Canonical extension: Difference between revisions

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.

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 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).
+* 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 ''natural'' 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 (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.