Cross-domain temperament merging: Difference between revisions

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Given an interval, comma basis, or mapping — anything that has an associated domain basis — it is possible to change it from one domain basis to another. We can accomplish this using a '''basis change matrix''', an object that works like a two-way bridge between two domain bases.

In fact, there is no real difference between a ''basis matrix'',' such as we've been using to represent nonstandard domain bases in terms of the primes, and a basis ''change'' matrix. We can think of the basis matrix as a basis change matrix, from whatever domain ''to the simplest prime-only basis''. And so we use the same symbol for both of these objects, <math>B</math>. When the change is from a basis to the simplest prime-only basis, the subscript on <math>B</math> can simply be the basis being described; if however, the basis change matrix is from one nonprime basis to another nonprime basis, such as 2.49/3.5 to 2.7/3.5, then the subscript should contain the names of both bases, ideally with a "↔" symbol between them, and the superspace on the left and the subspace on the right, as that corresponds to their positions as labels on the matrix.

In fact, there is no real difference between a ''basis matrix'', such as we've been using to represent nonstandard domain bases in terms of the primes, and a basis ''change'' matrix. We can think of the basis matrix as a basis change matrix, from whatever domain ''to the simplest prime-only basis''. And so we use the same symbol for both of these objects, <math>B</math>. When the change is from a basis to the simplest prime-only basis, the subscript on <math>B</math> can simply be the basis being described; if however, the basis change matrix is from one nonprime basis to another nonprime basis, such as 2.49/3.5 to 2.7/3.5, then the subscript should contain the names of both bases, ideally with a "↔" symbol between them, and the superspace on the left and the subspace on the right, as that corresponds to their positions as labels on the matrix.

Elsewhere, these types of matrices have been called [[subgroup basis matrices]], but that terminology will not be used here, for the same reasons as are described in the last section of this article (here: [[Domain basis#Terminology: domain basis vs. subgroup]]).

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 Given an interval, comma basis, or mapping — anything that has an associated domain basis — it is possible to change it from one domain basis to another. We can accomplish this using a '''basis change matrix''', an object that works like a two-way bridge between two domain bases.
-In fact, there is no real difference between a ''basis matrix'',' such as we've been using to represent nonstandard domain bases in terms of the primes, and a basis ''change'' matrix. We can think of the basis matrix as a basis change matrix, from whatever domain ''to the simplest prime-only basis''. And so we use the same symbol for both of these objects, <math>B</math>. When the change is from a basis to the simplest prime-only basis, the subscript on <math>B</math> can simply be the basis being described; if however, the basis change matrix is from one nonprime basis to another nonprime basis, such as 2.49/3.5 to 2.7/3.5, then the subscript should contain the names of both bases, ideally with a "↔" symbol between them, and the superspace on the left and the subspace on the right, as that corresponds to their positions as labels on the matrix.
+In fact, there is no real difference between a ''basis matrix'', such as we've been using to represent nonstandard domain bases in terms of the primes, and a basis ''change'' matrix. We can think of the basis matrix as a basis change matrix, from whatever domain ''to the simplest prime-only basis''. And so we use the same symbol for both of these objects, <math>B</math>. When the change is from a basis to the simplest prime-only basis, the subscript on <math>B</math> can simply be the basis being described; if however, the basis change matrix is from one nonprime basis to another nonprime basis, such as 2.49/3.5 to 2.7/3.5, then the subscript should contain the names of both bases, ideally with a "↔" symbol between them, and the superspace on the left and the subspace on the right, as that corresponds to their positions as labels on the matrix.
 Elsewhere, these types of matrices have been called [[subgroup basis matrices]], but that terminology will not be used here, for the same reasons as are described in the last section of this article (here: [[Domain basis#Terminology: domain basis vs. subgroup]]).