User:Mike Battaglia/Mike's working page for graph theory, group theory, topology

Some more thoughts on the mysterious connection that seems to exist seeing graph theory, group theory, and topology, all of which is converging on scales and temperaments. Some of this is a more focused review of the past discussion, other parts are new:

1) Say you start off with a JI group and some "complexity" metric on it, then you temper down and induce the quotient metric on that group. I will call this quotient metric the "modulation metric" for the temperament, such that the distance between any note and 1/1 is that note's "modulation complexity." Distance within this metric will be called "modulation distance" in general.

2) For any scale in your tempered group, you can come up with a fully connected weighted graph in which notes are vertices, and the weight of each edge is the modulation distance between the pair of notes it touches.

3) If you "quantize" this graph by keeping all edges under some maximum distance and then throwing the weights away, you recover Gene Smith's graph of a scale, assuming our initial JI metric was the Weil metric made octave-equivalent.

4) An alternate way to recover the graph is to quantize first by taking the Cayley graph of the entire tempered group, with the set of odd-limit consonances used as a generating set, and then look at the subgraph corresponding to your scale.

5) Both of these provide a connection between graph theory and group theory, which we'll milk as much as we can.

6) Gene's also introduced topology into the fold by writing about the genus of a graph, which deals with its embeddings into compact orientable 2-manifolds.

7) We can take a variant of this idea and relate it back to group theory - but rather than looking at embeddings of graphs into compact surfaces, we'll look at embeddings of entire tempered groups into pretori.

8) The entirety of any tempered group T can be embedded into a pretorus such that the codimension of T is the dimension of the embedded torus; I'll call the smallest such pretorus the "minimal pretorus." The embedding also naturally defines a group action of T on the pretorus.

9) For instance, here's an example of meantone being embedded into the cylinder. This ties the present discussion into ideas that Paul Erlich and Joe Monzo seem to have anticipated a long time ago.

10) In fact, we can go even further and start considering isometries: for whatever metric we put on our original group, we can consider the tight span of that metric space. I conjecture that this is going to be isometrically isomorphic to the pretorus the group embeds into, with a suitable metric put on it (i.e. if you had the T1 metric on your group, you get a T1 metric on your pretorus). This seems obvious to me, but I haven't formally proven it yet.

11) Finally, we can think about isometric embeddings of the tight span into a higher-dimensional real-coordinate metric space. So if I'm correct above about this tight span always being the minimal pretorus, then we can ask whether or not an isometry exists embedding this pretorus into higher-dimensional real-coordinate space with some suitable metric (i.e. if you had the T1 metric on the torus, you get it in higher-dimensional real-coordinate space too).

12) I conjecture that this always exists, so you can always think of Gene's graph as representing a set of notes that are connected if their distance, after being isometrically embedded into the surface of some manifold, which you can further visualize as being embedded into ordinary Euclidean space (but without the Euclidean metric) is smaller than some threshold value.

13) From a purely topological standpoint, we can musically "hear" the 'holes' in this space by the way that modulations work, and how you can move out in some direction, keep going, and then suddenly end up back at the tonic again. For topological spaces, the concept of the "fundamental group" encapsulates this property perfectly, where closed paths from a base point are sometimes homotopic to the identity and sometimes not.

14) It turns out that the fundamental group of the minimal pretorus of T has exactly the musical interpretation of being something like a "group of comma pump classes" (the latter being, in a certain precise way, a subset of the set of homotopy classes of the pretorus), except the comma pumps are continuous and move by infinitesimal steps (but this turns out not to matter in practice).

15) Homotopy theory may add structure we don't need. A way to study the comma pumps that may not involve adding so much structure is homology theory, and think about our original group (or the various graphs obtainable from it) as triangulations of the torus. This may prove a more useful setting for thinking about the modulatory structure in a tempered space with the modulation metric, since real-life comma pumps move in discrete steps. Also, jumping back to graph theory, it's noteworthy that the graph of a scale has an associated simplicial complex, the clique complex of the graph. Additionally, there's the otonal "hypergraph" of a scale, which is also a simplicial complex.

16) #14 and #15 seem heavily related to Petr Parizek's work on comma pumps.

My knowledge ends here (so far), but the field of study that seems to link every single one of these things together is Algebraic Topology, and that's where I will go. Geometric Group Theory may also be of interest.