Canonical extension: Difference between revisions
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Consider a regular temperament on a JI group. The following are both informal concepts. | Consider a regular temperament on a JI group. The following are both informal concepts. | ||
* A strong [[extension]] of said temperament on a JI group | * A strong [[extension]] of said temperament on a larger JI group is '''natural''' if the commas tempered out by the temperament induce the presence of the added prime(s). | ||
* Such an extension is (more weakly) '''canonical''' if it is the most efficient (accurate and low-complexity) strong extension of the temperament to the larger subgroup. | * Such an extension is (more weakly) '''canonical''' if it is the most efficient (accurate and low-complexity) strong extension of the temperament to the larger subgroup. | ||
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* 5-limit Kleismic -> 2.3.5.13 Kleismic | * 5-limit Kleismic -> 2.3.5.13 Kleismic | ||
* 5-limit Porcupine -> 2.3.5.11 Porcupine | * 5-limit Porcupine -> 2.3.5.11 Porcupine | ||
* 5-limit Wurschmidt -> 2.3.5.23.47.49 Wurschmidt | |||
=== Canonical but non-natural extensions === | === Canonical but non-natural extensions === | ||
* 5-limit Meantone -> 7-limit Septimal Meantone (12 & 19) | * 5-limit Meantone -> 7-limit Septimal Meantone (12 & 19) | ||
* 5-limit Tetracot -> 2.3.5.13 Tetracot | * 5-limit Tetracot -> 2.3.5.13 Tetracot | ||
Revision as of 15:00, 4 February 2026
This page or section is a work in progress. It may lack sufficient justification, content, or organization, and is subject to future overhaul.
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.
Consider a regular temperament on a JI group. The following are both informal concepts.
- A strong extension of said temperament on a larger JI group is natural if the commas tempered out by the temperament induce the presence of the added prime(s).
- Such an extension is (more weakly) canonical if it is the most efficient (accurate and low-complexity) strong extension of the temperament to the larger subgroup.
Examples
Natural extensions
- 5-limit Kleismic -> 2.3.5.13 Kleismic
- 5-limit Porcupine -> 2.3.5.11 Porcupine
- 5-limit Wurschmidt -> 2.3.5.23.47.49 Wurschmidt
Canonical but non-natural extensions
- 5-limit Meantone -> 7-limit Septimal Meantone (12 & 19)
- 5-limit Tetracot -> 2.3.5.13 Tetracot
