Sane and insane temperaments: Difference between revisions

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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 [[Temperament_Mapping_Matrices_(M-maps)|mapping matrix]] for your larger temperament, and multiply it by the [[Subgroup_Mapping_Matrices_(V-maps)|subgroup matrix]] for your subgroup in question, that the result is not [[contorted]] ~~(or in other words, the result is [[defactored]])~~.

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 [[Temperament_Mapping_Matrices_(M-maps)|mapping matrix]] for your larger temperament, and multiply it by the [[Subgroup_Mapping_Matrices_(V-maps)|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).

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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 [[~~Saturation~~|unsaturated]] ~~(or [[enfactored]])~~, 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.

It turns out that an equivalent definition is that a temperament is insane iff its kernel is [[Mathematical theory of saturation|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:

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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.

[[Category:Regular temperament theory]]

[[Category:Math]]

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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 [[Temperament_Mapping_Matrices_(M-maps)|mapping matrix]] for your larger temperament, and multiply it by the [[Subgroup_Mapping_Matrices_(V-maps)|subgroup matrix]] for your subgroup in question, that the result is not [[contorted]] (or in other words, the result is [[defactored]]).
+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 [[Temperament_Mapping_Matrices_(M-maps)|mapping matrix]] for your larger temperament, and multiply it by the [[Subgroup_Mapping_Matrices_(V-maps)|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).
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 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 [[Saturation|unsaturated]] (or [[enfactored]]), 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.
+It turns out that an equivalent definition is that a temperament is insane iff its kernel is [[Mathematical theory of saturation|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:
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 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.
+[[Category:Regular temperament theory]]
+[[Category:Math]]