Projection: Difference between revisions

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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]]).

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#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>.

Revision as of 17:00, 30 January 2023 view source Cmloegcmluin (talk \| contribs) autopatrolled, emailconfirmed 5,170 edits →The unchanged-interval basis: add bolded key term ← Older edit		Revision as of 17:55, 30 January 2023 view source Cmloegcmluin (talk \| contribs) autopatrolled, emailconfirmed 5,170 edits →Derivation of generator embedding matrix from unchanged-interval basis and mapping Newer edit →
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	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]]).		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#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>.