Subgroup Temperament Families, Relationships, and Genes

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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.

In all these examples, we can unambiguously denote a subgroup temperament in a variety of ways, one of which is to first specify the subgroup and then specify a comma basis for the kernel. For instance, meantone is 2.3.5 81/80, archy is 2.3.7 64/63, machine is 2.7.9.11 64/63 & 99/98, and so on. We will use this notation quite frequently.

Preliminaries

The various "subgroups" we talk about are all subgroups of the group of JI intervals, which is the free abelian group of strictly positive multiplicative rationals. However, mathematically, it is often much easier for us to choose some arbitrarily large but finitely-generated subgroup to formalize everything in. 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.

Support

Given two subgroup temperaments A and B, temperament A is said to support temperament B if and only if:

  • Temperament B's subgroup is a "sub-subgroup" of temperament A's subgroup
  • Temperament B's kernel is a subgroup of temperament A's kernel

For instance, the 11-limit 22p patent val, treated as a subgroup temperament, is 2.3.5.7.11 22 35 51 62 76]. This temperament supports all of the following other subgroup temperaments:

  • 5-limit porcupine: 2.3.5 250/243
  • 7-limit pajara: 2.3.5.7 50/49 * 64/63
  • 2.7.9.11 machine: 2.3.7.9 64/63 & 99/98
  • 2.7.9.11 11p: 2.7.9.11 64/64 & 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 essentially tempered chord that is playable in B is also playable in A.

This is most often looked at in the setting of equal temperaments supporting various higher-rank temperaments, but the same principle can also be used for rank-2 temperaments supporting rank-3 and so on. This perspective often arises when treating some rank-3 temperament as the starting point from which further temperaments are derived, as though it were JI. One good example is the 2.3.7.13/5 91/90 "Biome" temperament, which, although it seems to be on a very strange subgroup at first glance, happens to be the unique temperament that is generated by the chords 6:7:9 and 10:13:15 (and 2/1), and tempers things such that 6:7:9 and 10:13:15 are inverses of one another (e.g. 10:13:15 is equated to 14:18:21), typically by widening the supermajor third somewhat to be closer to 13/10 (which is perhaps useful as 10:13:15 is perhaps slightly more concordant than 14:18:21). The resulting lattice is very similar to 5-limit JI - it is a rank-3 lattice, for instance, with "Fokker blocks" and everything, except where the 5/4 and 6/5 are instead replaced with these tempered subminor and supermajor/ultramajor thirds. If we simply use this temperament as a starting point, we can derive additional temperaments that "support" this one simply by tempering further.

Although "supporting" relationships between temperaments are very useful, it turns out that we can go much farther. We can strengthen this notion to get subgroup "extensions" and "restrictions" - the main idea in our organizing of the subgroup temperament universe - although we will first define "expansions" and "retractions" as an intermediate step just to make extensions easier to explain.

Expansions and Retractions

Suppose that we have some temperament A on some subgroup S. If R is a "sub-subgroup" of S, we can always take the tempered intervals of A and remove all interpretations except for those in R. We can also then drop any intervals which are left with no mapping at all, so that the result is another true temperament (e.g. remove contorsion and rank-deficiencies), although it can also be useful to look at the situation where we leave contorsion in. If we do remove contorsion, then a basically equivalent perspective is to take the kernel of A and remove all commas which aren't also in R, i.e. to take the intersection of A with R. So if our original temperament was [math]A = S/K[/math] (meaning subgroup S mod kernel K), our new temperament is [math]B = R/(K \int R)[/math].

We can do this for any subgroup temperament A and new subgroup R. The new temperament we obtain in this way is called the retraction of A to the subgroup R, and A is said to be an expansion of B. Note that while the retraction is always unique, in general there are many expansions of B to any larger subgroup.

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 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 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 81/80 & 128/125 to the 2.3.5 subgroup you get 2.3.5 81/80 & 128/125 rather than 2.3.5 81/80. Put another way, 2.3.5 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.

Extensions and Restrictions

Suppose that we have some subgroup temperament A for which temperament B is a retraction to some subgroup. If B is the same rank as A, then it is said to be a restriction of A, and A is said to be an extension of B. We will again assume contorsion is being removed.

Restrictions (and retractions) can be easily computed using Subgroup basis matrices. For instance, if [math]M[/math] is some mapping matrix for the temperament [math]A[/math], and [math]N[/math] is a subgroup basis matrix representing some basis for [math]R[/math] (expressed in the coordinate system of [math]S[/math]), then we can easily compute the restriction of [math]A[/math] to [math]R[/math] by first taking the matrix product

[math] M \cdot S [/math]

If the resulting matrix isn't rank-deficient, then it represents a restriction. If it is rank-deficient, then it represents a retraction. (Rank-deficiency can be corrected using either the Hermite normal form, which will leave one or more rows of zeros at the bottom of the normalized matrix that can be removed.)

Even if the resulting matrix isn't rank-deficient, it may still be contorted. For instance, the restriction of 2.3.5.7.11 81/80 & 121/120 & 126/125 (11-limit Mohajira) to the 2.3.5 subgroup yields a contorted mapping matrix in which the perfect fifth is split in half to an "unmapped" neutral third. This contortion can be removed (Smith Normal Form can be useful for this) to yield the relevant restriction.

Strong and Weak Extensions/Expansions

Above, we saw that some temperaments will have contorsion when naively restricted to a smaller subgroup, so that if the contorsion is removed the generators change. Put another way, some temperament extensions cause the generators to split, and some don't.

If some temperament extension doesn't cause the generators to split, it is called a strong extension, and if it does, it's called a weak extension. So mohajira is a weak extension of meantone, whereas meanpop is a strong extension. Likewise, meantone is a weak extension of the 2.9.5 81/80 temperament. Likewise, if A is a "strong extension" of B, then B is a "strong restriction" of A, or else it is a "weak restriction."

A temperament extension is strong if and only if the matrix multiplication above yields an uncontorted matrix.

The same definition applies for strong and weak expansions and strong and weak retractions.

Families

Given some temperament, we can look at the set of all extensions of that temperament. This set is called the family generated by that temperament. The temperament is called the origin of the family.

If we look at strong extensions only, then we have the strong family or immediate family of the temperament. If we also look at the weak extensions, we have the general family or extended family (or just "family.")

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 Genomes

'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...'

We can systematically search for the most important subgroup temperament families - with either strong extensions only, or both strong and weak extensions - within the universe.

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 genome 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.

Genes, and base pairs in particular, are meant to generalize commas, which are ratios tempered out by some temperament. Any subgroup temperament could hypothetically be a gene, much like any ratio of any kind could hypothetically be a comma, but in both situations the term has the added interpretation of suggesting that the "gene" or "comma" is particularly musically useful as the nucleation point for a subgroup family, i.e. it is low-complexity, low-error, and in this case, on a relatively simple subgroup. So much like 81/80, 64/63, and 243/242 are JI intervals that make for good commas, we have that 2.3.5 81/80, 2.3.7 64/63, 2.3.11 243/242 are subgroup temperaments that make for good genes.

Note that being a good gene is quite a bit stronger criterion than being a good comma. For instance, if we are only looking at rank-2 genes, note that the comma 64/63 does very well, with at least two good genes right away - 2.3.7 64/63 and 2.7.9 64/63. The "simpler" comma 36/35, on the other hand, doesn't lead to any good rank-2 genes at all. This is because there simply is not any decent rank-3 subgroup that has 36/35 in it to begin with; the simplest is 2.3.35. Another instance is 1029/1024, which does very well on the 2.3.7 subgroup with 2.3.7 1029/1024 (slendric), whereas the simplest rank-2 genes for the much "simpler" comma 126/125 are 14.3.5 126/125 and 2.5.63 126/125. Put another way, 126/125 really is kind of a rank-3 comma; the 2.3.5.7 126/125 (starling) temperament is very good, but it simply doesn't do well in the rank-2 setting unless some other comma is also added.

Sparsity

As a very coarse rule of thumb - one which is not perfect, but an interesting starting point - the commas which tend to lead to good genes are those which are "sparse"; e.g. whose monzos have few nonzero coefficients. 1029/1024 is [-10 1 0 3, with the 5's coefficient being zero, so we know that we can drop it and get some interesting genes from this comma on the 2.3.7 subgroup. 36/35 is [2 2 -1 -1, though, so there's no simple subgroup like that. In general, if a comma is n-sparse, e.g. it has only n nonzero coefficients, it will tend to lead to good rank-(n-1) genes (as long as the nonzero coefficients are for relatively simple primes).

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 [-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 Genomes 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.

We can take this idea further and look at the contorted restriction of mohajira to the 2.3 subgroup, which yields a contorted version of 2.3 JI in which the 3/2 is split in half. Likewise, the contorted restriction of porcupine to the 2.3 subgroup is the contorsion of 2.3 JI in which the 4/3 is split into three parts.

In general, given any rank-r subgroup temperament on subgroup S, we can look at its contorted restriction to any other JI subgroup T that is:

  1. a "sub-subgroup" of S
  2. of the same rank-r as the subgroup temperament

For instance, marvel is a rank-3 temperament on the 2.3.5.7 subgroup, so we can look at its contorted restriction to any other rank-3 JI subgroup that is a subgroup of 2.3.5.7. If we do this with the mapping matrix, the restriction will either be a contorted subgroup JI similar to the above, or it will be a rank-deficient matrix (e.g. a lower-rank temperament). For instance, the contorted restriction of 2.3.5 128/125 (a rank-2 temperament) to the 2.3 subgroup yields a contorted JI in which the octave is split into three equal parts, but the contorted restriction of 2.3.5 128/125 to the 2.5 subgroup doesn't yield any contorted subgroup JI at all (it's just the 2.5 128/125 temperament).

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 genomes 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 pergenome).

The Gene Spectrum of the JI Universe

In general, for any JI uninverse that the various subgroups are drawn from, we can choose any gene signature (or subset of signatures) and look at the set of all genes within the universe which have that signature. This is called the gene spectrum of that signature. The set of all possible gene spectra is the gene spectrum of the universe.

Again, while technically the gene spectrum of a signature is just the set of all r-dimensional and c-codimensional subgroup temperaments in the universe - much like a "comma" could potentially be any rational number at all - the interpretation is that good genes are supposed to be the origin points of good subgroup families, and that there is also some kind of associated score or ranking measuring how good they are, and that we only really care about the best ones.

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.

Note there is some nuance to the interpretations above - with the "commas" we typically interpret those as the origins of "extended families," whereas with "pergens" we typically interpret those as the origins of "immediate families" only. Similarly, if we look at "expansions" rather than "extensions," then the codimension-1 gene spectra also get associated with things like The Archipelago and The Biosphere. Note also that some other people have talked about clans built on codimension-2 genes rather than codimension-1.

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

The various notions above - "support," "extension," "expansion," and so on, lead to a partial order on the set of subgroup temperaments for which we can compute various meets, joins, and so on. For the "support" relationship, if we have that A ≤ B means that A is supported by B, the "join" of any two temperaments is the simplest temperament supporting both (direct sum of subgroups and kernels), and the "meet" is the most complex temperament supported by both (intersection of subgroups and kernels).

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. 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.