Mathematics and group theory in music

Athanase Papadopoulos (1)

1 IRMA - Institut de Recherche Mathématique Avancée

The purpose of this paper is to show through particular examples how group theory is used in music. The examples are chosen from the theoretical work and from the compositions of Olivier Messiaen (1908-1992), one of the most influential twentieth century composers and pedagogues. Messiaen consciously used mathematical concepts derived from symmetry and groups, in his teaching and in his compositions. Before dwelling on this, I will give a quick overview of the relation between mathematics and music. This will put the discussion on symmetry and group theory in music in a broader context and it will provide the reader of this handbook some background and some motivation for the subject. The relation between mathematics and music, during more than two millennia, was lively, widespread, and extremely enriching for both domains.

Mots clés: Olivier Messiaen; Group theory; mathematics and music; Greek music; non-retrogradable rhythm; symmetrical permutation; mode of limited transposition; Pythagoras; Olivier Messiaen.

Domaines: Histoire et perspectives sur les mathématiques [math.HO]

Athanase Papadopoulos. Mathematics and group theory in music. Handbook of Group Actions, Vol. II (ed. L. Ji, A. Papadopoulos and S.-T. Yau), 32, International Press and Higher Education Press, p. 525-572., 2015, Advanced Lectures in Mathematics, 978-7-04-041363-2. ⟨hal-01027418⟩