Temperament addition: Difference between revisions

Line 307:

The first step to addabilization defactoring is inspired by [[Pernet-Stein defactoring]]: we find the value of the enfactoring factor (the "greatest factor") by following this algorithm until the point where we have a square transformation matrix, but instead of inverting it and multiplying by it to ''remove'' the defactoring, we simply take this square matrix's determinant, which is the factor we were about to remove. If that determinant is 1, then we're already defactored; if not, then we need to take do some additional steps.

It turns out that you can always<ref>This conjecture was first suggested by Mike Battaglia, but it has not yet been mathematically proven. Sintel and Tom Price have done some experiments but nothing complete yet. Douglas Blumeyer's test cases in the [[RTT library in Wolfram Language]] ~~suggest~~ this is true, though.</ref> isolate the greatest factor in the single final vector of the matrix — the linearly independent vector — through linear combinations of the vectors in the <math>L_{\text{dep}}</math>.

It turns out that you can always<ref>This conjecture was first suggested by Mike Battaglia, but it has not yet been mathematically proven. Sintel and Tom Price have done some experiments but nothing complete yet. Douglas Blumeyer's test cases in the [[RTT library in Wolfram Language]] have emiprically proven that this is true, though.</ref> isolate the greatest factor in the single final vector of the matrix — the linearly independent vector — through linear combinations of the vectors in the <math>L_{\text{dep}}</math>.

The example that will be worked through in this section below is as simple as it can get: the <math>L_{\text{dep}}</math> consists of only a single vector, so we simply add some number of this single linearly dependent vector to the linearly independent vector. However, if there are multiple vectors in the <math>L_{\text{dep}}</math>, the linear combination which surfaces the greatest factor may involve just one or potentially all of those vectors, and the best approach to finding this combination is simply an automatic solver. An example of this approach is demonstrated in the [[RTT library in Wolfram Language]], here: https://github.com/cmloegcmluin/RTT/blob/main/main.m#L477

@@ Line 307: / Line 307: @@
 The first step to addabilization defactoring is inspired by [[Pernet-Stein defactoring]]: we find the value of the enfactoring factor (the "greatest factor") by following this algorithm until the point where we have a square transformation matrix, but instead of inverting it and multiplying by it to ''remove'' the defactoring, we simply take this square matrix's determinant, which is the factor we were about to remove. If that determinant is 1, then we're already defactored; if not, then we need to take do some additional steps.
-It turns out that you can always<ref>This conjecture was first suggested by Mike Battaglia, but it has not yet been mathematically proven. Sintel and Tom Price have done some experiments but nothing complete yet. Douglas Blumeyer's test cases in the [[RTT library in Wolfram Language]] suggest this is true, though.</ref> isolate the greatest factor in the single final vector of the matrix — the <span style="color: #B6321C;">linearly independent vector</span> — through linear combinations of the vectors in the <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span>.
+It turns out that you can always<ref>This conjecture was first suggested by Mike Battaglia, but it has not yet been mathematically proven. Sintel and Tom Price have done some experiments but nothing complete yet. Douglas Blumeyer's test cases in the [[RTT library in Wolfram Language]] have emiprically proven that this is true, though.</ref> isolate the greatest factor in the single final vector of the matrix — the <span style="color: #B6321C;">linearly independent vector</span> — through linear combinations of the vectors in the <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span>.
 The example that will be worked through in this section below is as simple as it can get: the <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span> consists of only a single vector, so we simply add some number of this <span style="color: #3C8031;">single linearly dependent vector</span> to the <span style="color: #B6321C;">linearly independent vector</span>. However, if there are multiple vectors in the <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span>, the linear combination which surfaces the greatest factor may involve just one or potentially all of those vectors, and the best approach to finding this combination is simply an automatic solver. An example of this approach is demonstrated in the [[RTT library in Wolfram Language]], here: https://github.com/cmloegcmluin/RTT/blob/main/main.m#L477