Canonical extension: Difference between revisions
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Consider a regular temperament on a JI group. | Consider a regular temperament on a JI group. | ||
* A strong [[extension]] of said temperament on a larger JI group is '''natural''' (TODO: change term) if the commas tempered out by the temperament induce the presence of the added prime(s). It has the following sub-concepts that can be defined formally: | * A strong [[extension]] of said temperament on a larger JI group is '''natural''' (TODO: change term) if the commas tempered out by the temperament induce the presence of the added prime(s). It has the following sub-concepts that can be defined formally: | ||
** More formally: Suppose the base temperament tempers out a comma that is a product of powers of consecutive [[square-superparticular]]s with nonzero exponents. A strong extension is ''endoparticular'' if it additionally tempers out the individual square-superparticulars. | ** More formally: Suppose the base temperament tempers out a comma that is a product of powers of consecutive [[square-superparticular]]s with nonzero exponents. A strong extension is '''endoparticular''' if it additionally tempers out the individual square-superparticulars. | ||
** Such an extension is ''paraparticular'' instead if it tempers out a square-superparticular ''adjacent'' to the square-superparticulars in question. | ** Such an extension is '''paraparticular''' instead if it tempers out a square-superparticular ''adjacent'' to the square-superparticulars in question. | ||
* 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 judgments. | * 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 judgments. | ||
== Examples == | == Examples == | ||
=== | === Endoparticular extensions === | ||
* 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 | ||
Revision as of 18:10, 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.
- A strong extension of said temperament on a larger JI group is natural (TODO: change term) if the commas tempered out by the temperament induce the presence of the added prime(s). It has the following sub-concepts that can be defined formally:
- More formally: Suppose the base temperament tempers out a comma that is a product of powers of consecutive square-superparticulars with nonzero exponents. A strong extension is endoparticular if it additionally tempers out the individual square-superparticulars.
- Such an extension is paraparticular instead if it tempers out a square-superparticular adjacent to the square-superparticulars in question.
- 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 judgments.
Examples
Endoparticular 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
