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.

If you can convince yourself of interpretation (*), then it follows that we always have a JI interpretation for the period p. 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(d*p, g) = d*W(p,g) = d, we have W(p,g) = 1. Ta-da!

If you can convince yourself of interpretation (*), then it follows that the period 1\d exists in the temperament space. Since the map defining the temperament is surjective, we always have a JI interpretation for the period p. 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(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.
-If you can convince yourself of interpretation (*), then it follows that we always have a JI interpretation for the period p. 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(d*p, g) = d*W(p,g) = d, we have W(p,g) = 1. Ta-da!
+If you can convince yourself of interpretation (*), then it follows that the period 1\d exists in the temperament space. Since the map defining the temperament is surjective, we always have a JI interpretation for the period p. 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(d*p, g) = d*W(p,g) = d, we have W(p,g) = 1. Ta-da!
 == Truncation of wedgies ==