Introduction to the Mathematics of Music

Introduction to the Mathematics of Music is aimed at helping bridge the substantial gap between a classical musician culture and the universe of mathematical notions in music. It explains the necessary notions, starting from scratch, with rigour but without any unnecessary formalism. It was developed from a course given in Perpignan, France, for a bachelor in Music theory. After a mandatory outline of the seminal role of numbers in music, based on the equations of consonance, the book introduces the essential formalisation of pitch-classes and pc-sets as elements and subsets of the integers modulo 12. Transpositions and inversions, traditional musical operations, are formalized in that context. Symmetries and structures are studied efficiently with these tools — for instance linking Olivier Messiaen’s forgotten Modes of Limited Transposition with subgroups of the dihedral group D12. The book ends with a sampling of the geometrical models of musical spaces that mark the modern era in the discipline. A wealth of exercises (and indications of solutions) is provided, since the notions exposed are better assimilated with paper and pencil. This text is primarily intended for serious music students who intend to develop the ability to understand the current research in mathematical music. Some prior knowledge of music theory (non mathematical) is an asset. It is hoped that the style of presentation, together with the numerous exercises, might also be a source of pedagogical inspiration for teachers and students even in so called `pure’ mathematics.