The wedgie: Difference between revisions
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To find the '''generator''': Treat W(2, v) as a linear map where you plug in a JI vector v, and use the [https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm extended Euclidean algorithm] to find a linear combination g = a_1 q_1 + ... + a_n q_n such that W(2, g) = a_1 W(2, q_1) + ... a_n W(2,q_n) = d. | To find the '''generator''': Treat W(2, v) as a linear map where you plug in a JI vector v, and use the [https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm extended Euclidean algorithm] to find a linear combination g = a_1 q_1 + ... + a_n q_n such that W(2, g) = a_1 W(2, q_1) + ... a_n W(2,q_n) = d. | ||
For example, consider the wedgie <<1 4 10 4 13 12|| for meantone. We have W(2,3) = 1, W(2,5) = 4, W(2,7) = 10, so d = 1, and our period is 1\1. | For example, consider the wedgie W = <<1 4 10 4 13 12|| for meantone. We have W(2,3) = 1, W(2,5) = 4, W(2,7) = 10, so d = 1, and our period is 1\1. | ||
We already have W(2,3) = 1, so we can use 3/1 as our generator. Alternatively, W(2, 3/2) = W(2,3) - W(2, 2) = W(2, 3) = 1, so 3/2 is a valid generator for meantone as well. | We already have W(2,3) = 1, so we can use 3/1 as our generator. Alternatively, W(2, 3/2) = W(2,3) - W(2, 2) = W(2, 3) = 1, so 3/2 is a valid generator for meantone as well. | ||