Exploring Musical Spaces: A Synthesis of Mathematical Approaches (Oxford Studies in Music Theory)

Exploring Musical Spaces is a comprehensive synthesis of mathematical techniques in music theory, written with the aim of making these techniques accessible to music scholars without extensive prior training in mathematics. The book adopts a visual orientation, introducing from the outset a number of simple geometric models―the first examples of the musical spaces of the book's title―depicting relationships among musical entities of various kinds such as notes, chords, scales, or rhythmic values. These spaces take many forms and become a unifying thread in initiating readers into several areas of active recent scholarship, including transformation theory, neo-Riemannian theory, geometric music theory, 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. Musical examples ranging from Bach to the late twentieth century keep the underlying musical motivations close at hand. The book includes hundreds of figures to aid in visualizing the structure of the spaces, as well as exercises offering readers hands-on practice with a diverse assortment of concepts and techniques. Cover Series Exploring Musical Spaces Copyright Contents Preface Acknowledgments Part One Foundations of Mathematical Music Theory: Spaces, Sets, Graphs, and Groups 1. Spaces I: Pitch and Pitch-​Class Spaces 1.1 Pitch spaces 1.2 Pitch-​class spaces 1.3 Spaces generated by fifths and thirds 1.4 Tonnetz spaces Notes Suggested reading 2 Sets, Functions, and Relations 2.1 Sets 2.2 Ordered sets and multisets 2.3 Functions 2.4 Relations 2.5 Modular arithmetic 2.6 Relationships among modular spaces Notes Suggested reading 3. Graphs 3.1 Graphs 3.2 Isomorphism of graphs 3.3 Loops, multiple edges, and infinite graphs 3.4 Directed graphs 3.5 Transformation graphs and networks Notes Suggested reading 4. Spaces II: Chordal, Tonal, and Serial Spaces 4.1 Double-​circle spaces and related constructions 4.2 Tonnetz-​related chordal and tonal spaces 4.3 Generic and diatonic chordal spaces 4.4 Some additional models 4.5 Analytical examples Notes Suggested reading 5. Groups I: Interval Groups and Transformation Groups 5.1 The interval and transposition groups of pitch space 5.2 Definition of a group; additive, modular, and multiplicative groups 5.3 Abstract groups; further properties of groups 5.4 Interval groups and interval spaces 5.5 Transformation groups and group actions 5.6 The relation between intervals and transformations Notes Suggested reading Part Two Transformation Theory: Intervals and Transformations, including Neo-​Riemannian Theory 6 Groups II: Permutations, Isomorphisms, and Other Topics in Group Theory 6.1 Permutation groups 6.2 Group tables and Cayley diagrams 6.3 Isomorphism of groups 6.4 Direct-​product groups 6.5 Groups, equivalence relations, and symmetry 6.6 Quotient groups; considerations with noncommutative groups Notes Suggested reading 7 Intervals 7.1 Label functions for interval spaces 7.2 Homomorphisms and isomorphisms of interval spaces 7.3 Direct products of interval spaces 7.4 Quotients of interval spaces 7.5 Transposition operators and interval-​preserving mappings 7.6 Inversion operators and interval-​reversing mappings Notes Suggested reading 8. Transformations I: Triadic Transformations 8.1 Uniform triadic transformations 8.2 Riemannian UTTs and neo-​Riemannian analysis 8.3 Other topics in triadic transformation theory Notes Suggested reading 9 Transformations II: Transformation Graphs and Networks; Serial Transformations 9.1 Transformation graphs and networks: basic properties 9.2 Consistency properties 9.3 Isomorphism and isography 9.4 Klumpenhouwer networks 9.5 Serial transformations and UTTs 9.6 Transformations of pitch classes and order numbers Notes Suggested reading Part Three Geometric Music Theory: The OPTIC Voice-​Leading Spaces 10. Spaces III: Introduction to Voice-​Leading Spaces 10.1 The hexatonic triad graph as a continuous voice-​leading space 10.2 A larger space of three-​voice chords 10.3 The OPTIC relations 10.4 Normal forms in OPTIC spaces Notes Suggested reading 11. Spaces IV: The Geometry of OPTIC Spaces 11.1 Manifolds and orbifolds; one-​voice spaces 11.2 Two-​voice spaces 11.3 Three-​voice OP-​space 11.4 Three-​voice T-​, PT-​, PTI-​, OPT-​, and OPTI-​space 11.5 Four-​voice OP-​space 11.6 Four-​voice T-​, OPT-​, and OPTI-​space Notes Suggested reading 12. Distances 12.1 Interval functions and measures of distance 12.2 Distance functions; real and modular interval spaces as distance spaces 12.3 Distance functions defined by graphs or groups 12.4 Distance functions on product spaces 12.5 Distance functions on quotient spaces; OPTIC spaces as distance spaces Notes Suggested reading Part Four Theory of Scales: Diatonic and Beyond 13 Scales I: Diatonic Spaces 13.1 Diatonic and generic scales as musical spaces 13.2 Diatonic scales in chromatic space 13.3 Signature transformations 13.4 Genus and species Notes Suggested reading 14 Scales II: Beyond the Diatonic 14.1 Seven-​note scales and spelled heptachords 14.2 Maximal evenness and the geometry of scales 14.3 Beyond the chromatic: other specific cardinalities Notes Suggested reading Appendix 1 List of Musical Spaces Appendix 2 List of Sets and Groups References Index