A study of musical scales (2017)

Nearly jumped out of my seat when I saw this! Look at how similar our approaches are: http://welliam.github.io/molts/ (apologies for how poorly written it is, it's been a few years!)

Not that it's a super novel concept, but we both used this for the modes of limited transposition. For me it was an efficient way of generating MOLTs for any equal temperament scale.

And form.

It's been awhile since I've had the chance to dig into more advanced music theory but this is such an interesting framing of all of the possibilities present in both named and unnamed scales and the relationship of notes. The idea to lay it out mathematically like this and how you frame your overall thinking about scales is great. Bach would appreciate this work.

Also reminds me a bit of how folks like Jacob Collier think about harmonies and new ways to arrive at different notes. The more you can internalize these kinds of mathematics, the more you can improvise in new and different ways.

Well done! Really enjoyed this.

pretty good, this has motivated me to do an online post about my own take at similar issues. However I do not assume 12-tet, just the ocatave, so it's slightly different.

Exactly. I would like to see this done using integer ratios to define the frequency intervals.

You can still use the octave as a 'start' and 'end' point for the scale to make things simple (although there's no axiom anywhere that says this has to be the case, it's just a good candidate because it's the simplest and most easily recognizable interval, aside from the unison).

Since you'd be dealing with a potentially infinite number of intervals, you can start with the 'simplest' 20 or so (using lowest possible integer values to create the distinct ratios). Some would argue that the simplest 40 can be considered musical although that would get you into some seriously dissonant territory, not to mention the amount of possibilities you would have available at that point.