Subgroup temperament families, relationships, and genes - Revision history

ArrowHead294 at 02:32, 26 January 2026

2026-01-26T02:32:07Z

← Older revision		Revision as of 02:32, 26 January 2026
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	== Support ==		== Support ==
	Given two subgroup temperaments A and B, temperament A is said to ''support'' temperament B if and only if:		Given two subgroup temperaments ''A'' and ''B'', temperament ''A'' is said to ''support'' temperament ''B'' if and only if:
	* ~~Temperament~~ B's JI subgroup is a "sub-subgroup" of ~~temperament~~ A's JI subgroup		* ''B''{{'s}} JI subgroup is a "sub-subgroup" of ''A''{{'s}} JI subgroup
	* ~~Temperament~~ B's kernel is a subgroup of ~~temperament~~ A's kernel		* ''B''{{'s}} kernel is a subgroup of ''A''{{'s}} kernel

	For instance, the 11-limit 22p patent val, treated as a subgroup temperament, is 2.3.5.7.11 {{val\|22 35 51 62 76}}. This temperament supports all of the following other subgroup temperaments:		For instance, the 11-limit 22p patent val, treated as a subgroup temperament, is 2.3.5.7.11 {{val\|22 35 51 62 76}}. This temperament supports all of the following other subgroup temperaments:
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	Expansions are very similar to extensions—probably the more important notion, which is given below—but where the rank is permitted to increase. So 2.3.5.7 {{nowrap\|81/80 & 126/125}} is an expansion of 2.3.5 81/80 (and an extension), but 2.3.5.7 81/80 is also an expansion of 2.3.5 81/80 (which is not an extension).		Expansions are very similar to extensions—probably the more important notion, which is given below—but where the rank is permitted to increase. So 2.3.5.7 {{nowrap\|81/80 & 126/125}} is an expansion of 2.3.5 81/80 (and an extension), but 2.3.5.7 81/80 is also an expansion of 2.3.5 81/80 (which is not an extension).

	If ''A'' is an expansion of ''B'', then ''B'' clearly "supports" ''A''. The rank of ''A'' is also greater than or equal to the rank of ''B''. However, these two properties are not sufficient to support an expansion; ''B'' must also be the (unique) retraction of ''A'' to ''B''{{'}}s subgroup. For instance, the rank-3 2.3.5.7.11 {{nowrap\|81/80 & 128/125}} temperament, which is basically 5-limit 12p with two additional "independent" generators for 7/1 and 11/1, is ''not'' an expansion of 2.3.5 81/80, because if you retract 2.3.5.7.11 {{nowrap\|81/80 & 128/125}} to the 2.3.5 subgroup you get 2.3.5 {{nowrap\|81/80 & 128/125}} rather than 2.3.5 81/80. Put another way, 2.3.5 {{nowrap\|81/80 & 128/125}} (12p) also isn't an "expansion" of 2.3.5 81/80 at all.		If ''A'' is an expansion of ''B'', then ''B'' clearly "supports" ''A''. The rank of ''A'' is also greater than or equal to the rank of ''B''. However, these two properties are not sufficient to support an expansion; ''B'' must also be the (unique) retraction of ''A'' to ''B''{{'s}} subgroup. For instance, the rank-3 2.3.5.7.11 {{nowrap\|81/80 & 128/125}} temperament, which is basically 5-limit 12p with two additional "independent" generators for 7/1 and 11/1, is ''not'' an expansion of 2.3.5 81/80, because if you retract 2.3.5.7.11 {{nowrap\|81/80 & 128/125}} to the 2.3.5 subgroup you get 2.3.5 {{nowrap\|81/80 & 128/125}} rather than 2.3.5 81/80. Put another way, 2.3.5 {{nowrap\|81/80 & 128/125}} (12p) also isn't an "expansion" of 2.3.5 81/80 at all.

	We can strengthen our notion of expansion and retraction to get to extension and restriction.		We can strengthen our notion of expansion and retraction to get to extension and restriction.
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	* The rank-1, codimension-0 gene spectrum is the set of "monzos" of the universe, except where monzos of different sign are identified;		* The rank-1, codimension-0 gene spectrum is the set of "monzos" of the universe, except where monzos of different sign are identified;
	* The rank-1, arbitrary-codimension gene spectrum is the set of projective "tempered monzos" of the universe		* The rank-1, arbitrary-codimension gene spectrum is the set of projective "tempered monzos" of the universe
	* The rank-n, codimension-{{nowrap\|(''n'' − 1)}} gene spectrum is the set of rank-n temperaments of the universe		* The rank-''n'', codimension-{{nowrap\|(''n'' − 1)}} gene spectrum is the set of rank-''n'' temperaments of the universe

	== Partial order ==		== Partial order ==

Sintel: /* Preliminaries */ link wikipedia

2025-04-23T20:53:29Z

Preliminaries: link wikipedia

← Older revision		Revision as of 20:53, 23 April 2025
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	== Preliminaries ==		== Preliminaries ==
	The various "subgroups" we talk about are all subgroups of the group of all possible JI intervals, which can be identified with the set of positive rational numbers, which forms an infinite-rank [[free abelian group]]. However, mathematically, it is often much easier for us to choose some arbitrarily large but still finite-rank subgroup to formalize everything in, typically the p-limit for some "large enough" choice of p. To that extent we will choose some sufficiently large JI group, which we will call the ''universe group'' or simply the ''universe'', which all the subgroups are subgroups of.		The various "subgroups" we talk about are all subgroups of the group of all possible JI intervals, which can be identified with the set of positive rational numbers, which forms an infinite-rank {{w\|free abelian group}}. However, mathematically, it is often much easier for us to choose some arbitrarily large but still finite-rank subgroup to formalize everything in, typically the p-limit for some "large enough" choice of p. To that extent we will choose some sufficiently large JI group, which we will call the ''universe group'' or simply the ''universe'', which all the subgroups are subgroups of.

	Sometimes it can be useful to choose a deliberately small universe just to see how the general system is structured. Most of the regular temperaments on the wiki fit into a 13-limit universe, with a few instances of 17 and 19 here and there. However, in theory, you can simply go as high as you want, with any sufficiently large prime-limit as the universe.		Sometimes it can be useful to choose a deliberately small universe just to see how the general system is structured. Most of the regular temperaments on the wiki fit into a 13-limit universe, with a few instances of 17 and 19 here and there. However, in theory, you can simply go as high as you want, with any sufficiently large prime-limit as the universe.

ArrowHead294: /* Support */

2025-04-01T12:33:29Z

Support

← Older revision		Revision as of 12:33, 1 April 2025
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	* 7-limit pajara: 2.3.5.7 {{nowrap\|50/49 & 64/63}}		* 7-limit pajara: 2.3.5.7 {{nowrap\|50/49 & 64/63}}
	* 2.7.9.11 machine: 2.3.7.9 {{nowrap\|64/63 & 99/98}}		* 2.7.9.11 machine: 2.3.7.9 {{nowrap\|64/63 & 99/98}}
	* 2.7.9.11 11p: 2.7.9.11 {{nowrap\|64/64 & 99/98 & 352/343}}		* 2.7.9.11 11p: 2.7.9.11 {{nowrap\|64/63 & 99/98 & 352/343}}

	In short, if temperament A supports temperament B, then any interval that appears in B also appears in A (although perhaps more heavily tempered), and any comma that vanishes in B also vanishes in A. Thus, any [[comma pump]] or [[dyadic chord\|essentially tempered chord]] that is playable in B is also playable in A.		In short, if temperament A supports temperament B, then any interval that appears in B also appears in A (although perhaps more heavily tempered), and any comma that vanishes in B also vanishes in A. Thus, any [[comma pump]] or [[dyadic chord\|essentially tempered chord]] that is playable in B is also playable in A.

FloraC: +categories

2025-04-01T11:00:58Z

+categories

← Older revision		Revision as of 11:00, 1 April 2025
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	That being said, while some of the terminology is not set in stone at all, the next step is to totally push pause on the terminology, do the real math that is needed compute this stuff, see what the structure looks like, which will clarify the terminology if need be. There is much more heavy-duty math to be posted which will be next, and which I'm sure will lead to many revisions of this.		That being said, while some of the terminology is not set in stone at all, the next step is to totally push pause on the terminology, do the real math that is needed compute this stuff, see what the structure looks like, which will clarify the terminology if need be. There is much more heavy-duty math to be posted which will be next, and which I'm sure will lead to many revisions of this.

	[[Category:~~In progress~~]]		{{Todo\|WIP}}
			[[Category:Regular temperament theory]]
			[[Category:Subgroup]]
			[[Category:Essays]]

ArrowHead294 at 12:23, 31 March 2025

2025-03-31T12:23:22Z

Show changes

Sintel: -legacy

2025-03-29T18:14:10Z

-legacy

← Older revision		Revision as of 18:14, 29 March 2025
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	~~{{Legacy}}~~
	The universe of subgroup temperaments, though quite vast, has some very interesting structure that makes it possible to better map out the relationships between subgroup temperaments and the families they fall into. We will explore some useful ways to quantify this relationship here.		The universe of subgroup temperaments, though quite vast, has some very interesting structure that makes it possible to better map out the relationships between subgroup temperaments and the families they fall into. We will explore some useful ways to quantify this relationship here.

ArrowHead294: ArrowHead294 moved page Subgroup Temperament Families, Relationships, and Genes to Subgroup temperament families, relationships, and genes: WP:NCCAPS

2025-02-24T13:59:01Z

ArrowHead294 moved page Subgroup Temperament Families, Relationships, and Genes to Subgroup temperament families, relationships, and genes: WP:NCCAPS

← Older revision	Revision as of 13:59, 24 February 2025
(No difference)

VectorGraphics at 03:42, 21 February 2025

2025-02-21T03:42:19Z

← Older revision		Revision as of 03:42, 21 February 2025
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	= Preliminaries =		= Preliminaries =
	The various "subgroups" we talk about are all subgroups of the group of all possible JI intervals, which can be identified with the set of positive rational numbers, which forms an infinite-rank free abelian group. However, mathematically, it is often much easier for us to choose some arbitrarily large but still finite-rank subgroup to formalize everything in, typically the p-limit for some "large enough" choice of p. To that extent we will choose some sufficiently large JI group, which we will call the ''universe group'' or simply the ''universe'', which all the subgroups are subgroups of.		The various "subgroups" we talk about are all subgroups of the group of all possible JI intervals, which can be identified with the set of positive rational numbers, which forms an infinite-rank [[free abelian group]]. However, mathematically, it is often much easier for us to choose some arbitrarily large but still finite-rank subgroup to formalize everything in, typically the p-limit for some "large enough" choice of p. To that extent we will choose some sufficiently large JI group, which we will call the ''universe group'' or simply the ''universe'', which all the subgroups are subgroups of.

	Sometimes it can be useful to choose a deliberately small universe just to see how the general system is structured. Most of the regular temperaments on the wiki fit into a 13-limit universe, with a few instances of 17 and 19 here and there. However, in theory, you can simply go as high as you want, with any sufficiently large prime-limit as the universe.		Sometimes it can be useful to choose a deliberately small universe just to see how the general system is structured. Most of the regular temperaments on the wiki fit into a 13-limit universe, with a few instances of 17 and 19 here and there. However, in theory, you can simply go as high as you want, with any sufficiently large prime-limit as the universe.

Lériendil at 16:54, 17 February 2025

2025-02-17T16:54:26Z

← Older revision		Revision as of 16:54, 17 February 2025
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	{{Legacy~~\|Subgroup_Temperament_Families,_Relationships,_and_Genes~~}}		{{Legacy}}
	The universe of subgroup temperaments, though quite vast, has some very interesting structure that makes it possible to better map out the relationships between subgroup temperaments and the families they fall into. We will explore some useful ways to quantify this relationship here.		The universe of subgroup temperaments, though quite vast, has some very interesting structure that makes it possible to better map out the relationships between subgroup temperaments and the families they fall into. We will explore some useful ways to quantify this relationship here.

Lériendil: deploying new cat

2025-02-17T16:08:43Z

deploying new cat

← Older revision		Revision as of 16:08, 17 February 2025
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			{{Legacy\|Subgroup_Temperament_Families,_Relationships,_and_Genes}}
	The universe of subgroup temperaments, though quite vast, has some very interesting structure that makes it possible to better map out the relationships between subgroup temperaments and the families they fall into. We will explore some useful ways to quantify this relationship here.		The universe of subgroup temperaments, though quite vast, has some very interesting structure that makes it possible to better map out the relationships between subgroup temperaments and the families they fall into. We will explore some useful ways to quantify this relationship here.

← Older revision		Revision as of 12:33, 1 April 2025
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	* 7-limit pajara: 2.3.5.7 {{nowrap\|50/49 & 64/63}}		* 7-limit pajara: 2.3.5.7 {{nowrap\|50/49 & 64/63}}
	* 2.7.9.11 machine: 2.3.7.9 {{nowrap\|64/63 & 99/98}}		* 2.7.9.11 machine: 2.3.7.9 {{nowrap\|64/63 & 99/98}}
	* 2.7.9.11 11p: 2.7.9.11 {{nowrap\|64/64 & 99/98 & 352/343}}		* 2.7.9.11 11p: 2.7.9.11 {{nowrap\|64/63 & 99/98 & 352/343}}

	In short, if temperament A supports temperament B, then any interval that appears in B also appears in A (although perhaps more heavily tempered), and any comma that vanishes in B also vanishes in A. Thus, any [[comma pump]] or [[dyadic chord\|essentially tempered chord]] that is playable in B is also playable in A.		In short, if temperament A supports temperament B, then any interval that appears in B also appears in A (although perhaps more heavily tempered), and any comma that vanishes in B also vanishes in A. Thus, any [[comma pump]] or [[dyadic chord\|essentially tempered chord]] that is playable in B is also playable in A.