Mathematical theory of music

The aim of this book is to provide an introduction to the mathematical theory of music, with emphasis on the fascinating and original connections with musical analysis. Despite the complexity of the subject, we have tried to make this introduction accessible to a large audience. Many examples are given for a better understanding. Musicians and mathematicians are invited to contemplate some nontrivial aspects of the mathematical construction.
The book is divided into eleven chapters. In the first chapter, we introduce the language of harmonic models. After a review of some physical models based on the wave equation, which try to modelize the inharmonicity of the spectrum of certain frequencies, we look briefly at sonance models. These models are built around the notion of dissonance curves, which go back to the Hermann Helmholtz's book Sensation of Tones. The next section is devoted to affinity models introduced by Edmond Costere. Costere's theory could be generalized to other weight systems, but the crucial question is to know what is the best weight system for a given context and to give an algebraic demonstration of the classification obtained. Costere has just given the results for a particular weight system obtained from a computational software. Today, we have to understand what is behind these representations and to give the properties of the associated weight systems. In the last section, we review some geometric methods which lead to the theory of modulation.
Chapter 2 on Taxonomy of chords and scales, is devoted to the classification of pitch-class sets. We first introduced the notion of a group action, and review Polya's enumeration theory. Four main group actions are considered : the cyclic classification proposed by Edmond Costere (1954), the dihedral classification proposed by Hanson (1960), Zalewski and Forte (1972), the affine classification proposed by the Czech composer Pinos (1971)...