A Euclidean string is an interval vector that becomes a rotation of itself when its first element is incremented and its last decremented

Ellis et al. define a string P=(p0,…,p_{n-1}) of non-negative integers as a Euclidean string if incrementing p0 by 1 and decrementing p_{n-1} by 1 yields a rotation of P. Representing a rhythm by its inter-onset duration vector, many Euclidean rhythms turn out to be Euclidean strings — e.g. E(4,9)=(2223), where the operation gives (3222), a rotation. This provides an algebraic characterization paralleling the geometric one (maximal evenness). Not all Euclidean rhythms are Euclidean strings: some are reverse Euclidean strings (E(3,8)=(332)), and E(5,8)=(21212) is neither, yet all are maximally even. Toussaint notes a tantalizing correlation: Euclidean rhythms favoured in classical music and jazz tend to be Euclidean strings, while those characteristic of African music tend not to be.

Examples

E(4,9)=(2223): incrementing first / decrementing last gives (3222) = rotation, so it is a Euclidean string. E(5,8)=(21212) is neither a Euclidean nor a reverse Euclidean string.

Assessment

Take E(3,5)=(221): apply the increment-first/decrement-last operation and decide whether it is a Euclidean string, a reverse Euclidean string, or neither.