The wedgie: Difference between revisions

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Let d = gcd(W(2, q_1), ..., W(2, q_n)). This tells you that for any JI ratio v in your JI subgroup, W(2, v) = 2n(v) for some number n(v) [that depends linearly on v]. This equation is also true when we replace 2 with any JI ratio u that is equated to 2. This tells us that for W(p, g) = 1, we (up to some choices) need p to be an interval such that d*p is equated to 2/1, i.e. p represents 1/d of the octave.

Choose a basis e1, e2 for the temperament group and write (the image of) 2/1 as 2/1 = ~~a1 e1~~ + a2 e2. Then:

Choose a basis e_1, e_2 for the temperament group and write (the image of) 2/1 as 2/1 = a_1 e_1 + a2 e_2. Then:

*W(2/1, e1) = W(~~a2 e2~~, e1) = -a2 W(e1, e2) = -a2

*W(2/1, e_1) = W(a_2 e_2, e_1) = -a_2 W(e_1, e_2) = -a_2

*W(2/1,e2) = W(~~a1e1~~, e2) = a1 W(e1, e2) = a1.

*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, q1), ..., W(2, qn))) and the fact that e1 and e2 represent JI ratios in the 2.q1. ... qn 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). Snce 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). Snce 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!

== Truncation of wedgies ==

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 Let d = gcd(W(2, q_1), ..., W(2, q_n)). This tells you that for any JI ratio v in your JI subgroup, W(2, v) = 2n(v) for some number n(v) [that depends linearly on v]. This equation is also true when we replace 2 with any JI ratio u that is equated to 2. This tells us that for W(p, g) = 1, we (up to some choices) need p to be an interval such that d*p is equated to 2/1, i.e. p represents 1/d of the octave.
-Choose a basis e1, e2 for the temperament group and write (the image of) 2/1 as 2/1 = a1 e1 + a2 e2. Then:
+Choose a basis e_1, e_2 for the temperament group and write (the image of) 2/1 as 2/1 = a_1 e_1 + a2 e_2. Then:
-*W(2/1, e1) = W(a2 e2, e1) = -a2 W(e1, e2) = -a2
+*W(2/1, e_1) = W(a_2 e_2, e_1) = -a_2 W(e_1, e_2) = -a_2
-*W(2/1,e2) = W(a1e1, e2) = a1 W(e1, e2) = a1.
+*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, q1), ..., W(2, qn))) and the fact that e1 and e2 represent JI ratios in the 2.q1. ... qn 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). Snce 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). Snce 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!
 == Truncation of wedgies ==