distribute n pulses as evenly as possible across k equal steps —
the same algorithm euclid used to find greatest common divisors
The Euclidean rhythm E(n,k) distributes n onsets as evenly as possible across k equal time slots. The algorithm is identical to the one Euclid described for finding greatest common divisors — it was discovered to underlie world rhythm by Godfried Toussaint in 2005.
These patterns appear independently wherever humans distribute rhythm optimally over a cycle. E(3,8) is the tresillo of Cuban and Brazilian music. E(5,8) is the cinquillo. E(3,7) is the Bulgarian ruchenitza. E(4,9) is the Turkish aksak. E(7,8) is the Bulgarian horo.
Stack multiple layers to create polyrhythmic interference patterns. Full coincidence — all layers landing on beat 1 simultaneously — recurs every LCM(k₁, k₂, …) steps. Try E(3,8) + E(5,8) + E(7,12) and listen for when the grid snaps back.
E(3,8) = [●○○●○○●○] tresillo · Cuba, Brazil, West Africa E(5,8) = [●○●●○●●○] cinquillo · Cuba E(7,8) = [●○●●●●●●] horo · Bulgaria