Cross-domain temperament merging: Difference between revisions

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# Canonicalize.

The reason this works is that wherever the corresponding top-half columns are all zeros, this was achieved through linear combinations of vectors from both domain bases, which means the information below them represents vectors that are in both of them. In other words, if <math>(x, x) + (y, 0) = (0, z)</math> and <math>x</math> is in <math>B_1</math> and <math>y</math> is in <math>B_2</math>, then we must have <math>x + y = 0</math> and <math>z = x</math><ref>credit this explanation to Tom Price on Discord</ref>. We're sort of abusing HNF as a way to solve a system, kind of like [[Douglas Blumeyer's RTT ~~How-To~~#Nullspace|when we calculate the nullspace]]<ref>Credit this explanation to Sintel on Discord</ref>.

The reason this works is that wherever the corresponding top-half columns are all zeros, this was achieved through linear combinations of vectors from both domain bases, which means the information below them represents vectors that are in both of them. In other words, if <math>(x, x) + (y, 0) = (0, z)</math> and <math>x</math> is in <math>B_1</math> and <math>y</math> is in <math>B_2</math>, then we must have <math>x + y = 0</math> and <math>z = x</math><ref>credit this explanation to Tom Price on Discord</ref>. We're sort of abusing HNF as a way to solve a system, kind of like [[Dave Keenan & Douglas Blumeyer's guide to RTT: exploring temperaments#Nullspace|when we calculate the nullspace]]<ref>Credit this explanation to Sintel on Discord</ref>.

=== But first: a gentle introduction ===

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 # Canonicalize.
-The reason this works is that wherever the corresponding top-half columns are all zeros, this was achieved through linear combinations of vectors from both domain bases, which means the information below them represents vectors that are in both of them. In other words, if <math>(x, x) + (y, 0) = (0, z)</math> and <math>x</math> is in <math>B_1</math> and <math>y</math> is in <math>B_2</math>, then we must have <math>x + y = 0</math> and <math>z = x</math><ref>credit this explanation to Tom Price on Discord</ref>. We're sort of abusing HNF as a way to solve a system, kind of like [[Douglas Blumeyer's RTT How-To#Nullspace|when we calculate the nullspace]]<ref>Credit this explanation to Sintel on Discord</ref>.
+The reason this works is that wherever the corresponding top-half columns are all zeros, this was achieved through linear combinations of vectors from both domain bases, which means the information below them represents vectors that are in both of them. In other words, if <math>(x, x) + (y, 0) = (0, z)</math> and <math>x</math> is in <math>B_1</math> and <math>y</math> is in <math>B_2</math>, then we must have <math>x + y = 0</math> and <math>z = x</math><ref>credit this explanation to Tom Price on Discord</ref>. We're sort of abusing HNF as a way to solve a system, kind of like [[Dave Keenan & Douglas Blumeyer's guide to RTT: exploring temperaments#Nullspace|when we calculate the nullspace]]<ref>Credit this explanation to Sintel on Discord</ref>.
 === But first: a gentle introduction ===