Canonical extension

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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
• 5-limit Wurschmidt -> 2.3.5.23 Wurschmidt

Canonical but non-natural extensions

• 5-limit Meantone -> 7-limit Septimal Meantone (12 & 19)
• 5-limit Tetracot -> 2.3.5.13 Tetracot