Preface

Abstract

Extract Music unfolds on a vast playing field with a mathematical structure of great complexity and beauty. This book is a study of that structure. Mathematics as a tool for the study of music Music theory has in recent decades been enriched by an infusion of ideas from many other disciplines, among which mathematics has earned a particular preeminence. Mathematical techniques enhance our understanding of numerous aspects of music, from scale construction and rhythmic organization to triadic harmony and atonal pitch structure, sometimes building upon established music-analytical traditions but sometimes providing entirely new methodologies and paradigms. Today’s music theory journals publish many articles at least partly mathematical in nature; the Journal of Mathematics and Music began publication in 2007, and international conferences on “Mathematics and Computation in Music” are held biennially. It is not an exaggeration to say that mathematical music theory has blossomed into a sizable industry. The price of admission to the field, however, can seem exorbitant. A solid foundation in mathematics, traditionally expected of students in physics, engineering, and other disciplines that use mathematics, is rare among students of music. Important work in the field is scattered through countless articles and an armload of books, terminology and notation are not consistent, and much of this work is highly technical, not readily accessible to those with limited mathematical training, no matter their musical expertise. David Lewin’s Generalized Musical Intervals and Transformations (Lewin [1987] 2007), the pathbreaking volume that singlehandedly established the field of transformational music theory, has been a definitive resource for more than three decades, but it also remains a famously difficult read for music theorists who must contend from its first pages not only with abstract concepts such as semigroup congruences and homomorphisms but also with an uncompromisingly formal Definition-Theorem-Proof-Corollary-Remark style characteristic of advanced mathematics texts.1