8 ms·

Music and Geometry: Intervals and Scales

209 points by coloneltcb 2y ago

rekado 2y ago
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/offsetmodal/ https://www.stick.com/method/articles/parallel/ https://www.stick.com/method/articles/parallel/
- mlochbaum 2y ago
  My own take on relating scales geometrically: https://mlochbaum.github.io/BQN-Musician/theory/modulation.html 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!
pohl 2y ago
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_a_discovery_im_calling_it_the_color_tree/ https://www.reddit.com/r/musictheory/comments/1etydas/i_made...
- hammock 2y ago
  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
  - dbcurtis 2y ago
    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)
    - recursive 2y ago
      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.
  - recursive 2y ago
    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)
tugu77 2y ago
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.
- openrisk 2y ago
  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.
  - PaulDavisThe1st 2y ago
    > 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).
    - openrisk 2y ago
      > 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-harmony-singers-sing-in-just-intonation-or-tempered-tuning https://music.stackexchange.com/questions/41383/do-capable-h... [2] https://en.wikipedia.org/wiki/Harmonic_minor_scale https://en.wikipedia.org/wiki/Harmonic_minor_scale
ChocMontePy 2y ago
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.
- smitelli 2y ago
  That interval (B-F) would be the tritone, arguably the most dissonant one in the toolbox.
- kian 2y ago
  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.
Duanemclemore 2y ago
Amazing. A deeper dive of the methods John Coltrane used. https://www.openculture.com/2024/12/john-coltrane-draws-a-picture-illustrating-the-mathematics-of-music.html https://www.openculture.com/2024/12/john-coltrane-draws-a-pi... And https://www.openculture.com/2017/10/john-coltrane-draws-a-mysterious-diagram-illustrating-the-mathematical-mystical-qualities-of-music.html https://www.openculture.com/2017/10/john-coltrane-draws-a-my... With plenty of great links to dive deeper in both!
ziofill 2y ago
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.
niobe 2y ago
I am a technical musician for 40 years and I couldn't understand the points he was trying to make... poorly explained
- deleted 2y ago
  [deleted]
- Rochus 2y ago
  What is a "technical musician"?
brcmthrowaway 2y ago
How can I make music without knowing an instrument?
- Rochus 2y ago
  E.g. with a tool like this: https://nodalmusic.com/ https://nodalmusic.com/
  - brcmthrowaway 2y ago
    How does this differ to Max/MSP or PureData
    - Rochus 2y ago
      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 https://www.youtube.com/watch?v=tQpJi0AkFBQ. And here are some nice composition created with Nodal: https://www.youtube.com/watch?v=y1BzGaz62PE https://www.youtube.com/watch?v=y1BzGaz62PE, https://youtu.be/gi61bHLyDsU https://youtu.be/gi61bHLyDsU, https://www.youtube.com/watch?v=sTpyIT8dWIA https://www.youtube.com/watch?v=sTpyIT8dWIA.
      - TheOtherHobbes 2y ago
        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."
        Rochus 2y ago
        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.
shermantanktop 2y ago
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.
hilbert42 2y ago
"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 https://en.m.wikipedia.org/wiki/Pythagorean_comma
f1shy 2y ago
There is a classical book by about the topic https://link.springer.com/book/10.1007/978-3-319-64364-9 https://link.springer.com/book/10.1007/978-3-319-64364-9
tambarskjelve 2y ago
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 https://link.springer.com/article/10.1007/s00283-022-10260-4
BaculumMeumEst 2y ago
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.
- pd3wb6INJxN 2y ago
  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%2Fassets-global.website-files.com%2F61796230edfd36b26f605bf8%2F63a0780752e819174cdf8017_Img%25206%2520-%2520full%2520caged%2520chart.jpg&f=1&nofb=1&ipt=3ea96f85f649af7e19290a319064f19400835a6daf6ffcc1771cd88f29fdf3e5&ipo=images 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.
- lachlan_gray 2y ago
  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-guitar/ 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
jrdres 2y ago
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-temperaments-part-1/ https://johncarlosbaez.wordpress.com/2024/01/11/well-tempera... https://johncarlosbaez.wordpress.com/2023/10/07/pythagorean-tuning/ https://johncarlosbaez.wordpress.com/2023/10/07/pythagorean-...
giancaIta 2y ago
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/ 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/ https://www.fachords.com/circle-of-fifths-chord-shape/
- aanet 2y ago
  Thanks - this looks great! I might download / get one of the books! <3
aanet 2y ago
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).

209 points by coloneltcb 2y ago

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/offsetmodal/ https://www.stick.com/method/articles/parallel/ https://www.stick.com/method/articles/parallel/

mlochbaum 2y ago
My own take on relating scales geometrically: https://mlochbaum.github.io/BQN-Musician/theory/modulation.html 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!

pohl 2y ago

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_a_discovery_im_calling_it_the_color_tree/ https://www.reddit.com/r/musictheory/comments/1etydas/i_made...

hammock 2y ago
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
- dbcurtis 2y ago
  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)
  - recursive 2y ago
    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.
- recursive 2y ago
  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)

tugu77 2y ago

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.

openrisk 2y ago
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.
- PaulDavisThe1st 2y ago
  > 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).
  - openrisk 2y ago
    > 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-harmony-singers-sing-in-just-intonation-or-tempered-tuning https://music.stackexchange.com/questions/41383/do-capable-h... [2] https://en.wikipedia.org/wiki/Harmonic_minor_scale https://en.wikipedia.org/wiki/Harmonic_minor_scale

ChocMontePy 2y ago

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.

smitelli 2y ago
That interval (B-F) would be the tritone, arguably the most dissonant one in the toolbox.
kian 2y ago
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.

Duanemclemore 2y ago

Amazing. A deeper dive of the methods John Coltrane used. https://www.openculture.com/2024/12/john-coltrane-draws-a-picture-illustrating-the-mathematics-of-music.html https://www.openculture.com/2024/12/john-coltrane-draws-a-pi... And https://www.openculture.com/2017/10/john-coltrane-draws-a-mysterious-diagram-illustrating-the-mathematical-mystical-qualities-of-music.html https://www.openculture.com/2017/10/john-coltrane-draws-a-my... With plenty of great links to dive deeper in both!

ziofill 2y ago

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.

niobe 2y ago

I am a technical musician for 40 years and I couldn't understand the points he was trying to make... poorly explained

deleted 2y ago
[deleted]
Rochus 2y ago
What is a "technical musician"?

brcmthrowaway 2y ago

How can I make music without knowing an instrument?

Rochus 2y ago
E.g. with a tool like this: https://nodalmusic.com/ https://nodalmusic.com/
- brcmthrowaway 2y ago
  How does this differ to Max/MSP or PureData
  - Rochus 2y ago
    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 https://www.youtube.com/watch?v=tQpJi0AkFBQ. And here are some nice composition created with Nodal: https://www.youtube.com/watch?v=y1BzGaz62PE https://www.youtube.com/watch?v=y1BzGaz62PE, https://youtu.be/gi61bHLyDsU https://youtu.be/gi61bHLyDsU, https://www.youtube.com/watch?v=sTpyIT8dWIA https://www.youtube.com/watch?v=sTpyIT8dWIA.
    - TheOtherHobbes 2y ago
      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."
      - Rochus 2y ago
        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.

shermantanktop 2y ago

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.

hilbert42 2y ago

"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 https://en.m.wikipedia.org/wiki/Pythagorean_comma

f1shy 2y ago

There is a classical book by about the topic https://link.springer.com/book/10.1007/978-3-319-64364-9 https://link.springer.com/book/10.1007/978-3-319-64364-9

tambarskjelve 2y ago

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 https://link.springer.com/article/10.1007/s00283-022-10260-4

BaculumMeumEst 2y ago

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.

pd3wb6INJxN 2y ago
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%2Fassets-global.website-files.com%2F61796230edfd36b26f605bf8%2F63a0780752e819174cdf8017_Img%25206%2520-%2520full%2520caged%2520chart.jpg&f=1&nofb=1&ipt=3ea96f85f649af7e19290a319064f19400835a6daf6ffcc1771cd88f29fdf3e5&ipo=images 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.
lachlan_gray 2y ago
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-guitar/ 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

jrdres 2y ago

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-temperaments-part-1/ https://johncarlosbaez.wordpress.com/2024/01/11/well-tempera... https://johncarlosbaez.wordpress.com/2023/10/07/pythagorean-tuning/ https://johncarlosbaez.wordpress.com/2023/10/07/pythagorean-...

giancaIta 2y ago

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/ 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/ https://www.fachords.com/circle-of-fifths-chord-shape/

aanet 2y ago
Thanks - this looks great! I might download / get one of the books! <3

aanet 2y ago

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).