- Membership
- Publications
- Meetings
- Competitions
- Community
- Programs
- Students
- High School Teachers
- Faculty and Departments
- Underrepresented Groups
- MAA Awards
- MAA Grants

- News
- About MAA

Publisher:

Chapman & Hall/CRC

Publication Date:

2013

Number of Pages:

318

Format:

Paperback

Price:

59.95

ISBN:

9781439867099

Category:

Textbook

[Reviewed by , on ]

Donald L. Vestal

01/9/2014

According to the book blurb, “At first glance, mathematics and music seem to be from separate worlds — one from science, one from art.” But on a very fundamental level, mathematics and music are the same kind of entity: both are languages. And since math is a language that helps us examine and understand the real world, and music is an important part of that world, it seems reasonable that math can be used to study music. This isn’t a new idea by any means — it dates back at least as far as Pythagoras. But can math be used to explain why certain styles and approaches in music sound more pleasant than others?

In the first three chapters, the authors introduce music theory, with some mathematics for help. Logarithms are related to half-steps; modular arithmetic is related to the circle of fifths; transformations are related to transpositions and inversions. One really nice trait of this text is the use of specific well-known examples from music. For example, in section 3.3, the authors explore scale shifts using a few measures from Schubert’s *Der Lindenbaum*, Beethoven’s *Fifth Symphony*, J. S. Bach’s *Inventio 11*, and Alicia Keys’ *If I Ain’t Got You*. The authors manage to sneak in a theorem (which shows that the set of transpositions and chromatic inversions act transitively on the set of all major and minor chords) and even an appearance of the Fibonacci sequence (in counting the number of chord progressions in Kostka and Payne’s model).

The next two chapters bring in the physics of spectrograms. In modeling musical sounds and notes, the trigonometric functions are of course useful, so the authors spend some time with trigonometric sums and the modeling of tones. We then have some concrete examples of spectrograms from vocal performances (from Renée Fleming, Luciano Pavarotti, Alicia Keys, and the choir Sweet Honey in the Rock) and instrumentals (from Louis Armstrong, Beethoven, the Benny Goodman orchestra, and Jimi Hendrix).

The authors then return to music theory, analyzing musical qualities using geometry and number theory. Examples include a geometric interpretation for chromatic inversion, a modular arithmetic approach to rhythmic inversion, and using the Euclidean algorithm to create a pentatonic or a diatonic scale. And the geometry of the (neo-Riemannian) tonnetz torus is used to further explore pitch classes and transformations. The final chapter covers audio synthesis, with a brief discussion of phase vocoding (which is used by AutoTuner), time stretching and shrinking, and different types of MIDI sound synthesis.

Although this book is structured like a textbook (with exercises and extra material at the textbook website), it is much more. Even at just over 300 pages, there is so much more that could be explored (such as Dmitri Tymoczko’s study of chord progressions, or the authors’ dissonance score for a scales). 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 is an Associate Professor of Mathematics at South Dakota State University. His interests include number theory, combinatorics, spending time with his family, and working on his hot sauce collection. He can be reached at Donald.Vestal(AT)sdstate.edu.

**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

Logarithms

**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**

Singing

Instrumentals

Compositions

**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

Serialism

**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
Bibliography
Index**

- Log in to post comments