Just wondering  looking around I'm surprised this isn't a common thing to nail down:
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 easytofind just intervals by ear.
But, this is not trivial for larger scales. Consider a 10note 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>
Scale anayses of chords

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Scale anayses of chords
“All musicians are subconsciously mathematicians.”
 Thelonious Monk
 Thelonious Monk

 Posts: 289
 Joined: Fri Mar 30, 2018 6:09 pm
 Location: Detroit area
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Re: Scale anayses of chords
Hmmm, did I use the image insertion correctly? I'm trying to reference a .PNG on Wikipedia for that lattice example.
“All musicians are subconsciously mathematicians.”
 Thelonious Monk
 Thelonious Monk
Re: Scale anayses of chords
Maybe a complete digraph with 12 notenodes, then prune the dissonant connections and find all mscales.

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Re: Scale anayses of chords
I'm looking for a more generalized way of thinking about this than sticking strictly to the standard 12note scale of perfect fifths. I want this to apply to potentially *any* scale.
After diving into some basic introductions to graph theory, I'm pretty sure what I'm actually looking for is: given an undirected graph (that represents consonant intervals in some scale), find all connected subgraphs of size n.
It looks like there isn't any simple calculation to just find the number of such subgraphs . . . either some software like Mathematica or a custom searching algorithm programmed by yours truly made be required . . .
The closest thing I've found so far is:
https://mathematica.stackexchange.com/q ... subgraphs
Unfortunately, the link to the paper linked in one of the comments looks dead.
After diving into some basic introductions to graph theory, I'm pretty sure what I'm actually looking for is: given an undirected graph (that represents consonant intervals in some scale), find all connected subgraphs of size n.
It looks like there isn't any simple calculation to just find the number of such subgraphs . . . either some software like Mathematica or a custom searching algorithm programmed by yours truly made be required . . .
The closest thing I've found so far is:
https://mathematica.stackexchange.com/q ... subgraphs
Unfortunately, the link to the paper linked in one of the comments looks dead.
“All musicians are subconsciously mathematicians.”
 Thelonious Monk
 Thelonious Monk
Re: Scale anayses of chords
In "C"...
C "major"  you need C, E, and B (or A)
C "minor"  depending on source of tonality, C Eb A, C Eb B or C Eb Bb... or whatever.
The rest is just "how do I get from C to E, E to B, E to B...
C "dominant", CEBb
etc...
It's not about total possibilities, it's about what absolute necessities you can reduce a "sound" to. There's an infinite number ways you can navigate between two or three notes depending on how much you want to stretch or contract your resolution points.
Music has never been about scales.. scales are just an easy way of organizing sounds.
C "major"  you need C, E, and B (or A)
C "minor"  depending on source of tonality, C Eb A, C Eb B or C Eb Bb... or whatever.
The rest is just "how do I get from C to E, E to B, E to B...
C "dominant", CEBb
etc...
It's not about total possibilities, it's about what absolute necessities you can reduce a "sound" to. There's an infinite number ways you can navigate between two or three notes depending on how much you want to stretch or contract your resolution points.
Music has never been about scales.. scales are just an easy way of organizing sounds.