Consider the major and minor chords in Western music. There are 24 of them altogether (with the normal twelve-tone equal temperament tuning system usually used): one major chord and one minor chord for each of the 12 notes in the chromatic scale (A-G and 5 accidentals).
Each of these 24 chords consists of 3 notes. Pick one of the notes in the chord and throw it away, leaving two notes. Given these two notes, one can always find two chords (one major and one minor) which use those two notes. So given any major chord one can find three minor chords which share two notes with it, and given any minor chord one can find three major chords which share two notes with it.
That suggests that the 24 chords form a sort of network or graph, with vertices corresponding to chords and edges corresponding to pairs of notes. In three dimensions this network can be arranged on the surface of a torus (Waller's Torus). You can arrange the vertices in such a way that all the edges are the same length, but some vertices will have different sets of angles between edges than others.
In 4 dimensions, the symmetries of the network are more apparent - the vertices can be arranged such that each edge is the same length and all the vertices have the same set of angles between edges. The resulting object isn't a polychroron because only three edges meet at each vertex but it isn't a polyhedron either because it does extend into all 4 dimensions. The analogous figure in 3D would might be something like a star shape - if you draw it in 2D the inner points will be closer to each other than the outer points but in 3D you can arrange the points on the surface of a cylinder (sort of like the points one might find at the top of a paper party hat from a Christmas cracker). Here's what it looks like:
Using this figure, one could generate a nonlinear mapping from points in a 4-dimensional space to chords. Not just major and minor chords either - other chords are represented by points other than the vertices. At each vertex you can move in the direction of one of the edges to change the pitch of one of the notes. Since there are only three degrees of freedom in a chord (one for the pitch of each note) there is also a direction at each point in 4-space which leaves the chord unchanged as you move along it.
Is this author still here? Would love to see the pictures that went with this article. I currently just see blank space where the pictures should be. I'm in love with this idea and the paper is so interesting. My brain is running overtime thinking about networking tetrads.
Looks like the YouTube embed got broken with a recent update of either YouTube or WordPress or both. I've updated it so you should be able to see it now!
WOW THANKS. Very intriguing stuff, thanks for your work.