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| When specifying a temperament by the list of commas it tempers out, the list should be [[defactored]] so it presents the intervals in their simplest, most direct form. | | When specifying a temperament by the list of commas it tempers out, the list should be [[defactored]] so it presents the intervals in their simplest, most direct form. |
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| === Reduced row echelon form ===
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| {{main| Wikipedia: Row echelon form }}
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| If you have a routine which will compute the reduced row echelon form of a matrix with rows consisting of vals using rational number arithmetic, this can be computed and any rows consisting of the zero val stripped off. The result is a unique identifier for the abstract temperament which is closely related to the normal val list. The intervals of the abstract temperament may be found in the same way, by applying the mappings (which are fractional vals) to monzos; or put in another way, by matrix multiplication of the monzos by the reduced row echelon form matrix.
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| For example, if we feed [{{val| 22 35 51 62 }}, {{val| 31 49 72 87 }}, {{val| 84 133 195 236 }}] into a reduced row echelon form routine, we obtain [{{val| 1 0 3 1 }}, {{val| 0 1 -3/7 8/7 }}, {{val| 0 0 0 0 }}]. Stripping off the zero val in the final row, we get E = [{{val| 1 0 3 1 }}, {{val| 0 1 -3/7 8/7 }}]. The monzo for 7/6 is {{monzo| -1 -1 0 1 }}, and E{{monzo| -1 -1 0 1 }} = [0 1/7]. Multiply by {{monzo| 1 0 0 0 }}, the monzo for 2, and the result is E{{monzo| 1 0 0 0 }}, which is [1 0]. We have in fact 7-limit orwell temperament, with period 2 and generator approximately 7/6.
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| == Translation between methods of specifying temperaments == | | == Translation between methods of specifying temperaments == |