Table des matières

Canons rythmiques mosaïques et conjecture de Fuglede

Samedi 2 février 2008

Ircam, Salle C. Shannon
1, place I. Stravinsky 75004 Paris
Entrée libre dans la mesure des places disponibles

Programme et résumés [PDF]

Programme de la journée


Moreno Andreatta & Carlos Agon - The Tiling Canon construction as a "mathemusical" problem: from Minkowski/Hajos to Fuglede Conjecture

We provide a survey of the Tiling Canon construction which emphasizes its " mathemusical " character. Rhythmic canons having the property to tile the time line are in fact natural musical constructions which can be taken, on one side, as an object study for purely mathematical research and whose mathematical formalizations open, at the same time, new interesting perspectives in the field of computer-aided composition.

Emmanuel Amiot – From Vuza canons and their mathematical models to the Spectral Conjecture (the return !)

In the study of rhythmic canons, after the seminal work of DT Vuza, several mathematical models emerge naturally: direct sum decompositions, DFT of characteristic functions and their zeroes, polynomials and their cyclotomic factors, interval vectors…

Looking for musically relevant features and transformations of rhythmic canons enabled to gain some valuable insight on tough 'pure' mathematical problems, such as the Spectral Conjecture.

Edouard Gilbert - Polynomial congruence and tiling canons

Rhythms can be represented as polynomials with integer coefficients. Considering a rhythm whose period is given, one can be interested in studying the polynomials from Z[X]/(Xn - 1) or F2 [X]/(Xn + 1). Usual transformation can easily be considered in those sets and most of them have some nice properties on the roots of their polynomials. Decomposition of such sets can also be used to start the search of tiles complements.

Franck Jedrzejewski - Cyclotomic properties of aperiodic Vuza canons

The factorization of the finite abelian group Zn into a direct sum of subsets (Zn=R+S) is a model for musical canons.

If Zn is a non-Hajos group, the decomposition uses only aperiodic tiles. This question was solved by Sands after the works of de Bruijn and Rédei.

More recently, Dan Vuza gave an algorithm to construct aperiodic canons. But unfortunately, it is well-known now that some aperiodic canons are not Vuza canons. In this talk, starting from a general aperiodic canon, we use the characteristic polynomial of each tile and their decomposition into cyclotomic polynomials to derive the index of the cyclotomic polynomials for a large set of aperiodic canons included Vuza canons.

Mihalis Kolountzakis - Tiling by translation: Fourier analysis, number theory and algorithms

We are going to present problems and methods that arise in the study of translational tiling by a single tile. We will show the connection to Fourier Analysis, where a tiling is studied in Fourier space by looking at the zeros of the Fourier Transform of the tile and the support of the Fourier Transform of the translation set. We will also see the specific case when the group (which is being tiled by a subset of it) is the cyclic group. In this case the cyclotomic polynomials play a major role, which we hope to exhibit. A specific kind of tiling problem is that of spectrality, and the so-called Fuglede conjecture will be explained in this context. The emphasis of this talk will be mostly on the algorithmic side of these questions, with the predominant question being to decide if a given finite set of integers is a tile or not. This question has not yet found a satisfying answer in terms of computational complexity. We will also mention analogous problems in dimension 2, where the computational questions are much more basic, as not even decidability has been proved yet.

Quelques repères bibliographiques

Pour plus d'information sur des problèmes de pavage en musique et quelques conjectures ouvertes en mathématiques, voir:
Mosaïques et pavages en théorie et composition musicales.