Projection: Difference between revisions

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This formula is used, among other places, in the ~~sum polytope~~ method for computing miniaverage tunings of regular temperaments (see here: [[Generator embedding optimization#Convert to generators]]).

This formula is used, among other places, in the [[zero-damage method]] for computing miniaverage tunings of regular temperaments (see here: [[Generator embedding optimization#Convert to generators]]).

And it turns out this formula works fine when subjected to a units analysis. <math>\mathrm{U}</math> has units of primes. And <math>M</math> has units of generators per primes. So we have <math>\small 𝗽\!·\!((𝗴</math>/<math>\small 𝗽)\!·\!𝗽)^{-1}</math>. The <math>\small 𝗽</math>'s on the inside of the parens cancel, and we're left with <math>\small 𝗽𝗴^{-1}</math>, or in other words, <math>\small 𝗽</math>/<math>\small 𝗴</math>, which are indeed the units of <math>G</math>.

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Projections and generator embeddings come 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 [[Generator embedding optimization#Pseudoinverse method]].

# those that use the pseudoinverse method, such as miniRMS and minimax-ES tunings: see [[Generator embedding optimization#Pseudoinverse method]].

# those that use the ~~sum polytope~~, such as miniaverage tunings: see [[Generator embedding optimization#~~Sum polytope~~ method]].

# those that use the zero-damage method, such as miniaverage tunings: see [[Generator embedding optimization#Zero-damage method]].

# tunings based on unchanged-intervals: see [[Generator embedding optimization#Unchanged-interval method]].

@@ Line 1,262: / Line 1,262: @@
-This formula is used, among other places, in the sum polytope method for computing miniaverage tunings of regular temperaments (see here: [[Generator embedding optimization#Convert to generators]]).
+This formula is used, among other places, in the [[zero-damage method]] for computing miniaverage tunings of regular temperaments (see here: [[Generator embedding optimization#Convert to generators]]).
 And it turns out this formula works fine when subjected to a units analysis. <math>\mathrm{U}</math> has units of primes. And <math>M</math> has units of generators per primes. So we have <math>\small 𝗽\!·\!((𝗴</math>/<math>\small 𝗽)\!·\!𝗽)^{-1}</math>. The <math>\small 𝗽</math>'s on the inside of the parens cancel, and we're left with <math>\small 𝗽𝗴^{-1}</math>, or in other words, <math>\small 𝗽</math>/<math>\small 𝗴</math>, which are indeed the units of <math>G</math>.
@@ Line 1,740: / Line 1,740: @@
 Projections and generator embeddings come 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 [[Generator embedding optimization#Pseudoinverse method]].
+# those that use the pseudoinverse method, such as miniRMS and minimax-ES tunings: see [[Generator embedding optimization#Pseudoinverse method]].
-# those that use the sum polytope, such as miniaverage tunings: see [[Generator embedding optimization#Sum polytope method]].
+# those that use the zero-damage method, such as miniaverage tunings: see [[Generator embedding optimization#Zero-damage method]].
 # tunings based on unchanged-intervals: see [[Generator embedding optimization#Unchanged-interval method]].