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Exploring Musical Spaces

A Synthesis of Mathematical Approaches

Julian Hook

Publication Date - November 2022

ISBN: 9780190246013

680 pages
7 x 10 inches

Retail Price to Students: $65.00


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.


  • Offers comprehensive and up-to-date coverage of mathematical approaches to music theory
  • Adopts a visual, geometric orientation, using the idea of a musical space as a unifying concept
  • Written for readers with no prior mathematical knowledge beyond basic algebra
  • Features hundreds of exercises spanning a wide range of difficulty levels

About the Author(s)

Julian Hook holds PhDs in both mathematics and music theory, as well as graduate degrees in architecture and piano performance. His work involving mathematical approaches to the study of music has appeared primarily in music theory journals but also at conferences of the American Mathematical Society and in the pages of Science. Since 2003 he has taught at Indiana University, where he is a former chair of the music theory department. He is a past president of Music Theory Midwest and was the founding reviews editor of the Journal of Mathematics and Music.


"The 'mathy' quality of much recent music theory has long been a barrier to its comprehension. No more. Julian Hook is a master explainer and, thanks to this book, music theorists and interested musicians now have an effective on-ramp not only to understanding but also to deep enjoyment of the rich regularities that can be heard to underpin musical experience." -- Joseph Straus, CUNY Graduate Center

"Exploring Musical Spaces draws together the most important results in algebraic and geometric music theory of the last fifty years. Julian Hook's treatise, featuring the author's signature clarity and depth of insight, will open this dazzling field to a new generation of scholars." -- Ian Quinn, Yale University

Table of Contents

    Part I Foundations of Mathematical Music Theory: Spaces, Sets, Graphs, and Groups
    Chapter 1: Spaces I: Pitch and Pitch-Class Spaces
    Chapter 2: Sets, Functions, and Relations
    Chapter 3: Graphs
    Chapter 4: Spaces II: Chordal, Tonal, and Serial Spaces
    Chapter 5: Groups I: Interval Groups and Transformation Groups
    Part II Transformation Theory: Intervals and Transformations, including Neo-Riemannian Theory
    Chapter 6: Groups II: Permutations, Isomorphisms, and Other Topics in Group Theory
    Chapter 7: Intervals
    Chapter 8: Transformations I: Triadic Transformations
    Chapter 9: Transformations II: Transformation Graphs and Networks; Serial Transformations
    Part III Geometric Music Theory: The OPTIC Voice-Leading Spaces
    Chapter 10: Spaces III: Introduction to Voice-Leading Spaces
    Chapter 11: Spaces IV: The Geometry of OPTIC Spaces
    Chapter 12: Distances
    Part IV Theory of Scales: Diatonic and Beyond
    Chapter 13: Scales I: Diatonic Spaces
    Chapter 14: Scales II: Beyond the Diatonic
    Appendix 1: List of Musical Spaces
    Appendix 2: List of Sets and Groups