Projection: Difference between revisions

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Supposing one desires to transform from a pair of <math>M_1</math> and <math>G_1</math> to another pair of <math>M_2</math> and <math>G_2</math> where both pairs multiply to the same <math>P</math>, or — said another way — you wish to keep your <math>M</math> and <math>G</math> ''in sync'', the simplest approach would be to — for each elementary row operation you apply to <math>M</math> you must apply the opposite elementary column operation to <math>G</math>, e.g. if you add three times the second row to the first row of <math>M</math>, then you must ''subtract'' three times the second ''column'' from the first ''column'' of <math>G</math>. This is along the same lines as the explanations provided for manipulating generator form by changing forms of <math>M</math>, which you can find here: [[Generator form manipulation]].

For example, if we have <math>M_1</math> = {{~~ket~~|{{bra|1 1 0}} {{bra|0 1 4}}}} and <math>G_1</math> = {{~~bra~~|{{ket|1 0 0}} {{ket|0 0 <math>\frac14</math>}}}}, then <math>M_1</math> and <math>G_1</math> are in sync because they're both in the form where <math>g_1</math> is ~2 and <math>g_2</math> is ~3/2. Or if we have <math>M_2</math> = {{~~ket~~|{{bra|1 0 -4}} {{bra|0 1 4}}}} and <math>G_2</math> = {{~~bra~~|{{ket|1 0 0}} {{ket|1 0 <math>\frac14</math>}}}} then they're still in sync because they're both <math>g_1</math> ~2 and <math>g_2</math> ~3 here. But if we mismatched those, they'd be out of sync. Those are both <math>M</math>'s for meantone, and both <math>G</math>'s that can work for quarter-comma meantone, but if you mismatch them with respect to the generator form information, you won't find the same <math>P</math> by matrix multiplication <math>GM</math>.

For example, if we have <math>M_1</math> = {{rket|{{bra|1 1 0}} {{bra|0 1 4}}}} and <math>G_1</math> = {{rbra|{{ket|1 0 0}} {{ket|0 0 <math>\frac14</math>}}}}, then <math>M_1</math> and <math>G_1</math> are in sync because they're both in the form where <math>g_1</math> is ~2 and <math>g_2</math> is ~3/2. Or if we have <math>M_2</math> = {{rket|{{bra|1 0 -4}} {{bra|0 1 4}}}} and <math>G_2</math> = {{rbra|{{ket|1 0 0}} {{ket|1 0 <math>\frac14</math>}}}} then they're still in sync because they're both <math>g_1</math> ~2 and <math>g_2</math> ~3 here. But if we mismatched those, they'd be out of sync. Those are both <math>M</math>'s for meantone, and both <math>G</math>'s that can work for quarter-comma meantone, but if you mismatch them with respect to the generator form information, you won't find the same <math>P</math> by matrix multiplication <math>GM</math>.

(This notion of "sync" is the same idea pointed out in the diagram at the start of the "Obtaining objects from the projection" section below, with the note on <math>G</math> reading "(the one matching M)". And for more information on generator form information, see the "Generator information types" below.)

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 Supposing one desires to transform from a pair of <math>M_1</math> and <math>G_1</math> to another pair of <math>M_2</math> and <math>G_2</math> where both pairs multiply to the same <math>P</math>, or — said another way — you wish to keep your <math>M</math> and <math>G</math> ''in sync'', the simplest approach would be to — for each elementary row operation you apply to <math>M</math> you must apply the opposite elementary column operation to <math>G</math>, e.g. if you add three times the second row to the first row of <math>M</math>, then you must ''subtract'' three times the second ''column'' from the first ''column'' of <math>G</math>. This is along the same lines as the explanations provided for manipulating generator form by changing forms of <math>M</math>, which you can find here: [[Generator form manipulation]].
-For example, if we have <math>M_1</math> = {{ket|{{bra|1 1 0}} {{bra|0 1 4}}}} and <math>G_1</math> = {{bra|{{ket|1 0 0}} {{ket|0 0 <math>\frac14</math>}}}}, then <math>M_1</math> and <math>G_1</math> are in sync because they're both in the form where <math>g_1</math> is ~2 and <math>g_2</math> is ~3/2. Or if we have <math>M_2</math> = {{ket|{{bra|1 0 -4}} {{bra|0 1 4}}}} and <math>G_2</math> = {{bra|{{ket|1 0 0}} {{ket|1 0 <math>\frac14</math>}}}} then they're still in sync because they're both <math>g_1</math> ~2 and <math>g_2</math> ~3 here. But if we mismatched those, they'd be out of sync. Those are both <math>M</math>'s for meantone, and both <math>G</math>'s that can work for quarter-comma meantone, but if you mismatch them with respect to the generator form information, you won't find the same <math>P</math> by matrix multiplication <math>GM</math>.
+For example, if we have <math>M_1</math> = {{rket|{{bra|1 1 0}} {{bra|0 1 4}}}} and <math>G_1</math> = {{rbra|{{ket|1 0 0}} {{ket|0 0 <math>\frac14</math>}}}}, then <math>M_1</math> and <math>G_1</math> are in sync because they're both in the form where <math>g_1</math> is ~2 and <math>g_2</math> is ~3/2. Or if we have <math>M_2</math> = {{rket|{{bra|1 0 -4}} {{bra|0 1 4}}}} and <math>G_2</math> = {{rbra|{{ket|1 0 0}} {{ket|1 0 <math>\frac14</math>}}}} then they're still in sync because they're both <math>g_1</math> ~2 and <math>g_2</math> ~3 here. But if we mismatched those, they'd be out of sync. Those are both <math>M</math>'s for meantone, and both <math>G</math>'s that can work for quarter-comma meantone, but if you mismatch them with respect to the generator form information, you won't find the same <math>P</math> by matrix multiplication <math>GM</math>.
 (This notion of "sync" is the same idea pointed out in the diagram at the start of the "Obtaining objects from the projection" section below, with the note on <math>G</math> reading "(the one matching M)". And for more information on generator form information, see the "Generator information types" below.)