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

3.1.2 Canonic and Polycanonic Rhythms

One main open problem (for musicians) is : can I make a rhythmic canon with this given (finite) tile (i.e. inner rhythm) ?

Up to a paper by (Coven and Meyerowitz, 1999) it was still unknown what one could get, even with as simple as a 6 notes motif. Only the case $|A |= pa$ was known (Newman, 1977). I could not find Newman’s paper, but with the ideas below I could get an idea of the results: something like the following

Theorem 9 If $A$ tiles and $|A |= pa$ , where $p$ is a prime and $a (- N*$ , then for adequate $b,k (- N*$

b A(x) =_ kDpb(x) (mod xp - 1)

i.e. there is equirepartition modulo $b p$ .

Equirepartition modulo the size of $|A |$ is not mandatory, as $A$ may have factors $Ppb$ with $b /= a$ as seen in this example with overall period $n = 8$ :

A = {0,1,4,5} |A|= 4 B = {0,2} A(x) =_ 2(1+ x) = 2D2(x) (mod x2- 1)

Some progress has been made on the case $a b n = p .q$ and partial progress for $a b g n = p .q.r$ . The following criteria make heavy use of cyclotomic factors of the polynomials associated with the tiling, essentially showing that there are mostly cyclotomic polynomials as factors of $A$ and $B$ , with a rigid arrangement.

3.1.3 Condition $(T 1)$

Before stating the condition we explain what it is about. Say $A$ is tiling $Z$ . The period of the tiling must be a multiple of $|A |$ (as $n = |A| .|B |$ ). Moreover, any cyclotomic $Pd$ dividing $A(x)$ will divide $A(x).B(x)$ and $xn- 1$ , hence $Pd(x)|A(x)$ implies $d |n$ .

A number of cyclotomic polynomials may divide $A(x)$ (for all $1 /= d|n$ in the trivial case $A = {0,1,2,...,n- 1}$ ). Now by Lemma 2, only the few of these for which $d$ is a prime power will contribute to the value of $A(1)$ (remember this is the cardinality of the set $A$ ). But according to Lemma 3, all the prime powers $pa |n$ will be there and every one will contribute for the value $Ppa(1) = p$ . Thus $p$ occurs at least $m(p)$ times (multiplicity of $p$ in $n$ , and also the number of distinct powers of $p$ dividing $n$ ) in $A(1)$ or $B(1)$ . These being integers, with $prod A(1).B(1) = n = pm(p) p|n$ , it means that all non-cyclotomic factors contribute nought to the values of $A(1)$ or $B(1)$ ! We have just proved Theorem $(B1)$ of (Coven and Meyerowitz, 1999) (this is true for $A$ and $B$ ) :