Canonical extension

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 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 (more strongly) structurally induced 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:
  • 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 instead tempers out a square-superparticular adjacent to the square-superparticulars in question, or exoparticular instead if it instead tempers out a square-superparticular not adjacent to said square-superparticulars. Paraparticular and exoparticular extensions are generally not considered structurally induced.

Examples

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Endoparticular extensions

5-limit Porcupine -> 2.3.5.11 Porcupine is endoparticular: we have

$\begin{array}{rl}\frac{250}{243}& =(\frac{10}{9}{)}^3/\frac{4}{3}\\ & =\left(\frac{10}{9}\right)(\frac{11}{10}{)}^2\mathrm{S}(10{)}^2/\frac{4}{3}\\ & =\frac{\frac{10}{9}\cdot \frac{11}{10}\cdot \frac{12}{11}}{\frac{4}{3}}\cdot \mathrm{S}(10{)}^2\mathrm{S}(11)\\ & =\mathrm{S}(10{)}^2\mathrm{S}(11),\end{array}$

and 2.3.5.11 Porcupine indeed tempers out S10 and S11.

Other examples:

• 5-limit Kleismic -> 2.3.5.13 Kleismic
• 5-limit Diaschismic -> 2.3.5.17 Diaschismic
• 5-limit Würschmidt -> 2.3.5.23.47.49 Würschmidt

Paraparticular extensions

• 2.3.5.13 Kleismic -> 2.3.5.7.13 Catakleismic

Canonical but non-structurally-induced extensions

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

Conjectures

• Conjecture: There is at most one S-expression for a comma in a given extended subgroup. As a corollary, an endoparticular extension of a temperament tempering out one given comma to a given extended subgroup is unique if it exists.
• Stronger conjecture: An endoparticular extension for a given temperament of any rank and a given extended subgroup is unique if it exists (regardless of the choice of comma basis).