Generator gronut/Mathematics: Difference between revisions
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(Under construction) | (Under construction) | ||
== | == Grylinder == | ||
Let <math>m > 1</math> and <math>n > m, \gcd(n, m) = 1</math> be integer edos and fix a generator <math>g, 2 \leq g \leq m - 2</math> with <math>\gcd(g, m) = 1.</math> Let <math>h</math> be the modular inverse of <math>g</math> modulo <math>m.</math> | Let <math>m > 1</math> and <math>n > m, \gcd(n, m) = 1</math> be integer edos and fix a generator <math>g, 2 \leq g \leq m - 2</math> with <math>\gcd(g, m) = 1.</math> Let <math>h</math> be the modular inverse of <math>g</math> modulo <math>m.</math> | ||
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# Take <math>n</math> equally spaced points on said circle. For sake of canonicality one of the <math>n</math> points is taken to be a "glue point". | # Take <math>n</math> equally spaced points on said circle. For sake of canonicality one of the <math>n</math> points is taken to be a "glue point". | ||
# Unglue the line segments and put them back at their original positions. | # Unglue the line segments and put them back at their original positions. | ||
The resulting sequence of embeddings <math>\mathbb{Z}/n \to S^1\to C</math> is called the <math>(n, g\backslash m)</math>-''' | The resulting sequence of embeddings <math>\mathbb{Z}/n \to S^1\to C</math> is called the <math>(n, g\backslash m)</math>-'''grylinder'''. | ||
== | == Gronut == | ||
For each vertex | Take the grylinder. 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. | ||
Latest revision as of 13:54, 20 October 2026
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 and be integer edos and fix a generator with Let be the modular inverse of modulo
Consider the "cylinder" with coordinates = (along "longitude", along "meridian"). Follow the following procedure:
- Glue copies of the real interval ("line segment") end to end, making a (continuous) circle of circumference For every copy of the next copy is the one that is steps away along the longitude.
- Take equally spaced points on said circle. For sake of canonicality one of the points is taken to be a "glue point".
- Unglue the line segments and put them back at their original positions.
The resulting sequence of embeddings is called the -grylinder.
Gronut
Take the grylinder. 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.
