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.

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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	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.		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).		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).