A geometry of music

What Hacker News says

1. acjohnson55Jul 2026
   If you want to go way deep, check out Tymoczko's first book: https://dmitri.mycpanel.princeton.edu/geometry-of-music.html

   It's a challenging read, but extremely thorough and enlightening. Tymoczko wrote a follow up that extends and reformulates his theories, but I haven't gotten through it yet.

   thread
2. fuhsnnDec 2024
   Geometry of Music by Dmitri Tymoczko is a fun book of visualizing chromatic music theory with geometry models, some of it also covered in author's papers https://dmitri.mycpanel.princeton.edu/publications.html
   thread
3. RochusDec 2024
   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.

   Show morethread
4. lioetersDec 2024
   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

   Show morethread
5. lioetersMar 2023
   > the major and minor 'scales' are the same 'scale'

   Indeed, they are two modes of the same pattern. If you look at that pattern in a circle, sometimes called a "necklace", the major and minor scales are rotations of each other.

   For this way of looking at music, I recommend the book A Geometry of Music by Dmitri Tymoczko, who teaches composition and theory at Princeton.

   > A Geometry of Music provides an accessible introduction to a new, geometrical approach to music theory. The book shows how to construct simple diagrams representing voice-leading relationships among familiar chords and scales. This gives readers the tools to translate between the musical and visual realms, revealing surprising structure in otherwise hard-to-understand pieces.

   https://dmitri.mycpanel.princeton.edu/geometry-of-music.html

   ---

   As an intellectual companion, there's a book called The Geometry of Musical Rhythm.

   https://en.wikipedia.org/wiki/The_Geometry_of_Musical_Rhythm

   It's written by Godfried Toussaint, a computer scientist who discovered "Euclidean rhythms", a large set of rhythm patterns generated by a simple algorithm, many of which are common in world music traditions.

   > In 2004 he discovered that the Euclidean algorithm for computing the greatest common divisor of two numbers implicitly generates almost all the most important traditional rhythms of the world.

   The Euclidean algorithm generates traditional musical rhythm - http://cgm.cs.mcgill.ca/~godfried/publications/banff.pdf (PDF)

   Show morethread
6. lioetersApr 2022
   Here's a dense and fun book in that direction..

   A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice

   http://dmitri.mycpanel.princeton.edu/geometry-of-music.html

   thread