Subgroup temperament families, relationships, and genes: Difference between revisions
→Preliminaries: link wikipedia |
ArrowHead294 (talk | contribs) mNo edit summary |
||
| Line 9: | Line 9: | ||
== 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: | ||
| Line 32: | Line 32: | ||
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. | ||
| Line 115: | Line 115: | ||
* 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 == | ||