Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

The asymmetry of the pattern is to some extent intrinsic, in the sense that there exists no breaking point giving two parts of equal length. Every division of the pattern gives two unequal parts, »half minus one« on the one side, and »half plus one« on the other side. Note that the oddity property requires that the circle is divided into an even number of units, so that it is possible to find patterns of the »half minus one / half plus one« type. Many such patterns appear in Central African music, and this makes a challenging question of cognitive science, but we do not address this question here, since this paper is restricted to purely combinatorial aspects.

We describe an algorithm for enumerating all the patterns satisfying the rhythmic oddity property. The main idea of the construction is that patterns of this type must have an even number of three-unit elements, and that these elements must be placed nearly opposite around the circle. More precisely, if the units on the circle corresponding to a rhythmyc pattern of length $n$ with the rhythmic oddity property are numbered from 0 to $n- 1$ , and one three-unit element starts at $i$ then the other three-unit element starts either at $i+ n/2 - 1$ or $i+ n/2+ 1$ (modulo $n$ ). The construction is expressed in the paradigm of combinatorics on words, and we recall some basic notions from this domain.

2 The >>Rhythmic Oddity Property<<

A word is a sequence of symbols from a given alphabet. In this paper, we consider words over the alphabet $A = {2,3}$ . We denote as usual by $* A$ the set of all words over $A$ , and by $e$ the empty word. For a word $w$ , we denote by $|w |$ the length of $w$ , and by $|w |x$ the number of symbols equal to $x$ in $w$ . The concatenation of words is an associative operation and the empty word is the neutral element for concatenation.

A word $u$ is called a prefix (resp. suffix) of a word $w$ if there exists a word $y$ such that $w = uy$ (resp. $w = yu$ ). In order to introduce the cyclic shifts of a word, let $d$ be the permutation of A* defined by

d(e) = e d(au) = ua

with $a (- A$ and $u (- A*$ .

The cyclic shifts of $w$ are the words of the form $dk(w)$ for any integer $k$ . For instance, the cyclic shifts of $2223$ are $2223$ , $2232$ , $2322$ , and $3222$ . The height of a word $u$ , denoted by $h(u)$ , is the sum of its symbols, and $h$ is a morphism from $A*$ to $N$ . A word $w$ satisfies the rhythmic oddity property if and only if (i) $h(w)$ is even, and
(ii) no cyclic shift of $w$ can be factorized into words $uv$ such that $h(u) = h(v)$ .

Example 1 The condition $h(w)$ even is added to this definition, because if $h(w)$ is odd, the second condition is obviously satisfied for any word. In the Aka Pygmies example, one has $h(w) = 24$ , and the factorization of cyclic shifts gives