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. Bjorklund's construction for it is Euclid's greatest-common-divisor algorithm in disguise: Godfried Toussaint showed in 2005 that the patterns it produces are the ones world rhythm already uses. (Toussaint swaps the letters, E(k,n), but the onsets still come first: the tresillo is E(3,8) in both.)
These patterns appear independently wherever humans distribute rhythm evenly 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(5,16) is the Brazilian bossa nova. Plenty of other pairs make good rhythms without carrying a name. The presets fill whichever layer you last touched, and ◂ ▸ rotate a layer's start.
Stack multiple layers to create polyrhythmic interference patterns. Full coincidence, all layers landing on beat 1 together, 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(3,7) = [●○●○●○○] ruchenitza · Bulgaria