Tap three even beats with one hand and two with the other over the same span — three-against-two — and a rolling cross-rhythm appears that only lines back up at the very start. It’s the engine of West African drumming, jazz swing, and the ‘pass the god-damn butter’ mnemonic. Slide the two pulse trains and watch them drift and re-lock.
A polyrhythm layers pulse trains whose beat counts share no common factor over one span — the classic 3:2 puts three evenly-spaced hits against two. Over a bar of 6 equal subdivisions the ‘3’ lands on 0,2,4 and the ‘2’ on 0,3, so they COINCIDE only at 0 and the pattern repeats every LCM(3,2) = 6 subdivisions. The perceived complexity is exactly this non-alignment; larger coprime ratios (4:3, 5:4) give longer, richer cycles. A fail-loud self-check throws unless the combined cycle equals LCM(3,2) = 6 and the two trains coincide only at the downbeat. ◆ real rhythm theory, node-verified.
Idealised evenly-spaced pulses (the exact LCM cycle and single coincidence); real performance adds swing and micro-timing that make it groove — the coprime-counts-meet-only-at-the-top structure is exact.