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

What are the non cyclotomic factors doing ?
How are the >neutral< cyclotomic polynomials (not in $(T 1)$ or $(T 2)$ ) dispatched between $A$ or $B$ , which is probably linked to the major question:
How does one ensure that a product of such polynomials has only 0 or 1 as coefficients ?

Three years later, Granville, Laba and Wang partially managed to tackle the case of 3 prime factors (Granville et al., 2001):

Theorem 13 If $A o+ B = Z/nZ$ , $|A|= paqbrg$ , $|B |= pqr$ ; if $Pp,Pq,Pr$ all are factors of $A(x)$ , then so are $Ppq,Prq,Prp$ .

The proof takes 15 pages of heavy calculations. The number 15 plays a lighter part in the last section of this paper.

4 Polyrhythmic Canons and Future Results

Though many questions remain open concerning canons with ONE motif, still less is known when several motives (tiles) are allowed, even with just a motif and its reverse. But perhaps this greater complexity paves the way for new tools and deeper results on old questions.

4.1 Johnsons’s Question and the Number 15

About one year ago in Royan at the JIM (Johnson, 2001), Johnson began to try tiling with a motif : ${0,1,4}$ , and its augmentations. To continue working with integers, he selected augmentations by 2: $2.{0,1,4}= {0,2,8}$ , $4.{0,1,4}= {0,4,16}$ and so on.

Setting $J(x) = 1+ x + x4$ , the augmentations read $J(x2) = 1+ x2 + x8$ , $J(x4) = 1 + x4 + x16...$ , and the problem is to find 0-1 polynomials satisfying

A(x).J(x)+ B(x) .J(x2) [+C(x) .J(x4)+ ...] = Dn(x)

(1)

It is easy enough to find by hand the smallest tiling, of period 15:

2 8 10 2 2 (1+ x + x + x ).J(x) + x .J(x ) = D15(x)

Shortly after Johnson stated his problem, Andranik Tangian came up with a Fortran program and a list of solutions up to a given size (Tangian, 2001).

There was one salient fact: all periods were multiples of 15, 1 for length 15, 6 for length 30 (beginning with $J$ ), etc. Tom asked if this was general and whether could be proved.