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 Transpositionwith 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.

Preface

The importance of the relationship between mathematics and music is both undisputed and very poorly understood. More than two millennia old, it goes without saying; but only a handful of specialists are aware of its depth, to the point that it is often disparaged, either by musicians who believe that a few arithmetic relationships cannot encapsulate all the ineffable richness of an art form of divine origin, or by so-called serious mathematicians, who believe that the interaction stops at “elementary” mathematics, such as Diophantine approximation1. But these two extreme categories are guilty of ignorance: the scientific discipline of musical mathematics, sometimes called ‘mathemusic’, has never ceased to develop, and does so with the help of the most up-to-date tools of contemporary mathematics – algebraic topology, category theory, word theory, graph theory, automata theory, differential geometry, Galois theory, etc. – and hand to hand with the most innovative music, to the point that sometimes certain difficult mathematical problems have been solved through musical reinterpretation. As Leibniz famously declared:

Musica est exercitium arithmeticae, occultum nescientis se numerare animi2, it is important to note that the word arithmetica should be taken in a broader sense than the modern meaning of “science of integers”: certainly, Pythagoras and his school advocated that “Everything is numbers” (integers, or at least rational numbers), but already in the 17thcentury, mathematics was exploring new fields such as analysis, integral and differential calculus, Cartesian coordinates, approximation, and so on, thanks in particular to the boldness of Leibniz. Moreover, the second part of the famous phrase (often omitted) suggests that even in the use of mathematics in music, there remains a significant element of mystery – today we would evoke the Unconscious – which will not be disputed by all those who have attempted to elucidate the obscure mechanisms of mathematical creation/discovery, or musical composition.

Undoubtedly, after the golden ages of Greek antiquity, the Middle Ages, when music was one of the four major arts of the Quadrivium3, and the 18th century, when Leonhard Euler invented graph theory at the same time as he developed a theory of consonance, there was a period of estrangement. In the 19th century, the scientific aspect of music shifted towards physics, with the analysis of the nature of sound (Fourier series, Helmholtz resonators); romantic and scientific ideologies contributed to this divergence. Nevertheless, as always, there was progress, but it was more isolated (von¨Ottingen and Hugo Riemann, the musicologist, reviving Euler’s Tonnetz at the end of the 19th century; Krenek modelling the twelve notes with a cyclic group, and Joseph Schillinger’s complex system, which influenced a generation of famous composers in the USA), until a revival after the Second World War, with American Set Theory (Allen Forte, Milton Babbitt, David Lewin, John Rahn, and others) dissecting note collections down to their bare bones, and in Europe, in addition to efforts parallel to American theories, the revolutionary creations of Xenakis, who invested in his compositions the fairly advanced mathematical knowledge he had acquired in his architectural studies, up to the most general models of Guerino Mazzola and his school, imposing the use of category theory sometimes for philosophical rather than practical reasons. Despite ongoing resistance, the Society for Mathematics and Computation in Music was founded in Berlin in 20074 and brings together researchers from around the world every two years. The society’s journal, the Journal of Mathematics and Music5, publishes articles that demonstrate the richness and sophistication of the discipline, but papers can increasingly be found in more traditional publications (Revue d’Analyse musicale, now defunct, Journal of Music Theory, Music Theory Online, Perspectives of New Music) and more recent ones (MusMat, in Brazil). Finally, a growing number of doctoral theses are being defended under the label of “Mathematics and Music” or somesuch, even though the discipline is not yet officially listed in the administrative categories of French universities.

The aim of this book is not to explore the most recent and dizzying developments, which are obviously sometimes inaccessible to novices, but rather to present the most accessible concepts, which form the common foundation for specialists but are rarely compiled in English7. This content has been developed and tested for a module in the musicology degree curriculum at the University of Perpignan, whose students are not required to have any mathematical background. The aim is to provide a minimum level of knowledge that will enable students to grasp what is currently being done in the field, and even to tackle research articles or contemporary creations with the comfort of knowing the required concepts, which are often taken for granted: a humanistic culture and the basics needed to go further. In this sense, my ambition is to surpass the online resources that discuss music and mathematics and are often of high quality, but which often lack the generality and structure that are so valuable in true mathematical definitions. While the primary audience for this manual is musicians who wish to learn about the mathematical dimension of their art, it may also be useful to mathematics teachers, from secondary school to university bachelor’s degree level. Indeed, many musical concepts provide excellent examples and concrete situations for crucial mathematical concepts (fractions, modular arithmetic, Diophantine approximations, powers, polynomials, groups, rings, modules, quotient structures, etc.). It can therefore be useful for both students and teachers who are looking for convincing illustrations and mental images to help them grasp the most fundamental concepts. It should be noted that rigorous definitions and proofs from a mathematical point of view have often been relegated to the appendix, or even omitted or glossed over.