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, among all extensions to that 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 basis element(s). Any sub-extension (i.e. whose JI group is between the base JI group and the original extension's JI group) of a structurally induced extension is also considered structurally induced. Structural inducing 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 one or more arithmetic progressions in the harmonic series, the extensions are similarly called endoarithmetic and para-arithmetic. Endoarithmetic extensions are considered structurally induced.

Examples

This page or section is a work in progress. It may lack sufficient justification, content, or organization, and is subject to future overhaul.

Endoparticular extensions

We repeatedly use the identity

$\frac{k}{k-1}=\mathrm{S}(k)\frac{k+1}{k},\mathrm{S}(k):=\frac{k^2}{k^2-1}.$

Porcupine

The extension Porcupine = 5-limit[250/243] → 2.3.5.11[S10, S11] 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 Kleismic = 5-limit[15625/15552] → 2.3.5.13[S25, S26], 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 Diaschismic = 5-limit[2048/2025] → 2.3.5.17[S16, S17] we have

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

Würschmidt

For Würschmidt = 5-limit[393216/390625] → 2.3.5.23.47.49[S46, S47, S48, S49] we have

$\begin{array}{rl}\frac{393216}{390625}& =(\frac{16}{15}{)}^2/(\frac{25}{24}{)}^3\\ & =(\frac{48}{45}{)}^2/(\frac{50}{48}{)}^3\\ & =(\frac{46}{45}\cdot \frac{47}{46}\cdot \frac{48}{47}{)}^2/(\frac{49}{48}\cdot \frac{50}{49}{)}^3\\ & =(\frac{46}{45}\cdot \frac{47}{46}\cdot \frac{48}{47}{)}^2/[(\frac{49}{48}\cdot \frac{50}{49}{)}^2(\frac{49}{48}\cdot \frac{50}{49}\left)\right]\\ & =\mathrm{S}(48{)}^2(\frac{46}{45}\cdot \frac{47}{46}\cdot \frac{49}{48}{)}^2/\left[\right(\frac{49}{48}\cdot \frac{50}{49}{)}^2(\frac{49}{48}\cdot \frac{50}{49})]\\ & =\mathrm{S}(48{)}^2(\frac{46}{45}\cdot \frac{47}{46}{)}^2/\left[\right(\frac{50}{49}{)}^2(\frac{49}{48}\cdot \frac{50}{49})]\\ & =\mathrm{S}(47{)}^2\mathrm{S}(48{)}^4\mathrm{S}(49{)}^2(\frac{46}{45}{)}^2/(\frac{49}{48}\cdot \frac{50}{49})\\ & =\mathrm{S}(46{)}^2\mathrm{S}(47{)}^4\mathrm{S}(48{)}^6\mathrm{S}(49{)}^2(\frac{49}{48}{)}^2/(\frac{49}{48}\cdot \frac{50}{49})\\ & =\mathrm{S}(46{)}^2\mathrm{S}(47{)}^4\mathrm{S}(48{)}^6\mathrm{S}(49{)}^3.\end{array}$

Paraparticular extensions

• Kleismic = 2.3.5.13[S26, S27] → Catakleismic = 2.3.5.7.13[S26, S27, S28]

Canonical but non-structurally-induced extensions

• 5-limit Meantone = 5-limit[81/80] → Septimal Meantone (12 & 19) = 7-limit[81/80, 126/125]
• 5-limit Tetracot = 5-limit[20000/19683] → Add-13 Tetracot = 2.3.5.13[20000/19683, 512/507]

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).