Canonical extension: Difference between revisions
From Xenharmonic Reference
no more alphabet soup |
|||
| Line 5: | Line 5: | ||
(The page is marked as problematic because the definition of naturalness needs to be made more rigorous) | (The page is marked as problematic because the definition of naturalness needs to be made more rigorous) | ||
Consider a regular temperament on a JI group. The following are both informal concepts. | |||
* A strong [[extension]] | * A strong [[extension]] of said temperament on a JI group with one additional prime is '''natural''' if the commas tempered out by the temperament induce the presence of the added prime. | ||
* | * 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 == | == Examples == | ||
=== Natural extensions === | === Natural extensions === | ||
* 5-limit Kleismic -> | * 5-limit Kleismic -> 2.3.5.13 Kleismic | ||
* 5-limit Porcupine -> | * 5-limit Porcupine -> 2.3.5.11 Porcupine | ||
=== Canonical but non-natural extensions === | === Canonical but non-natural extensions === | ||
* | * 5-limit Meantone -> 7-limit Septimal Meantone (12 & 19) | ||
Revision as of 02:54, 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.
(The page is marked as problematic because the definition of naturalness needs to be made more rigorous)
Consider a regular temperament on a JI group. The following are both informal concepts.
- A strong extension of said temperament on a JI group with one additional prime is natural if the commas tempered out by the temperament induce the presence of the added prime.
- 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
Canonical but non-natural extensions
- 5-limit Meantone -> 7-limit Septimal Meantone (12 & 19)
