Canonical extension

This is an expert page. It either assumes experience with xen theory or involves fairly technical procedures.

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.
  • If the commas of the extension are not necessarily superparticulars but nevertheless involve an arithmetic progression in the harmonic series, the extensions are called endoarithmetic, para-arithmetic, and exoarithmetic. Endoarithmetic extensions are considered structurally induced.

Examples

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Endoparticular extensions

Porcupine

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 indeed, 2.3.5.11 Porcupine can be defined by tempering out S10 and S11.

Kleismic

For 5-limit Kleismic -> 2.3.5.13 Kleismic, we have

$\begin{array}{rl}\frac{15625}{15552}& =(\frac{25}{24}{)}^3/\frac{9}{8}\\ & =\left(\frac{25}{24}\right)(\frac{26}{25}{)}^2\mathrm{S}(25{)}^2/\frac{9}{8}\\ & =(\frac{25}{24}\cdot \frac{26}{25}\cdot \frac{27}{26})\mathrm{S}(25{)}^2\mathrm{S}(26)/\frac{9}{8}\\ & =\mathrm{S}(25{)}^2\mathrm{S}(26).\end{array}$

Diaschismic

For 5-limit Diaschismic -> 2.3.5.17 Diaschismic, we have

$\begin{array}{rl}\frac{2048}{2025}& =(\frac{16}{15}{)}^2/\frac{9}{8}\\ & =\left(\frac{16}{15}\right)\left(\frac{17}{16}\right)\mathrm{S}(16)/\frac{9}{8}\\ & =\left(\frac{16}{15}\right)\left(\frac{18}{17}\right)\mathrm{S}(16)\mathrm{S}(17)/\frac{9}{8}\\ & =\left(\frac{17}{16}\right)\left(\frac{18}{17}\right)\mathrm{S}(16{)}^2\mathrm{S}(17)/\frac{9}{8}\\ & =\mathrm{S}(16{)}^2\mathrm{S}(17).\end{array}$

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).