Projection: Difference between revisions

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The injection of these <math>F</math> matrices doesn't affect the temperament or tuning in any way. Think about it this way. 9/8 always goes to 193.157 cents in quarter comma meantone, whether you're using the octave-and-fifth form or the octave-and-fourth form or any other form. All the form does is tell you what your generator sizes themselves are. They generate (span) the same space regardless. <math>F</math> is only good for keeping the embedding and approximating parts of your projection in sync when you're changing the basis, that's all. The point is for it to have no effect on the general intervals.

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.

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. miniaverage, 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 = G_{\text{c}}M_{\text{c}}</math>, any <math>M_{\text{f}}</math> and <math>G_{\text{f}}</math> are viable so long as <math>M_{\text{c}} = F_{\text{f}}M_{\text{f}}</math> and <math>G_{\text{c}} = G_{\text{f}}F_{\text{f}}^{-1}</math> for some generator form <math>F_{\text{f}}</math>.

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Projections and generator bases comes up in some methods for calculating tunings according to commonly-used schemes:

#those that use the pseudoinverse, such as miniRMS and minimax-ES tunings: see [[Dave Keenan & Douglas Blumeyer's guide to RTT: tuning computation#Pseudoinverse method for a miniRMS tuning]] and [[Dave Keenan & Douglas Blumeyer's guide to RTT: all-interval tuning schemes#Minimax-ES]].

# those that use the sum polytope, such as ~~minimean~~ tunings: see [[Dave Keenan & Douglas Blumeyer's guide to RTT: tuning computation#Sum polytope method for a ~~minimean~~ tuning scheme]].

# those that use the sum polytope, such as miniaverage tunings: see [[Dave Keenan & Douglas Blumeyer's guide to RTT: tuning computation#Sum polytope method for a miniaverage tuning scheme]].

#tunings based on unchanged-intervals: see [[Dave Keenan & Douglas Blumeyer's guide to RTT: tuning computation#Unchanged-interval method]].

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 The injection of these <math>F</math> matrices doesn't affect the temperament or tuning in any way. Think about it this way. 9/8 always goes to 193.157 cents in quarter comma meantone, whether you're using the octave-and-fifth form or the octave-and-fourth form or any other form. All the form does is tell you what your generator sizes themselves are. They generate (span) the same space regardless. <math>F</math> is only good for keeping the embedding and approximating parts of your projection in sync when you're changing the basis, that's all. The point is for it to have no effect on the general intervals.
-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.
+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. miniaverage, 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 = G_{\text{c}}M_{\text{c}}</math>, any <math>M_{\text{f}}</math> and <math>G_{\text{f}}</math> are viable so long as <math>M_{\text{c}} = F_{\text{f}}M_{\text{f}}</math> and <math>G_{\text{c}} = G_{\text{f}}F_{\text{f}}^{-1}</math> for some generator form <math>F_{\text{f}}</math>.
@@ Line 1,741: / Line 1,741: @@
 Projections and generator bases comes up in some methods for calculating tunings according to commonly-used schemes:
 #those that use the pseudoinverse, such as miniRMS and minimax-ES tunings: see [[Dave Keenan & Douglas Blumeyer's guide to RTT: tuning computation#Pseudoinverse method for a miniRMS tuning]] and [[Dave Keenan & Douglas Blumeyer's guide to RTT: all-interval tuning schemes#Minimax-ES]].
-# those that use the sum polytope, such as minimean tunings: see [[Dave Keenan & Douglas Blumeyer's guide to RTT: tuning computation#Sum polytope method for a minimean tuning scheme]].
+# those that use the sum polytope, such as miniaverage tunings: see [[Dave Keenan & Douglas Blumeyer's guide to RTT: tuning computation#Sum polytope method for a miniaverage tuning scheme]].
 #tunings based on unchanged-intervals: see [[Dave Keenan & Douglas Blumeyer's guide to RTT: tuning computation#Unchanged-interval method]].