The diagrams look nice, but in the end of the day, they are merely nice visualizations of what's fundamentally algebra. There is not much geometry going on besides a quite simple group structure of order 12.
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.
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'.
- 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.
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.
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.
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.
Besides geometry there is not a lot of music either, in the sense that even this simple symmetry is kinda fake, effectively a forced resolution of an essentially unsolvable "problem": hammering the intervals into place to inject some "logic" into the task of dividing the octave in heptatonic scales. Despite the unrepentant Pythogereans across all ages, our musical brain is not mathematical except in a very loose way.
Besides aesthetics, though, there might be educational value given that the equal temperament tuning is a cornerstone of western music education.
> the task of dividing the octave in heptatonic scales.
that's not task. The octave has already been divided into twelve tones here.
The task is to pick sets of those twelve tones to serve some aesthetic purpose. The sets may be of various sizes, though 7 is common in a lot of european music.
If the task was to generate heptatonic scales from within the octave, there are a huge number of possibilities not described here, and most of them are rarely used (though many more than the ones based on a 12TET system are).
> The task is to pick sets of those twelve tones to serve some aesthetic purpose
Aesthetic in the musical or visual sense? The visual aspect is based on the Z12 symmetry and it is pleasant - like all symmetries.
The question is what does the visual experience have to do with the music experience?
The first disconnect with the musical experience is that the 12TET itself is not what people would, e.g., choose to sing in [1].
The second disconnect is that the Greek modes of the major scale are not remotely covering all the scales people enjoy, even adopting a Eurocentric point [2].
On singers and violinists adjusting their harmony to just intonation by fine-tuning to zero beat frequency, I wonder if anyone has made a keyboard that can do that.
If you mean a keyboard which includes a mechanism for causing strings to vibrate, you can tune ANY such keyboard to use just intonation.
What you cannot do is modulate between keys with a keyboard tuned to just intonation: it would have to be retuned for every key change. The scope of the mechanism that would be required to do this has not been implemented since the harpsichord was invented.
There are synthesizers that can be retuned in this way, because there is no physical mechanism to adjust. The results are ... odd. It is still challenging to play them because in addition to performing the notes, you need to signal the key change/retuning points.
Also, when singers and violinists do this, they are not "fine tuning to zero beat frequency". Either you sing in just intonation, in which case you cannot modulate between keys (because the Nth note of the scale has a different frequency depending on the root note), or you sing in some tempered scale (in which the frequencies of the notes have been adjusted to make modulation possible).
Ah yes, of course. 36T ... increasing the number of pitches per octave is a different approach to the problem, and works (at some cost to the performer :)
Singers and violinists can and do adjust intonation so each chord sounds (justly) in tune. The exception is if they were trained with equal tempered instruments (which is common nowadays - see Duffin, “How Equal Temperament Ruined Harmony”) or if they are playing with pre-quantized (fretted/keyed) instruments, in which case they would match the existing temperaments.
So the linked article, while it shows some beautiful shapes linked to 12s, has nothing to do with actually (justly) in tune music.
Source: master’s degree in the topic; am a professional singers specializing in music written before equal temperament was invented
I go to a lot of choir concerts. What I've found is that I much prefer (good) choirs singing without accompanying instruments, because when there are instruments involved the harmonies always fall into equal temperament. There is a quality when they sing a cappella which simply isn't there when they don't.
For a professional musician you are oddly singer/violin focused. Any instrument which can physically detune while playing has their musicians do this. On wind instruments it's via the mouthpiece, any string instrument beyond violin has some flexibility etc. It's only the piano that doesn't, essentially everybody else does.
But in practise, for many music styles, it doesn't really matter. Music is so much more than whether some chord is pitch perfect in tune.
Source: Jazz musician on 6 instrument types part time professional for 25 years (other part is software engineer).
I did speak a bit loosely and too simplistically. What I was trying to get to was the point you're making which is the same point the GP was making: when performers have the ability (because of the instrument they are using, including the voice), they will adjust to reduce beating.
Mathematicians regard 'symmetries' as algebra (as in group theory etc rather than high-school algebra) rather than geometry and I suspect this is about the use of the word geometry in part.
