Subgroup temperament families, relationships, and genes: Difference between revisions
| Line 26: | Line 26: | ||
= Expansions and Retractions = | = 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 \ | Suppose that we have some temperament A on some JI 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 JI subgroup S mod kernel K), our new temperament is <math>B = R/(K \cap 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. | 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. | ||