The complement of a Euclidean rhythm is also Euclidean

If E(k,n) is a Euclidean rhythm, then its bitwise complement — swapping every onset for a rest and every rest for an onset — is E(n-k,n), which is also Euclidean (a result of Taslakian and Toussaint). Euclidean rhythms therefore come in complementary pairs. The complement of E(3,8)=[x..x..x.] is E(5,8)=[x.xx.xx.], the Cuban cinquillo: the tresillo and cinquillo fill complementary positions, so layering one on onsets and the other on rests interlocks without collision (except at shared onset 0). This gives a quick way to derive a matching counter-rhythm for any Euclidean pattern.

Examples

Complement of E(3,8) is E(5,8). In Strudel s(“bd(3,8), hh(5,8)”) lands cinquillo hats in the gaps of the tresillo kick.

Assessment

Compute the complement of E(5,16) as an E(k,n) and confirm it is Euclidean. Name the two named Cuban rhythms that are complements of each other.