Douglas Blumeyer's RTT How-To: Difference between revisions

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First I’ll list the steps. Don’t worry if it doesn’t all make sense the first time. We’ll work through an example and go into more detail as we do. To be clear, what we're doing here is both more and less and different ways from the strict definition of the exterior product as you may see it elsewhere; I'm specifically here describing the process for finding the multicovector in the form you're going to be interested in for RTT purposes.

Take each combination of <math>r</math> primes where <math>r</math> is the rank, sorted in lexicographic order, e.g. <math>(2,3,5)</math>, <math>(2,3,7)</math>, <math>(2,5,7)</math>, <math>(3,5,7)</math>. Now convert each of those combinations to a square <math>r×r</math> matrix by slicing a column for each prime out of the mapping and putting them together. Now take each matrix's determinant. Extract the GCD from the resulting sequence of scalars, then set them inside <math>r</math> brackets, and you've got your multicovector.

Take each combination of <math>r</math> primes where <math>r</math> is the rank, sorted in [https://en.wikipedia.org/wiki/Lexicographic_order lexicographic order], e.g. if we're in the 7-limit, we'd have <math>(2,3,5)</math>, <math>(2,3,7)</math>, <math>(2,5,7)</math>, and <math>(3,5,7)</math>. Now convert each of those combinations to a square <math>r×r</math> matrix by slicing a column for each prime out of the mapping and putting them together. Now take each matrix's determinant. Extract the GCD from the resulting sequence of scalars, then set them inside <math>r</math> brackets, and you've got your multicovector.

Let’s work through the meantone example.

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 First I’ll list the steps. Don’t worry if it doesn’t all make sense the first time. We’ll work through an example and go into more detail as we do. To be clear, what we're doing here is both more and less and different ways from the strict definition of the exterior product as you may see it elsewhere; I'm specifically here describing the process for finding the multicovector in the form you're going to be interested in for RTT purposes.
-Take each combination of <span><math>r</math></span> primes where <span><math>r</math></span> is the rank, sorted in lexicographic order, e.g. <span><math>(2,3,5)</math></span>, <span><math>(2,3,7)</math></span>, <span><math>(2,5,7)</math></span>, <span><math>(3,5,7)</math></span>. Now convert each of those combinations to a square <span><math>r×r</math></span> matrix by slicing a column for each prime out of the mapping and putting them together. Now take each matrix's determinant. Extract the GCD from the resulting sequence of scalars, then set them inside <span><math>r</math></span> brackets, and you've got your multicovector.
+Take each combination of <span><math>r</math></span> primes where <span><math>r</math></span> is the rank, sorted in [https://en.wikipedia.org/wiki/Lexicographic_order lexicographic order], e.g. if we're in the 7-limit, we'd have <span><math>(2,3,5)</math></span>, <span><math>(2,3,7)</math></span>, <span><math>(2,5,7)</math></span>, and <span><math>(3,5,7)</math></span>. Now convert each of those combinations to a square <span><math>r×r</math></span> matrix by slicing a column for each prime out of the mapping and putting them together. Now take each matrix's determinant. Extract the GCD from the resulting sequence of scalars, then set them inside <span><math>r</math></span> brackets, and you've got your multicovector.
 Let’s work through the meantone example.