User:Frostburn/Geometric algebra for regular temperaments: Difference between revisions
→Decomposition: Add TODO indicating that the hand waving is bad. |
→Combining vals: Add a note about wedgies |
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=== Combining vals === | === Combining vals === | ||
Notice that it doesn't matter if we scale our val. <math>< 10, 16, 24 ]</math> represents only 5 unique steps within 10-tone equal temperament. There is no integral monzo (that is, no rational number) that would map to an odd number of steps in this rescaled version. Because the sizes don't matter when we add vals together we're producing a sort of average. Let's take the val for 7-tone equal temperament <math>< 7, 11, 16 ]</math> and add it to <math>< 5, 8, 12 ]</math>. The result is <math>< 12, 19, 28 ]</math> which just happens to line up with the val for our familiar 12-tone equal temperament. If we add the vals for 7-tone and 12-tone together we get <math>< 17, 27, 40 ]</math> which is different from the optimal ([[Patent val|patent]]) val for 17-tone equal temperament <math>< 17, 27, \mathbf{39} ]</math>. | Notice that it doesn't matter if we scale our val. <math>< 10, 16, 24 ]</math> represents only 5 unique steps within 10-tone equal temperament. There is no integral monzo (that is, no rational number) that would map to an odd number of steps in this rescaled version. Because the sizes don't matter when we add vals together we're producing a sort of average. Let's take the val for 7-tone equal temperament <math>< 7, 11, 16 ]</math> and add it to <math>< 5, 8, 12 ]</math>. The result is <math>< 12, 19, 28 ]</math> which just happens to line up with the val for our familiar 12-tone equal temperament. If we add the vals for 7-tone and 12-tone together we get <math>< 17, 27, 40 ]</math> which is different from the optimal ([[Patent val|patent]]) val for 17-tone equal temperament <math>< 17, 27, \mathbf{39} ]</math>. By insisting that the first non-zero component of a val is positive and that all of the components are in lowest terms (GCD = 1) we get 1-[[Wedgies and multivals|wedgies]] which are unique identifiers of tunings. | ||
=== Geometric interpretation === | === Geometric interpretation === | ||