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

Proposition 7 Let $f$ be a mapping from $E$ to $F$ , $=_ E$ and $=_ F$ equivalence relations on $E$ and $F$ respectively, and $D$ a subset of $E$ . If for any $x,y (- D$ , $f(x) =_ F f(y)$ is equivalent to $x =_ E y$ , there is a bijection between $D/ =_ E$ and $f(D)/ =_ F$ .

Proof: We define $f '$ from $D/ =_ E$ to $f(D)/ =_ F$ by $f '(cx) = cf(x)$ where $cx$ is the equivalence class of $x$ . It is possible since for any $y (- c /~\ D x$ , one has $c = c f(y) f(x)$ because $x =_ y E$ implies $f (x) =_ f(y) F$ . Then $f'(cx) = f'(cy)$ means $f(x) =_ f(y) F$ , which implies by hypothesis $x =_ y E$ , whence $c = c x y$ . Thus $f'$ is injective. $[]$

Considering the function $f$ from $A*$ to $B*$ , proposition 8 may be applied to the set $D$ of words $w = uv$ satisfying the rhythmic oddity property with $h(v) = h(u)+ 2$ . It proves that there exists a bijection between a cross-section of D and a cross-section of $f(D)$ for the conjugacy relation. Since $f(D)$ is the set of words of $B*$ having an odd number of symbols equal to $b$ , it is easy to compute a cross-section of $f (D)$ using Lyndon words. Thus one obtains a cross-section of $D$ .

Furthermore, every conjugacy classes of words satisfying the rhythmic oddity property contains at least one element of $D$ . Indeed, these words are either of the form $w = uv$ where $h(v) = h(u) +2$ thus being in $D$ , or $w = vu$ with the same condition thus being a cyclic shift of an element of $D$ . Finally, the computation of words satisfying the rhythmic oddity property is reduced to the computation of Lyndon words of $* B$ having an odd number of symbols equal to $b$ .

4 Counting the solutions

Let $n2$ and $n3$ be the number of two- and three-unit elements of a rhythmic pattern with the rhythmic oddity property. One has the following table, with $n2$ on the horizontal axis, and $n3$ on the vertical one. These values were obtained experimentally by a constraint-based program Chemillier and Truchet (2001).


	1	3	5	7	9	11	13	15	17

2	1	1	1	1	1	1	1	1	1
4	1	2	3	4	5	6	7	8	9
6	1	4	7	12	19	26	35	47	57
8	1	5	14	30	55	91	140	204	285
10	1	7	26	66	143	273	476	776	1197
12	1	10	42	132	335	728	1428	2586	4389

Let $X(p)$ be the number of words $w$ satisfying the rhythmic oddity property up to a cyclic shift, where $p$ is the length of $f(w)$ . Since $n3$ is even, we put $n3 = 2j$ and $p = n2 + j$ , where $j$ is the number of letters equal to $a$ in $f(w)$ .

Proposition 8 If $n3 = 2j$ is a power of $2$ , one has

X(p) = 1/pCjp

Proof: The computation of a solution consists in placing $j$ symbols equal to $a$ in a word of length p over the alphabet ${a,b}$ , and removing solutions that are