Canonical extension - Revision history

Inthar at 04:15, 24 February 2026

2026-02-24T04:15:56Z

← Older revision		Revision as of 04:15, 24 February 2026
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	{{~~Expert~~}}		{{Technical}}

	Consider a regular temperament on a JI group.		Consider a regular temperament on a JI group.

Inthar: /* Endoparticular extensions */

2026-02-24T04:12:53Z

Endoparticular extensions

← Older revision		Revision as of 04:12, 24 February 2026
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	{{Problematic}}		{{Problematic}}
	=== Endoparticular extensions ===		=== Endoparticular extensions ===
	We repeatedly use the identity k/(k - 1) = S(k)(k + 1)/k.		We repeatedly use the identity

			<math>\frac{k}{k - 1} = \mathrm{S}(k)\frac{k + 1}{k}, \ \mathrm{S}(k) := \frac{k^2}{k^2-1}.</math>
	==== Porcupine ====		==== Porcupine ====
	The extension Porcupine = 5-limit[250/243] → 2.3.5.11[S10, S11] is endoparticular: we have		The extension Porcupine = 5-limit[250/243] → 2.3.5.11[S10, S11] is endoparticular: we have

Inthar at 01:26, 10 February 2026

2026-02-10T01:26:09Z

← Older revision		Revision as of 01:26, 10 February 2026
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	Consider a regular temperament on a JI group.		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.		* 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. ~~where the extended~~ group is between the base group and the original ~~extended~~ group) of a structurally induced extension is also considered structurally induced. Structural inducing has the following sub-concepts that can be defined formally:		* 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-superparticular]]s with nonzero exponents. A strong extension is '''endoparticular''' if it additionally tempers out the individual square-superparticulars.		** Suppose the base temperament tempers out a comma that is a product of powers of consecutive [[square-superparticular]]s 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.		** 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.

Inthar: /* Endoparticular extensions */

2026-02-07T10:43:08Z

Endoparticular extensions

← Older revision		Revision as of 10:43, 7 February 2026
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	{{Problematic}}		{{Problematic}}
	=== Endoparticular extensions ===		=== Endoparticular extensions ===
	We repeatedly use the identity S(k)(k + 1)/k ~~= k/(k - 1)~~.		We repeatedly use the identity k/(k - 1) = S(k)(k + 1)/k.
	==== Porcupine ====		==== Porcupine ====
	The extension Porcupine = 5-limit[250/243] → 2.3.5.11[S10, S11] is endoparticular: we have		The extension Porcupine = 5-limit[250/243] → 2.3.5.11[S10, S11] is endoparticular: we have

Inthar: /* Endoparticular extensions */

2026-02-07T10:42:07Z

Endoparticular extensions

← Older revision		Revision as of 10:42, 7 February 2026
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	{{Problematic}}		{{Problematic}}
	=== Endoparticular extensions ===		=== Endoparticular extensions ===
			We repeatedly use the identity S(k)(k + 1)/k = k/(k - 1).
	==== Porcupine ====		==== Porcupine ====
	The extension Porcupine = 5-limit[250/243] → 2.3.5.11[S10, S11] is endoparticular: we have		The extension Porcupine = 5-limit[250/243] → 2.3.5.11[S10, S11] is endoparticular: we have

Inthar at 03:28, 7 February 2026

2026-02-07T03:28:41Z

← Older revision		Revision as of 03:28, 7 February 2026
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	Consider a regular temperament on a JI group.		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.		* 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 (where the extended group is between the base group and the original extended group) of a structurally induced extension is also considered structurally induced. Structural inducing has the following sub-concepts that can be defined formally:		* 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. where the extended group is between the base group and the original extended 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-superparticular]]s with nonzero exponents. A strong extension is '''endoparticular''' if it additionally tempers out the individual square-superparticulars.		** Suppose the base temperament tempers out a comma that is a product of powers of consecutive [[square-superparticular]]s 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.		** 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.

Inthar at 03:28, 7 February 2026

2026-02-07T03:28:12Z

← Older revision		Revision as of 03:28, 7 February 2026
Line 3:		Line 3:
	Consider a regular temperament on a JI group.		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.		* 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). It has the following sub-concepts that can be defined formally:		* 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 (where the extended group is between the base group and the original extended 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-superparticular]]s with nonzero exponents. A strong extension is '''endoparticular''' if it additionally tempers out the individual square-superparticulars.		** Suppose the base temperament tempers out a comma that is a product of powers of consecutive [[square-superparticular]]s 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.		** 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.

Inthar at 02:31, 7 February 2026

2026-02-07T02:31:32Z

← Older revision		Revision as of 02:31, 7 February 2026
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	Consider a regular temperament on a JI group.		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.		* 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). It has the following sub-concepts that can be defined formally:		* 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). 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-superparticular]]s with nonzero exponents. A strong extension is '''endoparticular''' if it additionally tempers out the individual square-superparticulars.		** Suppose the base temperament tempers out a comma that is a product of powers of consecutive [[square-superparticular]]s with nonzero exponents. A strong extension is '''endoparticular''' if it additionally tempers out the individual square-superparticulars.

Inthar at 02:30, 7 February 2026

2026-02-07T02:30:23Z

← Older revision		Revision as of 02:30, 7 February 2026
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	** Suppose the base temperament tempers out a comma that is a product of powers of consecutive [[square-superparticular]]s with nonzero exponents. A strong extension is '''endoparticular''' if it additionally tempers out the individual square-superparticulars.		** Suppose the base temperament tempers out a comma that is a product of powers of consecutive [[square-superparticular]]s 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.		** 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.
	** {{adv\|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.}}		** {{adv\|If the commas of the extension are not necessarily superparticulars but nevertheless involve one or more [[wikipedia:arithmetic progression\|arithmetic progression]]s in the harmonic series, the extensions are similarly called ''endoarithmetic'' and ''para-arithmetic''. Endoarithmetic extensions are considered structurally induced.}}

	== Examples ==		== Examples ==

Inthar at 02:29, 7 February 2026

2026-02-07T02:29:32Z

← Older revision		Revision as of 02:29, 7 February 2026
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	** Suppose the base temperament tempers out a comma that is a product of powers of consecutive [[square-superparticular]]s with nonzero exponents. A strong extension is '''endoparticular''' if it additionally tempers out the individual square-superparticulars.		** Suppose the base temperament tempers out a comma that is a product of powers of consecutive [[square-superparticular]]s 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.		** 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.
	** {{adv\|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'', ''para-arithmetic~~'', and ''exoarithmetic~~''. Endoarithmetic extensions are considered structurally induced.}}		** {{adv\|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 ==		== Examples ==