[Rochus]

> There is not much geometry going on

You can use the geometric representation ("necklace") to explain modulation of e.g. diatonic scales as axial mirroring. This can be regarded as a geometric operation. When you look at harmony in this way, some interesting insights open up. I've even built a tool to explore it: https://github.com/rochus-keller/MusicTools/tree/master.

[jvvw]

This sounds interesting. As a mathematician (in the sense that I have a PhD in group theory), is there a good guide to music theory for mathematicians?

There seems to be lots of stuff along the lines of 'if you understand music, here is some mathematics to help you think about it' but not much 'if you understand mathematics, but not so much about music, here is how to think about music'.

[Rochus]

There are many with various mathematical depth:

- Fauvel et al., Music and Mathematics - From Pythagoras to Fractals, 2003, Oxford UP

- Loy, Musimatics Volume 1, 2006 MIT Press

- Tymoczko, A Geometry of Music, 2011, Oxford UP

- Walker, Mathematics and Music, 2013, CRC Press

- Toussaint, The Geometry of Musical Rhythm, 2013, CRC Press

- Chew, Mathematical and Computational Modeling of Tonality, 2014, Springer

- Hook, Exploring Musical Spaces, 2023, Oxford UP

From my point of view, all titles can be appreciated by non-musicians with mathematical background (though I'm an engineer, not a mathematician, and very much involved with non-classical music). But for your specific requirement, maybe Loy is suited, but personally I consider the later books more interesting, especially Tymoczko and Hook. Book recommendations are always very subjective.

Also note that the music theory commonly taught at high schools and universities is barely able to describe music, or only a small fraction of it. And only a fraction of this theory has a mathematical fundament. Most of it is just a heuristic projection of existing music, only useful for recognizing and classifying elements, and not for deriving new music. In recent years, however, new theories have emerged that allow for both a more formal and a more practical approach.

[lioeters]

Great list of books on music and mathematics. It's an endlessly fascinating subject that appeals to the intellect and the heart. I remember years ago, reading Godfried Toussaint's paper, "The Euclidean Algorithm Generates Traditional Musical Rhythms". http://cgm.cs.mcgill.ca/~godfried/publications/banff-extende... (PDF)

Following the trail, I was glad to find he wrote a whole book, The Geometry of Musical Rhythm, where the article forms the basis of a chapter. It's one of my favorite books I keep returning to re-read different parts.

I hadn't seen "Exploring Musical Spaces", looking forward to reading it.

Dmitri Tymoczko's book is wonderful too, A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice. Rich with ideas and insights, I like how he tells the history and development of Western music theory.

Oh I just learned from the author's website that he has a new book released.

> I have just finished a second book, Tonality: An Owner's Manual, that proposes a new, hierarchical, and geometrical model of musical stucture.

> One interesting outgrowth is the musical programming language arca. This line of thinking has also led me to contemplate a third book about category theory and music.

https://dmitri.mycpanel.princeton.edu/index.html

---

This video of your live performance setup as a one-man band. Amazing.

https://www.youtube.com/shorts/S82hsEDY8Pc

[Rochus]

Thanks for the comment. The new book by Tymoczko is interesting, but written in a more philosophical, narrative than scientific/mathematical style, why I didn't mention it. There is no mention of a programming language in the book, but he has published some examples at https://www.madmusicalscience.com. I look forward to more of his books, but I find it a little regrettable that many free spirits hinder themselves by feeling they have to justify themselves and appease with the "old curia" for their findings; as a result, a lot of material gets into these books that distracts from the actual ideas and tries to follow the traditional, authority-based style instead of one shaped by independent science.

Thanks for the feedback. The video shows my setup ten years ago; my most recent musical results are here: http://rochus-keller.ch/?p=1317.

[Kye]

It depends on your goal.

Music theory is a way to encode and share the practice of music. The practice is largely unconcerned with and unaware of math. Any mathematical treatment that gets too far from the practice won't help you understand music.

If you want to understand and practice music, it's safest to limit your exposure to the body of work we call theory to scales, chords, and the circle of fifths and carefully expand from there. Theory can be useful, but the practice of theory can become too about itself and lose sight of the music.

Being too about theory is how you get people saying, confidently, that songs which use that common four chord progression are boring/hackish even though all the examples are of famous and beloved songs.