slightly off the prime topic, but I would argue that these are not the aspects of “non-repetitious” rhythms that are musically interesting. Breaking down a polymetric relationship with data points like these focus your attention on the numbers and not how the numbers affect a listener. Listening to a large number of short polymeteric phrases like the example you listed will feel repetitious because the cycles are so small, and while the listener will be able to feel different sets of 2-3 cycles against each other, they are still listening to the same short phrases over and over again. In my experience the key for polymeters to be really interesting is, & here’s the dirty word of the topic: subdivisions. Blasphemy, I know, but prime numbers are absolutely still applicable.
Take, for example, an instrument playing in 11. This could be subdivided a million ways, but say you break it up into a 6->5 feel:
| x _ _ x _ _ | x_ x _ _ |
Now take an instrument playing in 17. There’s a myriad of ways this can be broken up as well, but if you include your 11 pattern from above somewhere in that stretch, you’ll get a shifting relationship of the same pattern phasing against itself. & rather than subdivide the remaining 6 beats in two equal groups of 3, arrange them in an uneven way that will allow more space between the phasing.
| x _ _ _ | x _ _ x _ _ | x _ x _ _ | _ x |
The rest on the first beat of that last group of two can also psychologically blur the fact that you’ve repeated the 11 pattern within the 17 pattern. Now add a single beat phrase of 5, but offset to not make the start point so explicit:
| _ _ x_ _ |
This might only take the three instruments 935 beats to come back full circle, but now you have an interplay based on a common phrase and a straight prime cycle to keep interest beyond the pure “non-repetition”. This interplay, in my opinion, is what’s truly interesting about polymeters within or outside the realm of prime number.
