Euclidean rhythms spread k onsets as evenly as possible over n steps

A Euclidean rhythm takes a pattern length (e.g. 16 steps) and a beat count (e.g. 6) and distributes the beats as evenly as possible using an algorithm derived from Euclid’s Elements (~300 BC), promoted in music by Godfried Toussaint. The resulting patterns closely match world-music rhythms, especially Sub-Saharan African traditions, and especially once rotated in time to move where the downbeat falls. E(6,16) — equivalently notated (3,8) — yields 1001001010010010, the Cuban Tresillo, ‘the most fundamental and most prevalent duple-pulse rhythmic cell in Sub-Saharan African music traditions’. Because the distribution is trivial to compute, it embeds cheaply in hardware, which is why it is ubiquitous across modular gear. A common misreading is that Euclidean means random-but-nice; it is fully deterministic — a given (k,n) always gives the same pattern up to rotation.

Examples

E(6,16) = 1001001010010010 — the Cuban Tresillo/Habanera. Rotating the pattern (offsetting its start point) keeps the onset spacing but changes which step reads as the downbeat, changing the groove without changing k or n.

Assessment

Given E(6,16), write the binary pattern, name the traditional rhythm it matches, then describe what rotating the pattern does to the perceived groove and what it does not change.