The wedgie: Difference between revisions
m →Proof: typo |
m →Proof: the *extended* Euclidean algorithm finds the coefficients a_i |
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*W(2/1, e_1) = W(a_2 e_2, e_1) = -a_2 W(e_1, e_2) = -a_2 | *W(2/1, e_1) = W(a_2 e_2, e_1) = -a_2 W(e_1, e_2) = -a_2 | ||
*W(2/1, e_2) = W(a_1 e_1, e_2) = a_1 W(e_1, e_2) = a_1. | *W(2/1, e_2) = W(a_1 e_1, e_2) = a_1 W(e_1, e_2) = a_1. | ||
Divisibility (by d = gcd(W(2, q_1), ..., W(2, q_n))) and the fact that e1 and e2 represent JI ratios in the 2.q_1. ... q_n subgroup impliy that a_1 and a_2 are both divisible by d, and hence 2/1 is a dth power in M' (the temperament space). Since gcd(W(2, q_1), ..., W(2, q_n)) = d, we can always 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 using the Euclidean algorithm. Then since W(2, g) = W(d*p, g) = d*W(p, g) = d, we have W(p,g) = 1. Ta-da! | Divisibility (by d = gcd(W(2, q_1), ..., W(2, q_n))) and the fact that e1 and e2 represent JI ratios in the 2.q_1. ... q_n subgroup impliy that a_1 and a_2 are both divisible by d, and hence 2/1 is a dth power in M' (the temperament space). Since gcd(W(2, q_1), ..., W(2, q_n)) = d, we can always 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 using the extended Euclidean algorithm. Then since W(2, g) = W(d*p, g) = d*W(p, g) = d, we have W(p,g) = 1. Ta-da! | ||
== Truncation of wedgies == | == Truncation of wedgies == | ||