Subgroup temperament families, relationships, and genes: Difference between revisions
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In general, every temperament is part of many subgroup families, much like real people. For instance, the 2.3.5.7 64/63 & 81/80 temperament (dominant) is in both the 2.3.5 81/80 (meantone) and the 2.3.7 64/63 (archy) families. This is a feature, not a bug, and simply reflects the mathematical reality of the structure of the subgroup temperament universe. | In general, every temperament is part of many subgroup families, much like real people. For instance, the 2.3.5.7 64/63 & 81/80 temperament (dominant) is in both the 2.3.5 81/80 (meantone) and the 2.3.7 64/63 (archy) families. This is a feature, not a bug, and simply reflects the mathematical reality of the structure of the subgroup temperament universe. | ||
= Genes and Temperament | = Genes and Temperament Gene Spectra = | ||
''Note from Mike: these are some very preliminary ideas and results from the research we've done on this, just to get started and illustrate the general direction. I will add to this later...'' | ''Note from Mike: these are some very preliminary ideas and results from the research we've done on this, just to get started and illustrate the general direction. I will add to this later...'' | ||
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This is closely related to simply looking for the most important subgroup temperaments across all subgroups (rather than only within one subgroup). There are a few different ways to do this, but many of them tend to give the basic result of ranking very highly those relatively simple, but powerful temperaments that tend to have lots of important extensions. For now, we will identify any subgroup family with its origin, making the somewhat simplifying assumption that any good family has a good origin. | This is closely related to simply looking for the most important subgroup temperaments across all subgroups (rather than only within one subgroup). There are a few different ways to do this, but many of them tend to give the basic result of ranking very highly those relatively simple, but powerful temperaments that tend to have lots of important extensions. For now, we will identify any subgroup family with its origin, making the somewhat simplifying assumption that any good family has a good origin. | ||
A ''gene'' is a very simple subgroup temperament which is supported by some other subgroup temperament. The ''dimension'' or ''rank'' of a gene is simply its rank as a temperament, and the ''codimension'' is simply the rank of its kernel. The dimension and codimension of a gene form its ''signature''. If the restriction of the temperament to one of its genes doesn't change the generators, it's called a ''strong gene'', if it does it's a ''weak gene''. The '' | A ''gene'' is a very simple subgroup temperament which is supported by some other subgroup temperament. The ''dimension'' or ''rank'' of a gene is simply its rank as a temperament, and the ''codimension'' is simply the rank of its kernel. The dimension and codimension of a gene form its ''signature''. If the restriction of the temperament to one of its genes doesn't change the generators, it's called a ''strong gene'', if it does it's a ''weak gene''. The ''gene spectrum'' of a temperament is the set of genes that it supports. Genes are named in memory of Gene Smith, one of the main originators of most of the theory on this wiki (and its largest contributor by an enormous margin), who sadly passed in January of 2021 from COVID-19. | ||
The most important genes are codimension-1. A codimension-1 gene is just the pairing of one subgroup and a comma tempered out on that subgroup and is called a ''base pair.'' It is fairly natural to extend the definition of genes to codimension-2, codimension-3, etc, but for now I will primarily focus on codimension-1 below. | The most important genes are codimension-1. A codimension-1 gene is just the pairing of one subgroup and a comma tempered out on that subgroup and is called a ''base pair.'' It is fairly natural to extend the definition of genes to codimension-2, codimension-3, etc, but for now I will primarily focus on codimension-1 below. | ||
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However, the thing is, although the above criterion is sufficient, it isn't necessary. This is because if you look at subgroups that are generated by simple 'chords', you tend to get a larger list of subgroups than if you only look at those generated by simple 'dyads.' One good example is 2.3.7.13/5, which looks strange, but happens to be generated by 6:7:9, 10:13:15, and 2/1. Thus the comma 91/90, whose monzo representation is {{monzo|-1 2 -1 1 0 1}}, has five nonzero coefficients and would thus seem to naturally lead to only good rank-4 genes, but it so happens that the 2.3.7.13/5 91/90 temperament is very interesting - it's basically 7-limit JI, but where the inverse of 6:7:9 is tempered equal to 10:13:15 rather than 14:18:21, and other ways of ranking subgroup temperaments that look at chords rather than just dyads will reveal some "hidden gems" in this way which aren't immediately visible from the dyadic sparsity-based method. | However, the thing is, although the above criterion is sufficient, it isn't necessary. This is because if you look at subgroups that are generated by simple 'chords', you tend to get a larger list of subgroups than if you only look at those generated by simple 'dyads.' One good example is 2.3.7.13/5, which looks strange, but happens to be generated by 6:7:9, 10:13:15, and 2/1. Thus the comma 91/90, whose monzo representation is {{monzo|-1 2 -1 1 0 1}}, has five nonzero coefficients and would thus seem to naturally lead to only good rank-4 genes, but it so happens that the 2.3.7.13/5 91/90 temperament is very interesting - it's basically 7-limit JI, but where the inverse of 6:7:9 is tempered equal to 10:13:15 rather than 14:18:21, and other ways of ranking subgroup temperaments that look at chords rather than just dyads will reveal some "hidden gems" in this way which aren't immediately visible from the dyadic sparsity-based method. | ||
== Contorted | == Contorted Gene Spectra and Pergens == | ||
Many of the ideas above can be easily adapted to drop the restriction that we have to remove contorsion. For instance, we could simply say that the restriction of mohajira to the 2.3.5 subgroup simply is the contorted meantone which splits the fifth in half. We will call this the ''contorted restriction'' of mohajira to the 2.3.5 subgroup. | Many of the ideas above can be easily adapted to drop the restriction that we have to remove contorsion. For instance, we could simply say that the restriction of mohajira to the 2.3.5 subgroup simply is the contorted meantone which splits the fifth in half. We will call this the ''contorted restriction'' of mohajira to the 2.3.5 subgroup. | ||
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Notably, if you 'do' get a contorted subgroup JI, what you've gotten is basically a [[Pergen]] for that temperament with respect to the smaller subgroup, just without the usual pergen notation (and for any arbitrary set of primes or generating intervals at all). The mapping matrix you get from doing the above matrix notation is basically another notation for the Pergen of that temperament on that subgroup. | Notably, if you 'do' get a contorted subgroup JI, what you've gotten is basically a [[Pergen]] for that temperament with respect to the smaller subgroup, just without the usual pergen notation (and for any arbitrary set of primes or generating intervals at all). The mapping matrix you get from doing the above matrix notation is basically another notation for the Pergen of that temperament on that subgroup. | ||
In general, we can look at ''contorted genes'' of various signatures that temperaments can support, and we can look at the ''contorted | In general, we can look at ''contorted genes'' of various signatures that temperaments can support, and we can look at the ''contorted gene spectra'' they can support as well. If our temperament is r-dimensional, then the genes also of dimension-r and codimension-0 all correspond to the various possible pergens for the temperament on every possible supporting subgroup (which we may call the ''pergen spectrum''). | ||
== The Gene Spectrum of the JI Universe == | == The Gene Spectrum of the JI Universe == | ||