Given some scale, what is the set of chords of some size that only contain certain intervals?
This question seems trivial to ask, but I don't think it is. Even with a vanilla major scale, there are 7 choose 3 or 35 triads, another 35 quadrads, and so on, just using basic combinatoric math. This is a fair amount, although manageable, number of chords to list - and many of these will be dissonant. I went through this process manually a month ago to generate this:
and you can of course disagree on what I consider 'consonant'. That just depends on what you're trying to do with a piece of music. My concern here was to organize the chords by some arbitrary measure of 'consonance' that involves relatively easy-to-find just intervals by ear.
But, this is not trivial for larger scales. Consider a 10-note scale. 10 choose 3 is 120; 10 choose 4 is 210, and you see that trying to organize chords in a scale like that is a whole lot more work. So, my thought is, what is an algorithm, given some arbitrary scale of length m, to find all 'consonant' chords of length n within that scale? 'Consonant' in this case basically means whatever you want it to mean: pick which intervals you like.
As far as I can tell, this turns out to be a graph theory problem involving 'simple paths' on an 'undirected graph'. Which is to say, a bit more complicated and fun than I expected.
For example, you can extract a graph of the major scale from this lattice:
if you consider the notes ABCDEFG as vertices and that they're connected by edges, as per typical construction in graph theory. Edges represent what you want to be consonant intervals.
<Edit: Fixed image link>
