Sane and insane temperaments - Revision history

Sintel: -legacy

2025-03-29T18:13:31Z

-legacy

← Older revision		Revision as of 18:13, 29 March 2025
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	~~{{Legacy}}~~
	Given any subgroup temperament, we can define the restriction of that temperament to a smaller subgroup. If such a restriction does not change the generators, it is called a '''strong restriction''', otherwise it is called a '''weak restriction'''. Another way to state this criterion is that if you take the [[Temperament_Mapping_Matrices_(M-maps)\|mapping matrix]] for your larger temperament, and multiply it by the [[Subgroup_Mapping_Matrices_(V-maps)\|subgroup matrix]] for your subgroup in question, that the result is not [[contorted]].		Given any subgroup temperament, we can define the restriction of that temperament to a smaller subgroup. If such a restriction does not change the generators, it is called a '''strong restriction''', otherwise it is called a '''weak restriction'''. Another way to state this criterion is that if you take the [[Temperament_Mapping_Matrices_(M-maps)\|mapping matrix]] for your larger temperament, and multiply it by the [[Subgroup_Mapping_Matrices_(V-maps)\|subgroup matrix]] for your subgroup in question, that the result is not [[contorted]].

Lériendil at 16:55, 17 February 2025

2025-02-17T16:55:37Z

← Older revision		Revision as of 16:55, 17 February 2025
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	{{Legacy~~\|Sane_and_insane_temperaments~~}}		{{Legacy}}
	Given any subgroup temperament, we can define the restriction of that temperament to a smaller subgroup. If such a restriction does not change the generators, it is called a '''strong restriction''', otherwise it is called a '''weak restriction'''. Another way to state this criterion is that if you take the [[Temperament_Mapping_Matrices_(M-maps)\|mapping matrix]] for your larger temperament, and multiply it by the [[Subgroup_Mapping_Matrices_(V-maps)\|subgroup matrix]] for your subgroup in question, that the result is not [[contorted]].		Given any subgroup temperament, we can define the restriction of that temperament to a smaller subgroup. If such a restriction does not change the generators, it is called a '''strong restriction''', otherwise it is called a '''weak restriction'''. Another way to state this criterion is that if you take the [[Temperament_Mapping_Matrices_(M-maps)\|mapping matrix]] for your larger temperament, and multiply it by the [[Subgroup_Mapping_Matrices_(V-maps)\|subgroup matrix]] for your subgroup in question, that the result is not [[contorted]].

Lériendil: deploying new cat

2025-02-17T16:11:03Z

deploying new cat

← Older revision		Revision as of 16:11, 17 February 2025
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			{{Legacy\|Sane_and_insane_temperaments}}
	Given any subgroup temperament, we can define the restriction of that temperament to a smaller subgroup. If such a restriction does not change the generators, it is called a '''strong restriction''', otherwise it is called a '''weak restriction'''. Another way to state this criterion is that if you take the [[Temperament_Mapping_Matrices_(M-maps)\|mapping matrix]] for your larger temperament, and multiply it by the [[Subgroup_Mapping_Matrices_(V-maps)\|subgroup matrix]] for your subgroup in question, that the result is not [[contorted]].		Given any subgroup temperament, we can define the restriction of that temperament to a smaller subgroup. If such a restriction does not change the generators, it is called a '''strong restriction''', otherwise it is called a '''weak restriction'''. Another way to state this criterion is that if you take the [[Temperament_Mapping_Matrices_(M-maps)\|mapping matrix]] for your larger temperament, and multiply it by the [[Subgroup_Mapping_Matrices_(V-maps)\|subgroup matrix]] for your subgroup in question, that the result is not [[contorted]].

Lériendil: Changed protection level for "Sane and insane temperaments" ([Edit=Allow only administrators] (expires 17:01, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:01, 17 February 2025 (UTC)))

2025-02-17T16:01:20Z

Changed protection level for "Sane and insane temperaments" ([Edit=Allow only administrators] (expires 17:01, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:01, 17 February 2025 (UTC)))

← Older revision	Revision as of 16:01, 17 February 2025
(No difference)

FloraC: +categories

2021-11-18T14:11:34Z

+categories

← Older revision		Revision as of 14:11, 18 November 2021
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	The 2.9.5 restriction of 2.3.5 81/80 (meantone) is a weak restriction that is not insane. The resulting temperament is generated by the tempered 2/1 and 9/8. The original subgroup of 2.9.5 contains 9/1 as an interval but not 3/1. However, since 9/1 is a generator of this temperament, there is no way to "split" it further to obtain an unmapped 3/1, so it is not insane. The kernel is 81/80, which is a saturated lattice of the 5-limit, so we have a sane temperament.		The 2.9.5 restriction of 2.3.5 81/80 (meantone) is a weak restriction that is not insane. The resulting temperament is generated by the tempered 2/1 and 9/8. The original subgroup of 2.9.5 contains 9/1 as an interval but not 3/1. However, since 9/1 is a generator of this temperament, there is no way to "split" it further to obtain an unmapped 3/1, so it is not insane. The kernel is 81/80, which is a saturated lattice of the 5-limit, so we have a sane temperament.

			[[Category:Regular temperament theory]]
			[[Category:Math]]

FloraC: FloraC moved page Sane and Insane Temperaments to Sane and insane temperaments: WP:NCCAPS

2021-11-18T13:48:24Z

FloraC moved page Sane and Insane Temperaments to Sane and insane temperaments: WP:NCCAPS

← Older revision	Revision as of 13:48, 18 November 2021
(No difference)

Mike Battaglia at 04:14, 16 November 2021

2021-11-16T04:14:33Z

← Older revision		Revision as of 04:14, 16 November 2021
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	Note that the GCD criterion above does not change no matter which mapping matrix (and hence, set of generators) you choose for the temperament.		Note that the GCD criterion above does not change no matter which mapping matrix (and hence, set of generators) you choose for the temperament.

	It turns out that an equivalent definition is that a temperament is insane iff its kernel is [[~~Saturation~~\|unsaturated]], when expressed as a subgroup of the full-limit. This is the same problem that would typically lead to torsion if tempered out of the full-limit. Torsion can be gotten rid of by restriction the temperament to a smaller subgroup, but if you do so, you are instead guaranteed to get an insane temperament.		It turns out that an equivalent definition is that a temperament is insane iff its kernel is [[Mathematical theory of saturation\|unsaturated]], when expressed as a subgroup of the full-limit. This is the same problem that would typically lead to torsion if tempered out of the full-limit. Torsion can be gotten rid of by restriction the temperament to a smaller subgroup, but if you do so, you are instead guaranteed to get an insane temperament.

	Some further examples:		Some further examples:

Mike Battaglia: Protected "Sane and Insane Temperaments" ([Edit=Allow only administrators] (indefinite) [Move=Allow only administrators] (indefinite))

2021-11-15T03:07:11Z

Protected "Sane and Insane Temperaments" ([Edit=Allow only administrators] (indefinite) [Move=Allow only administrators] (indefinite))

← Older revision	Revision as of 03:07, 15 November 2021
(No difference)

Cmloegcmluin: remove temporary link to defactoring

2021-11-12T23:06:43Z

remove temporary link to defactoring

← Older revision		Revision as of 23:06, 12 November 2021
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	Note that the GCD criterion above does not change no matter which mapping matrix (and hence, set of generators) you choose for the temperament.		Note that the GCD criterion above does not change no matter which mapping matrix (and hence, set of generators) you choose for the temperament.

	It turns out that an equivalent definition is that a temperament is insane iff its kernel is [[Saturation\|unsaturated]] ~~(or [[enfactored]])~~, when expressed as a subgroup of the full-limit. This is the same problem that would typically lead to torsion if tempered out of the full-limit. Torsion can be gotten rid of by restriction the temperament to a smaller subgroup, but if you do so, you are instead guaranteed to get an insane temperament.		It turns out that an equivalent definition is that a temperament is insane iff its kernel is [[Saturation\|unsaturated]], when expressed as a subgroup of the full-limit. This is the same problem that would typically lead to torsion if tempered out of the full-limit. Torsion can be gotten rid of by restriction the temperament to a smaller subgroup, but if you do so, you are instead guaranteed to get an insane temperament.

	Some further examples:		Some further examples:

Cmloegcmluin: remove temporary link to defactoring

2021-11-12T22:59:07Z

remove temporary link to defactoring

← Older revision		Revision as of 22:59, 12 November 2021
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	Given any subgroup temperament, we can define the restriction of that temperament to a smaller subgroup. If such a restriction does not change the generators, it is called a '''strong restriction''', otherwise it is called a '''weak restriction'''. Another way to state this criterion is that if you take the [[Temperament_Mapping_Matrices_(M-maps)\|mapping matrix]] for your larger temperament, and multiply it by the [[Subgroup_Mapping_Matrices_(V-maps)\|subgroup matrix]] for your subgroup in question, that the result is not [[contorted]] ~~(or in other words, the result is [[defactored]])~~.		Given any subgroup temperament, we can define the restriction of that temperament to a smaller subgroup. If such a restriction does not change the generators, it is called a '''strong restriction''', otherwise it is called a '''weak restriction'''. Another way to state this criterion is that if you take the [[Temperament_Mapping_Matrices_(M-maps)\|mapping matrix]] for your larger temperament, and multiply it by the [[Subgroup_Mapping_Matrices_(V-maps)\|subgroup matrix]] for your subgroup in question, that the result is not [[contorted]].

	For example, the 2.3.11 restriction of 11-limit [[Mohajira]] is the 2.3.11 243/242 temperament, and a strong restriction since the generator of 11/9 does not change. The 2.3.5 restriction of 11-limit [[Meantone]] is the 2.3.5 81/80 temperament, which is a "weak" restriction since the generator changes (using the original generator leads to a contorted mapping).		For example, the 2.3.11 restriction of 11-limit [[Mohajira]] is the 2.3.11 243/242 temperament, and a strong restriction since the generator of 11/9 does not change. The 2.3.5 restriction of 11-limit [[Meantone]] is the 2.3.5 81/80 temperament, which is a "weak" restriction since the generator changes (using the original generator leads to a contorted mapping).