Projection: Difference between revisions

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Thus <math>\frac65</math> becomes {{vector|2 0 -3/4}}, and four of them equals an untempered <math>\frac{256}{125}</math>. The reader can do similar calculations to verify that <math>\frac21</math> is projected to <math>\frac21</math>, <math>\frac54</math> is projected to <math>\frac54</math>, and <math>\frac32</math> is projected to {{vector|0 0 1/4}}.

For ''third''-comma meantone~~, however~~, three fifths (<math>\frac{27}{8}</math>) are a full comma flat. That works out to <math>\frac{10}{3}</math>. Thus the generator is {{vector|1/3 -1/3 1/3}}. Again, the period is a pure octave. This gives us our generator embedding <math>G</math>. Multiply it by the same mapping <math>M</math> to find the projection matrix <math>P</math>:

For quarter-comma meantone, it's plain to see from the projection matrix that it's a rank-2 temperament: one of the rows is already all-zeros, so clearly it's not full-rank (rank-3)! This will not in general be true of projection matrices, however. For example, we can take a look at ''third''-comma meantone's projection.

For third-comma meantone, three fifths (<math>\frac{27}{8}</math>) are a full comma flat. That works out to <math>\frac{10}{3}</math>. Thus the generator is {{vector|1/3 -1/3 1/3}}. Again, the period is a pure octave. This gives us our generator embedding <math>G</math>. Multiply it by the same mapping <math>M</math> to find the projection matrix <math>P</math>:

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 Thus <math>\frac65</math> becomes {{vector|2 0 -3/4}}, and four of them equals an untempered <math>\frac{256}{125}</math>. The reader can do similar calculations to verify that <math>\frac21</math> is projected to <math>\frac21</math>, <math>\frac54</math> is projected to <math>\frac54</math>, and <math>\frac32</math> is projected to {{vector|0 0 1/4}}.
-For ''third''-comma meantone, however, three fifths (<math>\frac{27}{8}</math>) are a full comma flat. That works out to <math>\frac{10}{3}</math>. Thus the generator is {{vector|1/3 -1/3 1/3}}. Again, the period is a pure octave. This gives us our generator embedding <math>G</math>. Multiply it by the same mapping <math>M</math> to find the projection matrix <math>P</math>:
+For quarter-comma meantone, it's plain to see from the projection matrix that it's a rank-2 temperament: one of the rows is already all-zeros, so clearly it's not full-rank (rank-3)! This will not in general be true of projection matrices, however. For example, we can take a look at ''third''-comma meantone's projection.
+For third-comma meantone, three fifths (<math>\frac{27}{8}</math>) are a full comma flat. That works out to <math>\frac{10}{3}</math>. Thus the generator is {{vector|1/3 -1/3 1/3}}. Again, the period is a pure octave. This gives us our generator embedding <math>G</math>. Multiply it by the same mapping <math>M</math> to find the projection matrix <math>P</math>: