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
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 Transformations—Scale Shifts
Diatonic Transformations—Inversions, Retrograde
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
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
Time Stretching and Time Shrinking
A Exercise Solutions
B Music Software
C Amplitude and Frequency Results