Canons rythmiques mosaïques et conjecture de Fuglede

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

Programme et résumés [PDF]

• 15h00 - 15h30 - Moreno Andreatta & Carlos Agon – The Tiling Canon construction as a "mathemusical" problem : from Minkowski/Hajos to Fuglede Conjecture (pdf)
• 15h30 - 16h15 - Emmanuel Amiot - From Vuza canons and their mathematical models to the Spectral Conjecture (pdf)
• 16h15 - 16h45 - Edouard Gilbert - Polynomial congruence and tiling canons

• 17h00 - 17h30 - Franck Jedrzejewski - Cyclotomic properties of aperiodic Vuza canons (pdf)
• 17h30 - 18h15 - Mihalis Kolountzakis - Tiling by translation: Fourier analysis, number theory and algorithms (pdf)
• Discussion

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.

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.

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.

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.

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.

• H. Minkowski , Geometrie der Zahlen, Leipzig, 1896.
• H. Minkowski, Diophantische Approximationen. Eine Einführung in die Zahlentheorie, Chelsea Publishing Company, New York, 1907.
• G. Hajos, " Über einfache und mehrfache Bedeckung des n-dimensionalen Raumes mit einem Würfelgitter ", Math. Zeit., 47, pp. 427–467, 1942.
• G. Hajos, " Sur le probléme de factorisation des groupes cycliques ", Acta. Math. Acad. Sci. Hung., 1, pp. 189–195, 1950.
• L. Redei, " Ein Beitrag zum Problem der Faktorisation von endlichen Abelschen Gruppen ", Acta Math. Acad. Sci. Hung., 1, 1950, p. 197-207.
• N. G. de Brujin, " On the factorization of finite abelian groups ", Indag. Math. Kon. Ned. Akad. Wetensch. Amsterdam, 15, pp. 258-264, 1953.
• A. Sands, " On the factorization of finite abelian groupes ", Acta Math. Acad. Sci. Hung., 8, p. 65-86, 1957.
• A. Sands, " The Factorisation of Abelian Groups ", Quart. J. Math. Oxford, vol. 2 n° 10, p. 81-91, 1959.
• S. K. Stein, " Algebraic Tiling ", Amer. Math. Month., 81, pp. 445–462, 1974.
• B. Fuglede, " Commuting self-adjoint partial differential operators and a group-theoretical problem ", J. Funct. Analysis, 16, 1974, 101-121
• D. J. Newman, " Tessellation of Integers ", Journal of Number Theory, 9, 1977, p. 107-111.
• D.T. Vuza, " Sur le rythme périodique ", Revue Roumaine de Linguistique-Cahiers de linguistique Théorique et Appliquée 23, n°1, p. 73-103, 1985.
• D. T. Vuza, " Supplementary Sets and Regular Complementary Unending Canons ", en quatre parties, dans Perspectives of New Music, n. 29(2), p. 22-49 ; 30(1), p. 184-207 ; 30(2), p. 102-125 ; 31(1), p 270-305, 1991-1992.
• S. Stein & S. Szabó, Algebra and Tiling, The Carus Mathematical Monographs, n°25, 1994.
• R. Tijedeman, " Decomposition of the Integers as a Direct Sum od two Subsets ", Séminaire de théorie des nombres de Paris, 1995, 261-276
• M. Andreatta, Gruppi di Hajos, Canoni e Composizioni, tesi di laurea, Dipartimento di matematica, Università di Pavia.
• E. M. Coven and A. Meyerovitch, " Tiling the integers with translates of one finite set ", Journal of Algebra, 212(1), p. 161-174, février 1999.
• H. Fripertinger, " Enumeration of non-isomorphic canons ", Tatra Mt. Math. Publ., 23, p. 47-57, 2001.
• T. Johnson, " Tiling the line (pavage de la ligne). Self-Replicating Melodies, Rhythmic Canons, and an Open Problem ", Les Actes des 8e Journées d’Informatique Musicale, Bourges, p. 147-152, 2001.
• A. Tangian, " The Sieve of Eratosthene for Diophantine equations in integer polynomials and Johnson’s problem ", Discussion paper No. 309, FernUniversity of Hagen, 2001.
• T. Noll, M. Andreatta, C. Agon, G. Assayag et D. Vuza, " The Geometrical Groove: rhythmic canons between Theory, Implementation and Musical Experiments ", Actes des Journées d’Informatique Musicale, Bourges, 2001, p. 93-98.
• I. Laba, " The spectral set conjecture and multiplicative properties of roots of polynomials ", Journal of the London Mathematical Society, 65(3), p. 661–671, 2002.
• M. N. Kolountzakis, " The study of translational tiling with Fourier Analysis ", Lectures given at the Workshop on Fourier Analysis and Convexity, Università di Milano-Bicocca, June 11-22, 2001 (version March 2003)
• Andranik Tangian, " Constructing Rhythmic Canons ", Perspectives of New Music, 41(2), 2003.
• M. Andreatta, " On group-theoretical methods applied to music: some compositional and implementational aspects ", Perspectives in Mathematical and Computational Music Theory, ed. G. Mazzola, T. Noll and E. Lluis-Puebla. (Electronic Publishing Osnabrück, Osnabrück), 2004, p. 169-193
• E. Amiot, " À propos des canons rythmiques ", Gazette des mathématiques, 106, Octobre 2005.
• E. Amiot, " Rhythmic canons and galois theory ", In H. Fripertinger and L. Reich (eds.), Proceedings of the Colloquium on Mathematical Music Theory, Grazer Mathematische Berichte, vol. 347, p. 1-25, Graz, Austria, 2005.
• H. Fripertinger, " Remarks on Rhythmical Canons ", In H. Fripertinger and L. Reich (eds.), Proceedings of the Colloquium on Mathematical Music Theory, Grazer Mathematische Berichte, vol. 347, p. 1-25, Graz, Austria, 2005, p. 73-90.
• H. Zuber, Vers une arithmétique des rythmes ?, mémoire de magistère, École normale supérieure de Cachan, 2005
• F. Jedrzejewski, Mathematical Theory of Music, Collection " Musique/Sciences ", Ircam-Delatour France, 2006.
• M. N. Kolountzakis and M. Matolsci, " Complex Hadamard matrices and the spectral set conjecture ", Collectanea Mathematica, Extra, p. 281–291, 2006.
• O. Bodini & E. Rivals, " Tiling an Interval of the Discrete Line ", LNCS, Springer, 2006, p. 117-128.
• M. Andreatta, " De la conjecture de Minkowski aux canons rythmiques mosaïques ", L’Ouvert, n° 114, Mars 2007, p. 51-61.
• E. Gilbert, Polynômes cyclotomiques, canons mosaïques et rythmes k-asymétriques, mémoire de Master ATIAM, mai 2007.

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.