Cross-domain temperament merging: Difference between revisions

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So, first, we do the first step of merging interval bases: concatenate them. That gets us 2.25/9.11/7.2.5/3.7.11.

The next step of merging is to canonicalize. To begin that, we convert our interval basis to a matrix <math>B</math>. Here, we've labeled the columns with the number-list representation of the interval basis, to help show the correspondence, as well as the rows with the ~~basis elements~~:

The next step of merging is to canonicalize. To begin that, we convert our interval basis to a matrix <math>B</math>. Here, we've labeled the columns with the number-list representation of the formal primes of this interval basis, to help show the correspondence, as well as the rows with the primes these formal primes factor into:

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Now, let's work through an example of a cross-interval basis map-merge: 22 equal temperament <math>T_1</math> with 17 equal temperament <math>T_2</math>, where 22-ET is in the interval basis 2.3.5.11, which we'll call <math>B_1</math>, and 17-ET is in the 2.9.7.11 interval basis, which we'll call <math>B_2</math>.

First we must intersect these two temperaments' interval bases. The first step of that is to convert them to matrices ~~with~~ the ~~same basis elements~~:

First we must intersect these two temperaments' interval bases. The first step of that is to convert them to matrices which have formal primes as columns and the merged set of the primes they factor into as the rows:

@@ Line 77: / Line 77: @@
 So, first, we do the first step of merging interval bases: concatenate them. That gets us 2.25/9.11/7.2.5/3.7.11.
-The next step of merging is to canonicalize. To begin that, we convert our interval basis to a matrix <math>B</math>. Here, we've labeled the columns with the number-list representation of the interval basis, to help show the correspondence, as well as the rows with the basis elements:
+The next step of merging is to canonicalize. To begin that, we convert our interval basis to a matrix <math>B</math>. Here, we've labeled the columns with the number-list representation of the formal primes of this interval basis, to help show the correspondence, as well as the rows with the primes these formal primes factor into:
@@ Line 1,145: / Line 1,145: @@
 Now, let's work through an example of a cross-interval basis map-merge: 22 equal temperament <math>T_1</math> with 17 equal temperament <math>T_2</math>, where 22-ET is in the interval basis 2.3.5.11, which we'll call <math>B_1</math>, and 17-ET is in the 2.9.7.11 interval basis, which we'll call <math>B_2</math>.
-First we must intersect these two temperaments' interval bases. The first step of that is to convert them to matrices with the same basis elements:
+First we must intersect these two temperaments' interval bases. The first step of that is to convert them to matrices which have formal primes as columns and the merged set of the primes they factor into as the rows: