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

Proposition 4 Let $(u,v) = a(e,e)$ for a word $* a (- B$ If $|a|b$ is even, then $h(v) = h(u)$ , and if $|a| b$ is odd, then $h(v) = h(u)+ 2$ .

Proposition 5 If $|a| b$ is even, then $a(r,s) = a(e,e).(r,s)$ and if $|a| b$ is odd, then $a(r,s) = a(e,e).(s,r)$ for all $(r,s) (- A* × A*$ .

Corollary 1 A word $w$ satisfies the rhythmic oddity property if and only if there exists a word $a (- B*$ with $|a| b$ being odd, such that $w = uv$ or $w = vu$ with $(u,v) = a(e,e)$ .

Let $D$ be the subset of $* A$ defined by

* {w = uv, E a (- B ,| a |bodd,(u,v) = a(e,e)}

The set $D$ does not include every words satisfying the rhythmic oddity property, but only part of them. Indeed, as expressed by corollary 6, a word $w$ satisfies the rhythmic oddity property if and only if $w$ belongs to $D$ , or $w$ is a cyclic shift of a word that belongs to $D$ ( $w = vu$ with $uv (- D$ )

For any $w (- D$ , we put $f(w) = a$ , which is possible since $a$ is unique. Indeed, as expressed by proposition 1, a word $w$ cannot have more than one factorization $w = uv$ with $h(v) = h(u)+ 2$ . Morover, equality $(u,v) = a(e,e) = b(e,e)$ for $* a,b (- B$ implies that $a = b$ thanks to the fact that

a(A*× A*) /~\ (A* × A*) = (3A* × 3A*) /~\ (A*× 2A*) = Ø

For any $w (- D$ , we put $f(w) = a$ , which is possible since $a$ is unique. Indeed, following proposition 1, a word $w$ cannot have more than one factorization $w = uv$ with $h(v) = h(u)+ 2$ . Moreover, the image $f(D)$ is the subset of words of $B*$ having an odd number of symbols equal to $b$ , and $f$ is injective, so that $f$ is a bijection from $D$ to $f(D)$ .

Example 2 The computation of the image $f(w)$ of $w = 332232322$ proceeds as follows. First, the factorization of $w$ into $uv$ such that $h(v) = h(u) +2$ gives $u = 3322$ and $v = 32322$ . Then, one has

(u,v) = (3322,32322) = a(322,2322) = ab(322,322) = aba(22,22) = abab(2,22) = ababb(2,2) = ababbb(e,2) = ababbbb(e,e),

which gives $f(332232322) = ababbbb$ .

Proposition 6 For any $w,w' (- D$ , $f(w')$ is a cyclic shift of $f(w)$ if and only if $w'$ is a cyclic shift of w.