Computation of words satisfying the
Rhythmic Oddity Property

Reprinted from Information Processing Letter, vol. 86, no. 5, Marc Chemillier, Charlotte Truchet, Computation of words satisfying the ”rhythmic oddity property” (after Simha Arom’s works), pp. 255-261, 2003, with permission from Elsevier.

1 Introduction

In 1952, ethnomusicologist Constantin Brailoiu wrote a paper on the combinatorics of asymmetric rhythmic patterns entitled »Le rythme aksak« (Brailoiu, 1952). These patterns are combinations of durations equal to two or three units, such as the famous Turkish rhythm (see Lothaire, 1983). The asymmetry lies in the fact that they are based on two different durations. In his paper, Brailoiu gave a table of 1884 distinct rhythms, enumerating all the combinations that can be made with up to nine successive two- or three-unit elements. The present paper is devoted to the enumeration of a particular type of African asymmetric patterns, satisfying the »rhythmic oddity property«, defined in Arom (1991). In Aka Pygmies music one can find the following rhythmic pattern

The rhythmic oddity property asserts that when placing onsets corresponding to the two- and three-unit elements of the sequence cyclically on a circle (thus expressing the fact that the pattern is played as a loop), one cannot break the circle into two parts of equal length whatever the chosen breaking point. There is always one unit lacking on one side.