Hypergraphs and weighted graphs of scales

I was thinking about two different ways to expand on Gene's work with
graphs of scales, both of which seem useful.

The first thing is that while the cliques of the (unweighted) graph of
a scale refer to dyadic chords, that constructing a hypergraph over
the same set can give us otonal chords, assuming our set of
consonances is the set of chords within some odd-limit. Any hypergraph
can be split up into a set of k-uniform partial hypergraphs over the
same set of vertices. For any k-uniform hypergraph, the set of cliques
now refer to generalizations of dyadic chords such that each k-size
subset is a k-size otonal chord; these we may call "triadic cliques,"
"tetradic cliques," etc. Note that any for any k, being a k-adic
clique is a necessary yet insufficient condition for being a
(k+1)-adic clique.

There will also be lots of chords which aren't cliques, but which are
almost cliques. Taken as hypergraphs, these chords are interesting in
their own right; one example would be the meantone chord C-E-G-A, with
the 9-limit otonal chords as consonances. This isn't a triadic clique
since C-E-A isn't a 9-limit chord, but it is a dyadic clique with a
huge consonant triad right in it. If we place any stock in
psychoacoustics at all these days, we might conjecture that the fact
that a lot of the sound of this chord stems from the combination of
these two things (and additionally the fact that this triad is
rooted).

The second thing is to generalize the graph of a scale to allow for a
weighted graph, where the whole thing is connected. There are two ways
of doing this that I can think of: the first is that the weight of
each edge is the (logarithmic) Tenney height of the minimum-complexity
interpretation of the interval between them, so that higher-weighted
edges are more discordant. The other way is to set the weight of each
edge to be the reciprocal of Tenney height and disconnecting vertices
from themselves, so that lower-weighted edges are more discordant.

However you do it, a few concepts generalize very nicely with weighted
edges, such as the concept of the adjacency matrix, which now takes
real values rather than values in {0, 1}, though I'm not sure how the
interpretation changes. Other things seem more tricky, such as
connectivity, since the whole thing ends up being connected by
definition, but some connections are stronger than others.

Weighted hypergraphs are, of course, also possible.

Has anyone thought about any of this stuff before?

-Mike