Projection: Difference between revisions

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In other words, the service it provides is rather to ''give us a way to speak of the generator form of a temperament with respect to a particular mapping and a particular generator embedding.'' In other words, if we possess a scheme for unambiguously determining a "home base" <math>M</math> — such as a [[canonical form]], which we ''do'' have — and we possess a scheme for unambiguously determining a particular embedding — such as any tuning scheme which gives tunings expressible as embeddings (e.g. minimean, miniRMS) — then we now also have a way to describe with cold, hard matrices (i.e. not fuzzier instructions like "octave-fifth") what exact form we wish our generators to be in. We can speak of a pair of a linked bases — the generator embedding treated as a basis, and the mapping treated as a mapping-row basis — as the (generator) '''form''' of a temperament. And so a tuning system can be more fully specified than it could previously, leveraging this conceit.

It may be helpful to define <math>M_{\text{c}}</math> as the canonical mapping, the one in canonical form, and <math>G_{\text{c}}</math> as the one in "corresponding form" form to the canonical form of the mapping, i.e. per whichever tuning scheme is being used, it's the <math>G</math> you get from <math>M_{\text{c}}</math>. In this case, then with respect to <math>~~P_{\text{c}}~~ = G_{\text{c}}M_{\text{c}}</math>, any <math>M_{\text{x}}</math> and <math>G_{\text{x}}</math> are viable so long as <math>M_{\text{c}} = F_{\text{x}}M_{\text{x}}</math> and <math>G_{\text{c}} = G_{\text{x}}F_{\text{x}}^{-1}</math> for some generator form <math>F_{\text{x}}</math>.

It may be helpful to define <math>M_{\text{c}}</math> as the canonical mapping, the one in canonical form, and <math>G_{\text{c}}</math> as the one in "corresponding form" form to the canonical form of the mapping, i.e. per whichever tuning scheme is being used, it's the <math>G</math> you get from <math>M_{\text{c}}</math>. In this case, then with respect to <math>P = G_{\text{c}}M_{\text{c}}</math>, any <math>M_{\text{x}}</math> and <math>G_{\text{x}}</math> are viable so long as <math>M_{\text{c}} = F_{\text{x}}M_{\text{x}}</math> and <math>G_{\text{c}} = G_{\text{x}}F_{\text{x}}^{-1}</math> for some generator form <math>F_{\text{x}}</math>.

So, for a concrete example, we can say that for meantone temperament, given that its canonical form is the octave-twelfth form (that was no accident that we chose it as our "home base" earlier!) this generator form matrix <math>F_{\text{8ave,4th}}</math> represents the octave-fourth form:

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 In other words, the service it provides is rather to ''give us a way to speak of the generator form of a temperament with respect to a particular mapping and a particular generator embedding.'' In other words, if we possess a scheme for unambiguously determining a "home base" <math>M</math> — such as a [[canonical form]], which we ''do'' have — and we possess a scheme for unambiguously determining a particular embedding — such as any tuning scheme which gives tunings expressible as embeddings (e.g. minimean, miniRMS) — then we now also have a way to describe with cold, hard matrices (i.e. not fuzzier instructions like "octave-fifth") what exact form we wish our generators to be in. We can speak of a pair of a linked bases — the generator embedding treated as a basis, and the mapping treated as a mapping-row basis — as the (generator) '''form''' of a temperament. And so a tuning system can be more fully specified than it could previously, leveraging this conceit.
-It may be helpful to define <math>M_{\text{c}}</math> as the canonical mapping, the one in canonical form, and <math>G_{\text{c}}</math> as the one in "corresponding form" form to the canonical form of the mapping, i.e. per whichever tuning scheme is being used, it's the <math>G</math> you get from <math>M_{\text{c}}</math>. In this case, then with respect to <math>P_{\text{c}} = G_{\text{c}}M_{\text{c}}</math>, any <math>M_{\text{x}}</math> and <math>G_{\text{x}}</math> are viable so long as <math>M_{\text{c}} = F_{\text{x}}M_{\text{x}}</math> and <math>G_{\text{c}} = G_{\text{x}}F_{\text{x}}^{-1}</math> for some generator form <math>F_{\text{x}}</math>.
+It may be helpful to define <math>M_{\text{c}}</math> as the canonical mapping, the one in canonical form, and <math>G_{\text{c}}</math> as the one in "corresponding form" form to the canonical form of the mapping, i.e. per whichever tuning scheme is being used, it's the <math>G</math> you get from <math>M_{\text{c}}</math>. In this case, then with respect to <math>P = G_{\text{c}}M_{\text{c}}</math>, any <math>M_{\text{x}}</math> and <math>G_{\text{x}}</math> are viable so long as <math>M_{\text{c}} = F_{\text{x}}M_{\text{x}}</math> and <math>G_{\text{c}} = G_{\text{x}}F_{\text{x}}^{-1}</math> for some generator form <math>F_{\text{x}}</math>.
 So, for a concrete example, we can say that for meantone temperament, given that its canonical form is the octave-twelfth form (that was no accident that we chose it as our "home base" earlier!) this generator form matrix <math>F_{\text{8ave,4th}}</math> represents the octave-fourth form: