Generator gronut/Mathematics

This is a technical or mathematical page. While the subject may be of some relevance to music, the page treats the subject in technical language.

(Under construction)

Grylinder

Let $m>1$ and $n>m,\gcd (n,m)=1$ be integer edos and fix a generator $g,2\le g\le m-2$ with $\gcd (g,m)=1.$ Let $h$ be the modular inverse of $g$ modulo $m.$

Consider the "cylinder" $C=\mathbb{\mathbb{Z}}/m\times [0,1)$ with coordinates $(x,y)$ = (along "longitude", along "meridian"). Follow the following procedure:

1. Glue copies of the real interval ("line segment") $[0,1)$ end to end, making a (continuous) circle of circumference $m.$ For every copy of $[0,1)$ the next copy is the one that is $h$ steps away along the longitude.
2. Take $n$ equally spaced points on said circle. For sake of canonicality one of the $n$ points is taken to be a "glue point".
3. Unglue the line segments and put them back at their original positions.

The resulting sequence of embeddings $\mathbb{\mathbb{Z}}/n\to S^1\to C$ is called the $(n,g\mathrm{\setminus }m)$-grylinder.

Gronut

For each vertex, corresponding to a note of n-edo, draw an edge corresponding to 1\n and two edges for the two nearest intervals in n-edo to g\m, which straddle g\m. The resulting graph is called the gronut graph or the gronut; every vertex of the gronut has degree 6, and the graph forms a triangulation of the torus.