Sane and insane temperaments

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Given any subgroup temperament, we can define the restriction of that temperament to a smaller subgroup. If such a restriction does not change the generators, it is called a strong restriction, otherwise it is called a weak restriction. Another way to state this criterion is that if you take the mapping matrix for your larger temperament, and multiply it by the subgroup matrix for your subgroup in question, that the result is not contorted.

For example, the 2.3.11 restriction of 11-limit Mohajira is the 2.3.11 243/242 temperament, and a strong restriction since the generator of 11/9 does not change. The 2.3.5 restriction of 11-limit Meantone is the 2.3.5 81/80 temperament, which is a "weak" restriction since the generator changes (using the original generator leads to a contorted mapping).

Now, consider the 2.3.25 temperament restriction of the 2.3.5 81/80 (meantone) temperament. This is a strong restriction, since the generator stays the same at 3/2, and hence it is not contorted. However, oddly, while the interval 5/1 does not exist in the subgroup the temperament is defined on, the interval of 25/1 does have a "square root" within the temperament.

Put another way, 25/1 maps to the meantone augmented fifth plus four octaves. If we take our generators to be the octave and the fifth, then in tempered monzo notation, this would map to the tmonzo |0 8>. Note, however, that the tmonzo |0 8> can be bisected into |0 4>, which is what the interval 5/1 would usually map to, if not for that 5/1 doesn't exist within the subgroup the temperament is defined on. So strangely, although there is a perfectly good mapping for 5/1 within the temperament, we are electing not to use it! It's as though our temperament basically a representation for sqrt(25), but we do not elect to call it 5.

Such temperaments are called insane, and can be defined rigorously given the following definition, first given by Keenan Pepper (lightly edited):

Definition: A temperament is "sane" if, for every rational number in the subgroup it's mapping, represented by a monzo v in the *standard* basis (the primes), the rational number represented by v / gcd(v[0],v[1],...,m[1],m[2],...) is also in the subgroup. Here v[0],v[1],... are the monzo entries of v and m[1],m[2],... are the entries of its mapping in the temperament, given any mapping matrix for that temperament. A temperament is "insane" iff it is not sane.

Note that the GCD criterion above does not change no matter which mapping matrix (and hence, set of generators) you choose for the temperament.

It turns out that an equivalent definition is that a temperament is insane iff its kernel is unsaturated, when expressed as a subgroup of the full-limit. This is the same problem that would typically lead to torsion if tempered out of the full-limit. Torsion can be gotten rid of by restriction the temperament to a smaller subgroup, but if you do so, you are instead guaranteed to get an insane temperament.

Some further examples:

The 2.3.25 restriction of the 2.3.5 12-EDO patent val (<12 19 28|) is <12 19 56|. This is a strong restriction that is insane. To see this, note that while 25/1 maps to 56 steps, the implied square root of 5/1 exists at 28 steps, but we do not call it 5/1, which is enough to establish that the temperament is insane. The kernel is generated by 648/625 and 2048/2025, which is unsaturated as it contains (81/80)^2 = 6561/6400 but not 81/80.

The 2.9.5 restriction of 2.3.5 81/80 (meantone) is a weak restriction that is not insane. The resulting temperament is generated by the tempered 2/1 and 9/8. The original subgroup of 2.9.5 contains 9/1 as an interval but not 3/1. However, since 9/1 is a generator of this temperament, there is no way to "split" it further to obtain an unmapped 3/1, so it is not insane. The kernel is 81/80, which is a saturated lattice of the 5-limit, so we have a sane temperament.