A rhythm necklace is an equivalence class of cyclic rhythms that disregards the starting point

Two rhythms belong to the same necklace if one is a rotation of the other around the cycle. The necklace abstracts away the performer’s choice of where in the cycle to begin, so a single underlying pattern can surface under many names in many cultures. Inter-onset duration vectors — e.g. (332) for the tresillo — give a compact, rotation-invariant representation of a necklace. In live coding and sequencing this matters because rotating a pattern changes its musical feel (which onset lands on the downbeat) while preserving the intervallic structure. A common confusion is to treat a rotation as a different rhythm: E(3,8)=[x..x..x.] and [..x..x.x] are the same necklace, not two rhythms.

Examples

E(3,8) = (332) necklace surfaces as tresillo (Cuba), the Kinka timeline (Togo), and the dochmiac (Ancient Greece). Starting E(3,8) on its second onset gives the 13th-century Indian Mathya-Tisra tala.

Assessment

Given two onset patterns, determine whether they belong to the same necklace by checking for a rotation. Give the (332) necklace’s inter-onset vector and one named world-music instance of a rotation of it.