Books on music theory intended for mathematicians

Question

Some time ago I attended a colloquium given by Princeton music theorist Dmitri Tymoczko, where he gave a fascinating talk on the connection between music composition and certain geometric objects (as I recall, the work of Chopin can naturally be viewed as walks among lattice points lying in some hyperbolic surface, which gives some sense of canonicity to his compositions). I am asking whether there exist books on music theory that is intended for an audience with a reasonably sophisticated mathematical background. This question is in the same spirit as this one regarding physics.

Any suggestions would be great!

Answer 1

There is the book by Tymoczko from which his lecture was no doubt drawn:

Tymoczko, Dmitri. A geometry of music: Harmony and counterpoint in the extended common practice. Oxford University Press, 2010.

                 

And this book is more combinatorial and algorithmic:

Toussaint, Godfried T. The Geometry of Musical Rhythm: What Makes a" good" Rhythm Good?. CRC Press, 2013. CRC link.

                 

And there is a nice AMS page on Mathematics & Music.

Finally, here is an impressive video by Chris Tralie on Geometric Analysis of Musical Audio Data. In particular, check out this YouTube video illustrating Purple Rain, We can work it out, among other well-known songs.

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                  ^{Purple Rain.}

Answer 2

This is a very wide subject so I'm mentioning a book specifically on one subject: tuning. I found the book Tuning, Timbre, Spectrum, Scale by William Sethares to be particularly interesting.

Briefly: it describes a mathematical model for dissonance and then shows that if you optimise the set of notes that make up a scale so as to avoid dissonance, then you're led to Western scales if you use instruments whose harmonics are all multiples of a fundamental frequency. He then shows how other tunings make sense with instruments that have different series of overtones (eg. metal bars). There's also a bit of discussion of the mathematical subtleties of piano tuning, a subject far more complex than I had realised, as well as of non-Western tunings.

One thing I found very interesting in the book is his argument that the apparently special status of the octave (ie. that notes with fundamental frequencies f and 2f sound similar) is a function of the choice of instrument, not an inevitable feature of human biology.

(It's not specifically for mathematicians but it was something I found as a result of searching for works on music with non-trivial predictions derived from mathematics.)

Answer 3

There is a relevant question on Math.SE: Mathematics and Music - reference request. I can second the top-voted answer there, which suggests the book Music: a Mathematical Offering by Dave Benson.

Answer 4

The book by Guerino Mazzola relies heavily on algebraic geometry:

The Topos of Music, Geometric Logic of Concepts, Theory, and Performance. Birkhäuser (2002).

Answer 5

I've only read the very beginning, but the book Generalized Musical Intervals and Transformations by David Lewin was recommended to me by a music professor when he heard I was studying mathematics. In it, the author analyzes the works of Wagner and others using group theory.

You can find two of his related papers here and here.

Answer 6

This is quite a nice coincidence, as I studied and did research with both Dr. Gary Don and Dr. James Walker back in my undergraduate for mathematics and music, authors of one of the math and music articles in the AMS website. These two professors created a textbook Mathematics and Music: Composition, Perception, and Performance published in 2013. (My name is in the Acknowledgements, and I have an autographed copy from Dr. Don!) My impression is that Dr. Walker specialized mainly on harmonic analysis before he wrote this book, so there's some mention of FFTs, for example.

In addition, there is a website for this textbook.

Answer 7

This one seemed to hit on some math topics, not very difficult math, but it offered explanations in math: What Makes Music Work by Philip Seyer.

Answer 8

Amiot, Emmanuel. 2016. Music Through Fourier Space. Springer.

From the Springer website:

This book explains the state of the art in the use of the discrete Fourier transform (DFT) of musical structures such as rhythms or scales. In particular the author explains the DFT of pitch-class distributions, homometry and the phase retrieval problem, nil Fourier coefficients and tilings, saliency, extrapolation to the continuous Fourier transform and continuous spaces, and the meaning of the phases of Fourier coefficients.

Perspectives of New Music 49/2 (Summer 2011) is devoted almost entirely to "Tiling Rhythmic Canons", including articles by many of the authors mentioned in other posts.¹

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¹ The origin of the mathematics in this book can be found in Dan Vuza's four-part article "Supplementary Sets and Regular Complementary Unending Canons" in Perspectives of New Music (vols. 29/2–31/1). It is the founding article of an area of research ("Tiling Rhythmic Canons") for many of the authors mentioned here, including Tom Johnson, Guerino Mazzola, Emmanuel Amiot, Carlos Agon, and Moreno Andreatta.