Canonical extension
From Xenharmonic Reference
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.
- 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).
- More formally: Suppose the base temp tempers out a comma that is a product of powers of consecutive square-superparticulars with nonzero exponents. A strong extension is natural if it additionally tempers out the individual square-superparticulars.
- Such an extension is (informally and more weakly) canonical if it is the most efficient (accurate and low-complexity) strong extension of the temperament to the larger subgroup. This is a more informal concept which is decided by using heuristics and qualitative assessments.
Examples
Natural extensions
- 5-limit Kleismic -> 2.3.5.13 Kleismic
- 5-limit Porcupine -> 2.3.5.11 Porcupine
- 5-limit Diaschismic -> 2.3.5.17 Diaschismic
- 5-limit Würschmidt -> 2.3.5.23.47.49 Würschmidt
Canonical but non-natural extensions
- 5-limit Meantone -> 7-limit Septimal Meantone (12 & 19)
- 5-limit Tetracot -> 2.3.5.13 Tetracot
