I have spent some time in the combinatorial music space approaching the subjects of Euclidean rhythms, rhythm canons, and evenness. The musical interest has motivated the definition and theoretical exploration of these objects, which might be of interest to some.

**Euclidean Rhythms**

Euclidean rhythms are an object that is both ubiquitous in music (see Godfried Tousaint's "The Euclidean Algorithm Generates Traditional Musical Rhythms"), and also has interesting structure to them in combinatorial terms. Rhythms are binary necklaces where the digits represent beats, 1s denote "hits," and 0s "rests." Euclidean rhythms are those that have the 1s and 0s maximally evenly distributed with one another (aka maximally even rhythms).

I wrote a review on Euclidean rhythms in the first part of an undergraduate research paper found on my website: here.

To summarize: many papers have shown Euclidean rhythms (ERs) to be unique for any weight and length. When weight and length are not coprime, the ER is a concatenation of smaller ERs with coprime weight and length (primitive concatenation property). Several distinct algorithms exist that generate Euclidean rhythms (see Demaine et al. "The Distance Geometry of Music").

There is also Bjorklund's generating algorithm. A few papers found results about the object generated by Bjorklund's algorithm, which is assumed to be the Euclidean rhythm; however, in my research, I found that nobody had proved this fact. So, I frame the lit review in a way to prepare the reader for my proof of the fact in a proceeding section. That is, I had an agenda while writing the review haha, so keep that in mind.

"Euclidean Strings" (2003) by Ellis, Ruskey, Sawada, and Simpson discusses various results about Euclidean rhythms with coprime weight and length. They connect them to fibonacci strings, the stern-brocot tree, and a bunch of other stuff. It's a good read, though they reproduce some results of the following paper.

"Maximally Even Sets" (1991) by Clough and Douthett discuss Euclidean rhythms in different, mathematical music theoretic terms. While they are more concerned with the musical implications on maximally even scales, it is a theoretical work full of proofs of some neat facts.

**Rhythm Canons**

It is possible to represent the layering of rhythms of multiple voices (instruments) in a 2D binary array, where rows are voices, and columns are beats. When these rhythms repeat, the array becomes toroidal and some interesting mathematical questions arise with implications in both music composition and analysis:
Given a rhythm, how might we tile this rhythm in each row such that the weight of each column is exactly 1. Given a set of distinct rhythms, are they tileable? What properties do tileable sets of rhythms have? etc.

I wrote a project proposal around this sort of tiling problem found here; the reference section might be of interest, as well as the definitions. A lot of fundamental work on rhythm layering appears to have been done by a Romanian mathematician D.T. Vuza.

**Evenness**

Continuing from rhythms, recall that Euclidean rhythms are maximally even. Along with characterizations of maximal and minimal evenness, there are measures and sorting algorithms for evenness as well. "The Distance Geometry of Music" linked above is a good place to find examples of such measures. Toussaint argued that the most common rhythms are those of high evenness.

My view:
Music is a high dimensional landscape with ubiquitous structures and symmetries. Music has a lot of mathematically unexplored territory, which can provide insight into mathematical objects, music creation, and music analysis.

I hope this reply was of interest.

A Geometry of Musicby Dimitri Tymoczko, but I don't think it illustrates what you seek. $\endgroup$ – Joseph O'Rourke Jan 24 '15 at 17:16