Generator gronut/Mathematics: Difference between revisions
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Consider the "cylinder" <math>C = \mathbb{Z}/m \times [0, 1)</math> with coordinates <math>(x, y)</math> = (along "longitude", along "meridian"). Follow the following procedure: | Consider the "cylinder" <math>C = \mathbb{Z}/m \times [0, 1)</math> with coordinates <math>(x, y)</math> = (along "longitude", along "meridian"). Follow the following procedure: | ||
# Glue copies of the real interval <math>[0, 1)</math> end to end, making a (continuous) circle of circumference <math>m.</math> For every copy of <math>[0, 1)</math> the next copy is the one that is <math>h</math> steps away along the longitude. | # Glue copies of the real interval ("line segment") <math>[0, 1)</math> end to end, making a (continuous) circle of circumference <math>m.</math> For every copy of <math>[0, 1)</math> the next copy is the one that is <math>h</math> steps away along the longitude. | ||
# 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 embedding of <math>\mathbb{Z}/n</math> into <math>C</math> is called the <math>(n, g\backslash m)</math>-'''gronut'''. | The resulting embedding of <math>\mathbb{Z}/n</math> into <math>C</math> is called the <math>(n, g\backslash m)</math>-'''gronut'''. | ||
Revision as of 01:44, 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)
Definition
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 embedding of into is called the -gronut.
