Subgroup temperament families, relationships, and genes: Difference between revisions
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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 | 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. | ||
== 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: | ||
* | * ''B''{{'s}} JI subgroup is a "sub-subgroup" of ''A''{{'s}} JI subgroup | ||
* | * ''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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* 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/ | * 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. | ||
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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''{{'}} | 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 == | ||
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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: | {{Todo|WIP}} | ||
[[Category:Regular temperament theory]] | |||
[[Category:Subgroup]] | |||
[[Category:Essays]] | |||