# On two open mathematical problems in music theory: Fuglede spectral conjecture and discrete phase retrieval

(Seminar, TU Dresden, 29 Novembre 2012)

**Abstract**

After briefly presenting the underlying philosophy of “mathemusical” research which we carry on at IRCAM/Paris (from music to mathematics to computer-aided music theory, analysis and composition), and providing a short list of theoretical problems on which we have been working in the last years (see [1] for a detailed presentation), we focus on two open mathematical problems: Fuglede (or Spectral) Conjecture and the Discrete Phase Retrieval problem.

The first conjecture, originally appeared in [2], states the equivalence between spectral property of a domain of the n-dimensional Euclidean space and its tiling character. The conjecture, which is false for n≥3, is still open for n=1 and n=2. We will discuss the case n=1 by showing its deep relationships with a musical compositional process – the rhythmic tiling canons construction – via a much older geometric and number-theoretical conjecture, i.e. Minkowski’s Conjecture. This conjecture, raised by Minkowski in [3] and solved by Hajós almost forty years later [4], states that every lattice tiling of the n-dimensional Euclidean space by unit hypercubes contains two cubes that meet in an (n-1)-dimensional face. (See [5] and [6] for some recent perspectives on this first mathematical conjecture and its music-theoretical ramifications).

The second mathematical problem deals with the possibility of reconstruct a set by knowing its inter-point distances. It shows the deep connections between the notion of Z-relation in music theory – i.e. the property of two subsets of a cyclic group of having the same interval content, as originally introduced by Hanson in [7] and successively formalized by Forte [8] and Lewin [9] – and the theory of homometric sets in crystallography [10]. We will describe some aspects of phase-retrieval approaches in music by focusing on the particular case of the cyclic groups (beltway problem) and discussing the extended phase retrieval for a generalized musical Z-relation [11]. Some musical examples will finally show the relationships between these two open mathematical problems via OpenMusic, a Visual Progamming Language for computer-aided music theory analysis and composition currently developed by IRCAM Music Representation Team [12].

**References:**

- M. Andreatta, Mathematica est exercitium musicae: la recherche "mathémusicale" et ses interactions avec les autres disciplines, Habilitation thesis, IRMA (Institut de Recherche Mathématique Avancée, University of Strasbourg), 2010 (pdf)
- B. Fuglede, "Commuting Self-Adjoint Partial Differential Operators and a Group Theoretic Problem." J. Func. Anal. 16, 101-121, 1974
- H. Minkowski, Minkowski, Diophantische Approximationen. Leipzig, Teubner, 1907.
- G. Hajós, G. "Über einfache und mehrfache Bedeckung des n-dimensionalen Raumes mit einen Würfelgitter." Math. Z. 47, 427-467, 1942.
- M. Andreatta, C. Agon (eds), Tiling Problems in Music, Special Issue of the Journal of Mathematics and Music, July 2009, 3(2), 63-70 (Guest Editor’s Foreword)
- M. Andreatta, “Constructing and Formalizing Tiling Rhythmic Canons : A Historical Survey of a ‘Mathemusical’ Problem,” Perspectives of New Music, Special Issue, 49(1-2), 2011 (pdf)
- H. Hanson,
*Harmonic Materials of Modern Musi*c, Appleton-Century-Crofts, New York, 1960. - A. Forte,
*The Structure of Atonal Music*, Yale University Press, New Haven, 1973. - D. Lewin,
*Generalized Musical Intervals and Transformations*, Yale University Press, New Haven, 1987 (2nd ed., Oxford University Press, Oxford, 2007). - J. Mandereau, D. Ghisi, E. Amiot, M. Andreatta, C. Agon, “Z-relation and homometry in musical distributions,” Journal of Mathematics and Music, 5(2), 83-98, 2011 (pdf)
- J. Mandereau, D. Ghisi, E. Amiot, M. Andreatta, C. Agon, “Discrete Phase Retrieval in Musical Structures,” Journal of Mathematics and Music, 5(2), 99-116 (pdf)
- OpenMusic : http://repmus.ircam.fr/openmusic/home

pdf version

**Contact**

Moreno ANDREATTA

CNRS Researcher

Music Representation Team

Ircam - CNRS UMR 9912 (STMS)

1, place I. Stravinsky

F-75004 Paris

tel:+33 (0)1 44781649

e-mail : Moreno.Andreatta[at]ircam.fr

(back to the Formal Concept Analysis applied to Music webpage)