Mathematical theory of music

Musique / Sciences

Mathematical Theory of Music

This book provides a unified account of mathematical tools for music theory and analysis, as alternatives to empirical methods. It constitutes a unique reference. The first chapters of the book take as its subject the staple (basic, essential, age-old...) topics in systematic musicology : harmonic and sonance models, Helmholtz research, Costère affinity models as well as chords and scales taxonomy using group action method. Messiaen's limited transposition modes are generalized to n-tone equal temperament.

Methods reviewed by the author briefly include the popular techniques of Forte set theory and Lewin generalized interval systems. The legacy of Hugo Riemann is evoked via the new definition of Riemannian transformations with permutations, which appear in mathematical music theory as carriers for models of some musical objects, as well as for the semantics of other types of musical logic. Neo-Riemannian investigations also provide new interpretations in the design of chords lattices and isographies.

Among the novel features of this book are diagrammatic representations of knots and braids. Schoenberg series are mapped to singular knots. The appendix gives the complete classification of the twelve tone rows in chord diagrams.

In the chapters devoted to tuning and temperaments, the hierarchical organization of contemporary tunings and historical temperaments is studied under various refinements of fundamental concepts : continued fractions, harmonic distance and tuning groups. In the last chapter, the relationships between tilings and canons are related to groups and topology.

This book will be of value to graduate students and research workers in systematic musicology.