67
$\begingroup$

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!

$\endgroup$
  • 6
    $\begingroup$ The Topos of Music: Geometric Logic of Concepts, Theory, and Performance by Guerino Mazzola might be something in that direction: amazon.com/Topos-Music-Geometric-Concepts-Performance/dp/… $\endgroup$ – Thomas Kalinowski Feb 26 '17 at 23:00
  • 2
    $\begingroup$ A 6 minute video related to Tymoczko's work: George Hart, Making music with a Möbius strip, August 12, 2013. $\endgroup$ – Rodrigo de Azevedo Feb 27 '17 at 0:05
  • 2
    $\begingroup$ Perhaps some relevant information can also be found in the posts tagged music-theory+book-recommendation on math.SE. (And maybe even other posts tagged with music-theory.) $\endgroup$ – Martin Sleziak Feb 27 '17 at 4:02
  • $\begingroup$ Piggyback question: I would appreciate suggestions that have significant discussions of post-tonal theory. Serialism, after all, is all about the various actions of Z/12Z on (Z/12Z)^12 (with particular attention paid to elements in a free orbit). $\endgroup$ – Alexander Woo Feb 27 '17 at 8:24
  • $\begingroup$ @AlexanderWoo I'm not sure how mathematical this is, but the classic text is Post-Tonal Theory. $\endgroup$ – Clarinetist Mar 1 '17 at 2:14
30
$\begingroup$

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.


                  PurpleRain
                  Purple Rain.

$\endgroup$
16
$\begingroup$

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.)

$\endgroup$
  • 1
    $\begingroup$ I find this book very interesting and helpful, too, but the dissonance model is too focused on place theory. Microrythmic models like this one (described in the book “Verschmelzung und neuronale Autokorrelation als Grundlage einer Konsonanztheorie” by Martin Ebeling) lead to similar results. $\endgroup$ – Tobias Schlemmer Feb 28 '17 at 18:04
  • 1
    $\begingroup$ I didn't think that Sethares committed himself to any particular model of the cochlea in this book, and he (briefly) discusses place theory along with alternatives (adding "it is probably safe to say that there is not enough evidence to decide between them"). He has a mathematical model of dissonance that is abstracted from any particular biological mechanism and runs with that. $\endgroup$ – Dan Piponi Mar 1 '17 at 0:11
14
$\begingroup$

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.

$\endgroup$
4
$\begingroup$

The book by Guerino Mazzola relies heavily on algebraic geometry:

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

$\endgroup$
  • 11
    $\begingroup$ This was mentioned also in a comment under the question. It's a hefty tome; do you have a pretty good acquaintance with it? It would be good to have some informed commentary from an insider on how useful or worthwhile it is (I've always been curious). $\endgroup$ – Todd Trimble Feb 27 '17 at 2:01
  • $\begingroup$ Sorry for late response. I have not yet read it very carefully, but I did learn something from the first few chapters. $\endgroup$ – Joe Franklin Dec 12 '17 at 3:00
2
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.