Projection: Difference between revisions
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This formula is used, among other places, in the [[ | This formula is used, among other places, in the [[Generator embedding optimization/Zero-damage method|zero-damage method]] for computing miniaverage tunings of regular temperaments (see here: [[Generator embedding optimization/Zero-damage method#Convert to generators|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>. | 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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== Applications == | == Applications == | ||
Projections and generator embeddings come up in some methods for calculating tunings according to commonly-used schemes: | Projections and generator embeddings come up in some methods for calculating tunings according to commonly-used schemes: | ||
# Those that use the pseudoinverse method, such as miniRMS and minimax-ES tunings: see [[Generator embedding optimization | # Those that use the pseudoinverse method, such as miniRMS and minimax-ES tunings: see [[Generator embedding optimization/Pseudoinverse method|Pseudoinverse method]]. | ||
# Those that use the zero-damage method, such as miniaverage tunings: see [[Generator embedding optimization | # Those that use the zero-damage method, such as miniaverage tunings: see [[Generator embedding optimization/Zero-damage method|Zero-damage method]]. | ||
# Tunings based on unchanged-intervals: see [[Generator embedding optimization# | # Tunings based on unchanged-intervals: see [[Generator embedding optimization#Only held-intervals method|Only held-intervals method]]. | ||
== See also == | == See also == | ||