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Music and Geometry: Intervals and Scales

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Re: Music and Geometry: Intervals and Scales

#5

Post by mlochbaum »

rekado wrote: post #2

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!

Re: Music and Geometry: Intervals and Scales

#6

Post by openrisk »

tugu77 wrote: post #4

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.

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.

Re: Music and Geometry: Intervals and Scales

#7

Post by ChocMontePy »

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.

Re: Music and Geometry: Intervals and Scales

#8

Post by Rochus »

tugu77 wrote: post #4

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.

Re: Music and Geometry: Intervals and Scales

#9

Post by hammock »

pohl wrote: post #3

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

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