New Investigations on Rhythmic Oddity

Abstract

The “rhythmic oddity property” (rop) was introduced by ethnomusicologist Simha Aron in the 1990s. The set of rop words is the set of words over the alphabet $\{2,3\}$ satisfying the rhythmic oddity property. It is not a subset of the set of Lyndon words, but is very closed. We show that there is a bijection between some necklaces and rop words. This leads to a formula for counting the rop words of a given length. We also propose a generalization of rop words over a finite alphabet $\mathcal {A} \subset \{1,2,\ldots ,s\}$ for some integer $s \ge 2$. The enumeration of these generalized rop words is still open.