Exploring Musical Spaces

Abstract

Abstract This book is a comprehensive synthesis of mathematical techniques in music theory, using the idea of musical spaces as a unifying thread. The spaces studied include an assortment of pitch and pitch-class spaces, tonnetz spaces, spaces of chords or key areas, spaces of twelve-tone rows, temporal spaces, voice-leading spaces, scalar spaces, and spaces of note names. Continuous spaces are examined alongside discrete ones, and generic (mod-7) spaces alongside chromatic (mod-12), with related emphasis on formal aspects of note spelling, the line of fifths, and enharmonic equivalence. The spatial approach facilitates the study of several areas of active recent scholarship in a unified theoretical framework; these areas include the theory of intervals and transformations, triadic transformations and neo-Riemannian theory, the geometric theory of voice-leading spaces, diatonic theory, and scale theory. Concepts and techniques from mathematical set theory, graph theory, group theory, geometry, and topology are introduced as needed to address musical questions.