Mathematics and Music: Composition, Perception, and Performance, 1st Edition (Paperback) book cover

Mathematics and Music

Composition, Perception, and Performance, 1st Edition

By James S. Walker, Gary W. Don

Chapman and Hall/CRC

342 pages | 343 B/W Illus.

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Paperback: 9781439867099
pub: 2013-04-10
Hardback: 9781138442689
pub: 2017-07-27
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pub: 2013-04-19
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At first glance, mathematics and music seem to be from separate worlds—one from science, one from art. But in fact, the connections between the two go back thousands of years, such as Pythagoras’s ideas about how to quantify changes of pitch for musical tones (musical intervals). Mathematics and Music: Composition, Perception, and Performance explores the many links between mathematics and different genres of music, deepening students’ understanding of music through mathematics.

In an accessible way, the text teaches the basics of reading music and explains how various patterns in music can be described with mathematics. The authors extensively use the powerful time-frequency method of spectrograms to analyze the sounds created in musical performance. Numerous examples of music notation assist students in understanding basic musical scores. The text also provides mathematical explanations for musical scales, harmony, and rhythm and includes a concise introduction to digital audio synthesis.

Along with helping students master some fundamental mathematics, this book gives them a deeper appreciation of music by showing how music is informed by both its mathematical and aesthetic structures.

Web Resource

On the book’s CRC Press web page, students can access videos of many of the spectrograms discussed in the text as well as musical scores playable with the free music software MuseScore. An online bibliography offers many links to free downloadable articles on math and music. The web page also provides links to other websites related to math and music, including all the sites mentioned in the book.


"… a valuable addition to the literature and an important piece of pedagogical material. The exercises and answers make this work immediately valuable to any college-level (or advanced high school) instructor. An enlightened music department might count a course based on the book toward an undergraduate mathematics requirement. However it might be used, the authors have done a service to mathematicians and musicians alike."

—Robert Rowe, Journal of Mathematics and the Arts, 2014

"Although this book is structured like a textbook (with exercises and extra material at the textbook website), it is much more. … And there is some fascinating music trivia scattered throughout, like the fact that Mozart never used the iii to IV chord progression, the fact that the Beatles’ Eleanor Rigby is a multi-modal composition, and the interesting remix story of the Beatles’ Strawberry Fields Forever. If you have at least a moderate background in both music and mathematics, there is some great fun here."

—Donald L. Vestal, MAA Reviews, January 2014

"This book strikes an interesting balance between theoretical development and practical, concrete material. A companion website contains audio, video and score files for many of the examples; supplementary reading; and many external links."

—David Warren Bulger, Mathematical Reviews, November 2013

Table of Contents

Pitch, Frequency, Musical Scales

Pitch and Frequency

Overtones, Pitch Equivalence, and Musical Scales

The 12-tone equal-tempered scale

Musical Scales within the Chromatic Scale


Basic Musical Notation

Staff Notation, Clefs, Note Positions

Time Signatures and Tempo

Key Signatures and The Circle of Fifths

Some Music Theory

Intervals and Chords

Diatonic Music

Diatonic Transformations—Scale Shifts

Diatonic Transformations—Inversions, Retrograde

Chromatic Transformations

Web Resources

Spectrograms and Musical Tones

Musical Gestures in Spectrograms

Mathematical Model for Musical Tones

Modeling Instrumental Tones

Beating and Dissonance

Estimating Amplitude and Frequency

Windowing the Waveform: Spectrograms

A Deeper Study of Amplitude Estimation

Spectrograms and Music




Analyzing Pitch and Rhythm

Geometry of Pitch Organization and Transpositions

Geometry of Chromatic Inversions

Cyclic Rhythms

Rhythmic Inversion

Construction of Scales and Cyclic Rhythms

Comparing Musical Scales and Cyclic Rhythms


A Geometry of Harmony

Riemann’s Chromatic Inversions

A Network of Triadic Chords

Embedding Pitch Classes within the Tonnetz

Other Chordal Transformations

Audio Synthesis in Music

Creating New Music from Spectrograms

Phase Vocoding

Time Stretching and Time Shrinking

MIDI Synthesis

A Exercise Solutions

B Music Software

C Amplitude and Frequency Results

D Glossary

E Permissions



About the Originator


Subject Categories

BISAC Subject Codes/Headings:
SCIENCE / Physics