Wedgie/Archived version: Difference between revisions
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=== Example === | === Example === | ||
Consider the wedgie W = <<1 4 4|| for 2.3.5 meantone (the 12&19 temperament). We have W(2,3) = 1 and W(2,5) = 4, so d = 1, and our period is 1\1. Further, we have that 1*W(2,3) + 0*W(2,5) = 1, so we can use 3^1 * 5^0 = 3/1 as our generator. | Consider the wedgie W = <<1 4 4|| for 2.3.5 meantone (the 12&19 temperament). We have W(2,3) = 1 and W(2,5) = 4, so d = 1, and our period is 1\1. Further, we have that 1*W(2,3) + 0*W(2,5) = 1, so ''c''<sub>1</sub> = 1, ''c''<sub>2</sub> = 0 is one solution, and we can use 3^1 * 5^0 = 3/1 as our generator. | ||
Note that -3*W(2,3) + 1*W(2,5) = -3*1 + 1*4 = 1, so '' | Note that -3*W(2,3) + 1*W(2,5) = -3*1 + 1*4 = 1, so ''c''<sub>1</sub> = -3, ''c''<sub>2</sub> = 1 is another solution to the equation. Thus 5/27 is also a valid generator. This octave reduces to the [[40/27]] grave fifth, which is equated to 3/2 in meantone. | ||
=== Proof (a bit technical) === | === Proof (a bit technical) === | ||