Amazing. A deeper dive of the methods John Coltrane used.<p><a href="https://www.openculture.com/2024/12/john-coltrane-draws-a-picture-illustrating-the-mathematics-of-music.html" rel="nofollow">https://www.openculture.com/2024/12/john-coltrane-draws-a-pi...</a><p>And<p><a href="https://www.openculture.com/2017/10/john-coltrane-draws-a-mysterious-diagram-illustrating-the-mathematical-mystical-qualities-of-music.html" rel="nofollow">https://www.openculture.com/2017/10/john-coltrane-draws-a-my...</a><p>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:<p><a href="https://www.stick.com/method/articles/offsetmodal/" rel="nofollow">https://www.stick.com/method/articles/offsetmodal/</a>
<a href="https://www.stick.com/method/articles/parallel/" rel="nofollow">https://www.stick.com/method/articles/parallel/</a>
I don't know enough music to tell if this is insightful, or just neat pattern-matching.<p>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.)<p><a href="https://johncarlosbaez.wordpress.com/2024/01/11/well-temperaments-part-1/" rel="nofollow">https://johncarlosbaez.wordpress.com/2024/01/11/well-tempera...</a><p><a href="https://johncarlosbaez.wordpress.com/2023/10/07/pythagorean-tuning/" rel="nofollow">https://johncarlosbaez.wordpress.com/2023/10/07/pythagorean-...</a>
I love this. Here's another interesting thing I encountered. It's a way of organizing chromatic subsets by brightness<p><a href="https://www.reddit.com/r/musictheory/comments/1etydas/i_made_a_discovery_im_calling_it_the_color_tree/" rel="nofollow">https://www.reddit.com/r/musictheory/comments/1etydas/i_made...</a>
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.
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: <a href="https://link.springer.com/article/10.1007/s00283-022-10260-4" rel="nofollow">https://link.springer.com/article/10.1007/s00283-022-10260-4</a>
Oh that's my cup of tea, love this stuff!<p>I've created a 3d guitar fretboard here, where the height of the blocks corresponds to the height of the pitches:
<a href="https://www.fachords.com/guitar-fretboard-3d/" rel="nofollow">https://www.fachords.com/guitar-fretboard-3d/</a><p>And here are the shapes of the different chord qualities in the Circle Of Fifths:
<a href="https://www.fachords.com/circle-of-fifths-chord-shape/" rel="nofollow">https://www.fachords.com/circle-of-fifths-chord-shape/</a>
Fantastic resource! Thanks for sharing this. I love both the math, the music, and math&music combo. Tickles my inner geek. <3<p>I'd love to see if anyone has done this geometric / visualization for Indian classical (specifically, Hindustani) music??<p>Perhaps there's specific shapes / visualizations in certain Ragas that naturally emanate?<p>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.<p>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.
<i>"Pythagorean Temperament"</i><p>'Pythagorean Temperament' involves the Pythagorean Comma (aka Comma of Pythagoras). Whilst mentioned, it's not spelled out as such here: <a href="https://en.m.wikipedia.org/wiki/Pythagorean_comma" rel="nofollow">https://en.m.wikipedia.org/wiki/Pythagorean_comma</a>
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.
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 <a href="https://link.springer.com/book/10.1007/978-3-319-64364-9" rel="nofollow">https://link.springer.com/book/10.1007/978-3-319-64364-9</a>
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.