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.
Author(s): Julian Hook
Series: Oxford Studies in Music Theory
Publisher: Oxford University Press
Year: 2022
Language: English
Pages: 680
City: New York
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
