Amazing. A deeper dive of the methods John Coltrane used.
https://www.openculture.com/2024/12/john-coltrane-draws-a-pi...
And
https://www.openculture.com/2017/10/john-coltrane-draws-a-my...
With plenty of great links to dive deeper in both!
If you like this you may also be interested in Emmett Chapman's Offset Modal System:
https://www.stick.com/method/articles/offsetmodal/ https://www.stick.com/method/articles/parallel/
My own take on relating scales geometrically: https://mlochbaum.github.io/BQN-Musician/theory/modulation.h...
It does seem that I include all Chapman's scales (while saying nothing about chords), although oddly enough he's chosen to use the modes of harmonic major but not those of its inversion, harmonic minor?
Edit: In fact I found the second link (first one's pretty vague and wasn't enough for me to follow the diagram) relevant enough that I added a paragraph to point out the connection!
I am a technical musician for 40 years and I couldn't understand the points he was trying to make... poorly explained
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What is a "technical musician"?
I don't know enough music to tell if this is insightful, or just neat pattern-matching.
A few months ago, mathematician John Baez had a series on the mathematics of various temperament and keys. Of course he knows his math, but also music thanks to being a member of rather famous musical family. (More math in the second link.)
https://johncarlosbaez.wordpress.com/2024/01/11/well-tempera...
https://johncarlosbaez.wordpress.com/2023/10/07/pythagorean-...
I love this. Here's another interesting thing I encountered. It's a way of organizing chromatic subsets by brightness
https://www.reddit.com/r/musictheory/comments/1etydas/i_made...
That is cool and different. Glancing at the bottom several rows I tend to agree with the classification, as a trained musician. I wonder though, what is (the mathematical principle) behind it that is causing the brightness/ darkness in the sound of these chords?
Don't say "dissonance" (or explain what dissonance is)- that much is obvious, looking for something a bit more detailed, e.g. why 1-2-5 sounds brighter than 1-4-5
It is going to be the relative amplitudes of the overtones. Pure tone (like flute) is bright, many overtones present with the appropriate mix can sound dark (french horn). Next, you are going to ask me for the coefficients, but I don’t know that. Break into the nearest church with a proper pipe organ and start pulling and pushing stops to see what happens. (Or ask your friend the organist to take you so that you don’t get arrested)
It's a reference to this particular chart, not the general idea of brightness of a sound. And it's regarding chords or scales, not timbres.
I don't really understand this chart at all, but I think it's based on the idea that an upward movement of a fifth is bright, and a downward movement is dark. 1-2-5 can be built on two upward movements of a fifth, whereas 1-4-5 can be built be one upwards movement, and one downward (from the tonic)
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.
> 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.
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'.
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.
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.
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.
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.
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].
[1] https://music.stackexchange.com/questions/41383/do-capable-h...
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).
Eivind Groven developed a mechanical piano for playing in just intonation: http://www.joranrudi.no/mediefiler/The%20Just%20Intonation%2...
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 :)
This is not true.
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.
My words on this were wrong and misleading.
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
Tymoczko's take is far more interesting and educational than the article.
(I'm being polite.)
Whenever someone says "there's not much geometry going on" you've identified a person with little capacity for imagination
I'd love to see some imagination on that page beyond some star shapes.
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.
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Oh that's my cup of tea, love this stuff!
I've created a 3d guitar fretboard here, where the height of the blocks corresponds to the height of the pitches: https://www.fachords.com/guitar-fretboard-3d/
And here are the shapes of the different chord qualities in the Circle Of Fifths: https://www.fachords.com/circle-of-fifths-chord-shape/
Thanks - this looks great! I might download / get one of the books!
<3
I understand music and mathematics were much more closely related historically, to some extent practically regarded as the same subject, but new discoveries about this relationship are still happening in our time. One interesting finding is that the the Pythagoeran comma, i.e. tiny interval between to enharmonically equivalent notes can be constructed geometrically: https://link.springer.com/article/10.1007/s00283-022-10260-4
Fantastic resource! Thanks for sharing this. I love both the math, the music, and math&music combo. Tickles my inner geek. <3
I'd love to see if anyone has done this geometric / visualization for Indian classical (specifically, Hindustani) music??
Perhaps there's specific shapes / visualizations in certain Ragas that naturally emanate?
Note that in the Indian classical (Hindustani) music system, the Ragas are a "framework" for melody, not really a mode (as in Western music theory).
I never realized until now that in the the two different circles pictured (the Chromatic Circle and the Circle of Fifths) the pairs of notes opposite each other are the same in each circle. For example in both circles B is opposite from F.
And if you move around the Chromatic Circle, swapping every second pair of notes with its opposite on the other side of the circle, you have the Circle of Fifths.
That interval (B-F) would be the tritone, arguably the most dissonant one in the toolbox.
If you take the chromatic scale and then swap every other pair of notes on opposite sides of the circle, it yields the circle of fifths. You'll notice that on the circle of fifths notes that skip a step are a whole tone apart in the chromatic scale.
Although there have been some claims in these comments to the contrary, harmony is particularly mathematical. Symmetry and the breaking of within the integers mod 12 form the foundational principles of harmony.
"Pythagorean Temperament"
'Pythagorean Temperament' involves the Pythagorean Comma (aka Comma of Pythagoras). Whilst mentioned, it's not spelled out as such here: https://en.m.wikipedia.org/wiki/Pythagorean_comma
Somewhat unrelated: I’m looking for a comprehensive overview of why the CAGED system works on guitar. I see lots of mechanical explanations of how to use it to play various chords down the neck, but nothing explaining the theory behind it.
Open chord shapes are just the barre shapes but the nut becomes fret zero that's barred. Looking at it this way it's easy to see why they can be moved around the neck. The shape remains the same because the relationship of the strings and the notes on them remain the same. Like root-3rd-5th would remain the same relationship only with a different tone combination.
By the way I think the most difficult thing about CAGED is that it's way too overhyped. It promises too much (scales?) but delivers too little. (Doesn't even have minors)
I was super obsessed with this for a while! When you have a string instrument tuned in 4ths, there are 2D patterns that emerge which you can use to "derive" or "extrapolate" what a scale shape/pattern will look like across the whole neck
Using a 6-string bass as an example: https://bradleyfish.com/the-notes-on-the-6-string-bass-guita...
You can find a 2D pattern in the white notes (green notes in the pic) that you can use to understand how the pattern will extend from a given point. For example notice EF+BC always appear in the same 2x2 box shape. Also how those boxes repeat in a diagonal line, and how boxes are connected vertically by a "strip" of 3 notes ADG
The only difference for guitar is that you have to correct for the G/B strings which are separated by a 3rd instead of a 4th, by scooting the pattern on the B+E strings up by one fret
CAGED is descriptive, ie it's the result of someone noticing "we all know these chord shapes, and they have some useful properties which can contribute to your fretboard knowledge". Those properties are:
- those 5 chords are the first you learn, and mechanically easy to play in open position, so you know them by heart
See this diagram (https://external-content.duckduckgo.com/iu/?u=https%3A%2F%2F...):
- the CAGED 'order' (C then A then G etc) matches the order in which octaves of the same root appear as you go up the neck, therefore CAGED is a good way of visualising octaves (see how in the 'C-shapes' column the chords 'share' root notes when played in that order up the neck)
- each chord matches a root note position (C: top part of the neck box on the 5th string, A: bottom part of the neck box on the 5th string, etc), therefore if you're playing a scale, no matter what position you're playing it in, you can choose a CAGED chord to overlay on it and easily find the root, third, and fifths (see how in the 'C major scale' column, you can overlay each chord onto one way of playing the C major scale)
- learning these mnemonics should eventually help you 'unlock' the guitar neck, ie have an intuitive knowledge of what intervals you're playing and how to build melodic lines with them
Generally, music theory 'works' because it describes why things sound good. It's not theory that informs what sounds good, rather theory attempts to describe what sounds good and build patterns which will help theory learners, in turn, make music which sounds good.
Take a look at Fretboard Theory by Desi Serna - it spends a lot of time on how different scales are constructed and relating different patterns and chord forms back to the underlying concepts.
Thanks, this is exactly what I needed. It's amazing how much of this information other people decided to omit.
As a longtime guitarist, this is exactly the type of visual pattern crutch that the fretboard encourages and which is (for me) both a crutch and a trap. Geometry can help explain music but if it takes the lead, that’s the definition of formulaic.
There is a classical book by about the topic https://link.springer.com/book/10.1007/978-3-319-64364-9
I've been playing piano for 30+ years, and I only learned to use the circle of fifths this year. It's been short of a revelation and I can't recommend enough to practice scales and drills based on it.
How can I make music without knowing an instrument?
IMO the question doesn't make sense.
You can't make music without learning an instrument.
Maybe the instrument is a computer with PD or DAW software but it is still a musical instrument that you have to learn to play.
There is no way around spending hours learning and practicing whatever instrument you pick.
What makes a computer hard to learn to play is that it has too many options. You don't learn anything playing trumpet for a month, then changing to piano for 2 weeks, then trying a new guitar that just came out for 2 months. That though is basically what many computer musicians do.
You need to pick the software setup and then spend a long time learning it and don't switch instruments.
E.g. with a tool like this: https://nodalmusic.com/
How does this differ to Max/MSP or PureData
It has nothing in common with Max nor Pd. The latter specifiy signal flow and operators on the signal. Nodal instead represents events and time distances between them; this way you can design musical patterns; time is two dimensional, so you can draw loops; then you can associate musical information with the events and add logical operations so that your loops vary in time. It's a very nice experimenting and composition tool, especially if you don't play an instrument. See e.g. here for a tutorial: https://www.youtube.com/watch?v=tQpJi0AkFBQ.
And here are some nice composition created with Nodal: https://www.youtube.com/watch?v=y1BzGaz62PE, https://youtu.be/gi61bHLyDsU, https://www.youtube.com/watch?v=sTpyIT8dWIA.
With some persuasion, Max can do everything Nodal does. It's not convenient and you have to use numbers instead of line lengths. But it's untrue they have "nothing in common."
Well, that requires a demonstration ;-)
> you have to use numbers instead of line lengths
That's the whole point of Nodal; you represent musical patterns (and control flow) with graphical means. Entering numbers is close to Midi event lists and has little to do with Nodal's approach. Max/Pd are great tools, but focus on the processing and signal flow aspect (i.e. the machine which generates the music/noise), not the representation of the music.
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