Subgroup temperament families, relationships, and genes: Difference between revisions
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Many of the things we have talked about fit into the perspective of gene spectra. If r is the rank of the universe, then we have | Many of the things we have talked about fit into the perspective of gene spectra. If r is the rank of the universe, then we have | ||
* The codimension-1, rank-(r-1) gene spectrum is the set of "commas" of the subgroup, in which we don't care about the generators but only stuff like comma pumps; | |||
* The contorted codimension-0 genes are equivalent to the "pergens", in which we don't care about commas but only the generators; | |||
* The codimension-1 gene spectra of various ranks correspond to Gene's "clans", in which we care about both generators and commas; | |||
So that these are all special cases of gene spectra, each of which provides a unique family structure and way to view the temperaments of the universe. | So that these are all special cases of gene spectra, each of which provides a unique family structure and way to view the temperaments of the universe. | ||
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If we drop the interpretation of genes as representing temperaments, but simply treat them as mathematical entities which are subquotients of the universe, we also get: | If we drop the interpretation of genes as representing temperaments, but simply treat them as mathematical entities which are subquotients of the universe, we also get: | ||
* 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-n, codimension-(n-1) gene spectrum is the set of rank-n temperaments of the universe | |||
= Partial Order = | = Partial Order = | ||
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= Mike's In-Progress Thoughts (to be edited later) = | = Mike's In-Progress Thoughts (to be edited later) = | ||
This page is in-progress. The above summarizes much of the talk about subgroup extensions, restrictions, universes, and families that has been posted the past 10 years, much of it on Facebook and offlist. There is still plenty more, but this gives the basic idea and structure. Most of the terminology here (e.g. "support", "extension", "restriction", "family") has been in use for years. There is also terminology like "expansion" and "retraction" which, at this point, is pretty old (a group of us came up with that back in 2013), but hasn't been talked about all that much or posted on the wiki. And much of the terminology about genes is quite new, and I'm still kind of waiting to see where some of it goes - in particular, while I've been focusing on the interpretation of a "gene" as the origin of a family, i.e. generalizing comma, but there may be some use in taking a really abstract birds-eye view as just an arbitrary subquotient of the universe, i.e. generalizing monzo. | This page is in-progress. The above summarizes much of the talk about subgroup extensions, restrictions, universes, and families that has been posted the past 10 years, much of it on Facebook and offlist. There is still plenty more, but this gives the basic idea and structure. Most of the terminology here (e.g. "support", "extension", "restriction", "family") has been in use for years. There is also terminology like "expansion" and "retraction" which, at this point, is pretty old (a group of us came up with that back in 2013), but hasn't been talked about all that much or posted on the wiki. And much of the terminology about genes is quite new, and I'm still kind of waiting to see where some of it goes - in particular, while I've been focusing on the interpretation of a "gene" as the origin of a family, i.e. generalizing comma, but there may be some use in taking a really abstract birds-eye view as just an arbitrary subquotient of the universe, i.e. generalizing monzo. Similarly, while genes are currently "projective" objects (kind of), | ||
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]] | [[Category:in-progress]] | ||