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

h(32222) = 11 h(322222) = 13 h(22223) = 11 h(222223) = 13 ... ...

Proposition 1 If $w$ satisfies the rhythmic oddity property, then at least one of the following conditions is satisfied : (i) there exists a unique pair $(u,v)$ with $h(v) = h(u)+ 2$ such that $w = uv$

(ii) there exists a unique pair $(u,v)$ with $h(v) = h(u)+ 2$ such that $w = vu$

Proof: The uniqueness of the factorizations of conditions (i) and (ii) is trivial. To prove the existence of such a factorization, let $u$ be the longest prefix of $w$ such that $h(u) < h(v)$ , where $v$ is the corresponding suffix with $w = uv$ . We denote by $x$ the first symbol of $' v = xv$ . The two possible values for $x$ are $2$ and $3$ . One has $' h(u) < h(v )+ x$ . Since $u$ is maximal, one has $' h(u)+ x >= h(v )$ , but the rhythmic oddity property implies $' h(u)+ x > h(v)$ , thus $' |h(u)h(v )|< x$ .

If $x = 2$ , $h(w)$ being even implies that $|h(u) - h(v')|$ is even, thus equal to zero, and $h(v) = h(u)+ 2$ so that condition (i) is satisfied by the pair $(u,v)$ and condition (ii) by the pair $(v',2u)$ .
If , being even implies that is odd, thus equal to one. This gives two remaining cases.
- If $' h(v) = h(u)- 1$ , the equality $' h(v) = h(v )+ 3$ implies that $h(v) = h(u)+ 2$ so that condition (i) of the property is satisfied by the pair $(u,v)$ .
- If $h(v') = h(u)+ 1$ , the factorization $w = (u3)v'$ is such that $h(u3) = h(v')+ 2$ , so that condition (ii) of the property is satisfied by the pair $(v',u3)$ . $[]$

Proposition 1 implies that $2h(u) +2 = h(w) = n$ (with $n$ being even) so that $h(u) = n/2 - 1$ and $h(v) = n/2 + 1$ , thus expressing the »half minus one / half plus one« characterization of these patterns given in Arom (1991).

We introduce the notion of asymmetric pair that will be the key of our construction. Let $u$ , $v$ be words over $A$ . We say that $(u,v)$ is an asymmetric pairs if no pair of prefixes $' ' (u ,v)$ of $u$ and $v$ respectively exist such that $' ' h(v ) = h(u) +1$ . For instance, $(3322,32322)$ is an asymmetric pair, but $(3322,32232)$ is not.

Proposition 2 A word $w$ satisfies the rhythmic oddity property if and only if there exists an asymmetric pair $(u,v)$ such that $w = uv$ or $w = vu$ with $h(v) = h(u)+ 2$ .

Proof: Let $w = uv$ or $w = vu$ with $h(v) = h(u)+ 2$ . The existence of a pair of prefixes $(u',v')$ of $u$ and $v$ such that $u = u'u''$ and $v = v'v''$ with $h(v') = h(u') +1$ is equivalent to the existence of a cyclic shift $v''u'u''v'$ of $w$ (whatever being the