Canonical extension

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)

Let T be a regular temperament on JI group G and let H be a JI group containing G, but of one rank higher. The following are both informal concepts.

A strong extension U on H is natural if the commas tempered out by T induce the presence of the added basis element of H.
A strong extension U on H is (more weakly) canonical if it is the most efficient (accurate and low-complexity) strong extension of T to H.

Examples