4.3 Prospects

Many things should be attempted, and several might be affordable:

• Counting solutions of the different problems above.
• Exploring this new concept of polyrhythmic canons: this would be more difficult, but new ideas will arise, like Galois theory, which might help with the old questions (such as » is necessary and sufficient ?«). Already H. Fripertinger (Fripertinger, 2001) made some interesting remarks on generalizing the techniques used on the above problem.

  Specifically, rhythmic augmentations in terms of polynomials are obtained by a Frobenius automorphism, which means that Galois theory on finite fields is relevant.

• Another kind of polyrhythmic canon uses just a tile and its reverse. When is it possible to tile with these two ?
• It would be useful to have a database of all canons (of given order), such as Sloane’s famous encyclopedia of integer sequences online. For instance, it would help
• find all Vuza canons of given order (meaning aperiodic canons, not only the ones provided by his algorithm).
• Also it is necessary to develop a set of computer tools for all basic transforms on rhythmic canons: reductions to canonical (!) forms, expansion, reduction, affine transforms...This is currently under development and will eventually find its natural abode in environments like OpenMusic or Rubato.

I can only hope it will be noticed how a musician’s look on these problems often helps understand better the mathematics involved. Conversely, a musician’s questions to the mathematician are so unexpected that they seldom fail to suggest solutions, if only to other problems.

The musical culture might help solve difficult questions; for instance, no really good upper bound for the period of a canon is known from the width of the motif (). As noticed by (Coven and Meyerowitz, 1999), the pigeonhole principle used in demonstration of Theorem 3 yields an upper bound of about , though musical experience would suggest that , but this is as yet a conjecture.

In an other direction, perhaps musical intuition might help solve difficult problems, like the spectral conjecture (at least in dimension 1). To me it looks like musical considerations provide new ideas, concepts, and insights for non trivial mathematical problems.

I will thank again Moreno Andreatta, Tom Johnson, Harald Fripertinger, Andranik Tangian for prompting my interest in these fascinating matters, Thomas Noll for useful Mathematica notebooks, the reviewers for very helpful advice, and last but not least, Guerino Mazzola for kindly inviting me to Third International MaMuTh-Seminar.