The wedgie: Difference between revisions

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In the language of linear algebra, the wedgie is an "alternating bilinear form"; this means that it acts like the operation of finding the determinant of two vectors on the space of intervals of your rank-2 temperament. In geometric terms, given JI ratios u and v, and wedgie W, the number W(u,v) is the signed area of the parallelogram spanned by (tempered versions of) u and v. The entries of the wedgie give the values of the wedgie on the basis elements of the JI subgroup that the temperament is on. By the alternating property [i.e. W(u, v) = -W(v, u)] and bilinearity [W is linear in each argument separately], specifying the values on basis elements of the JI subgroup is enough to define W as an alternating bilinear form on all of the JI subgroup.

The musical interpretation of the parallelogram spanned by u and v is: If you want to consider intervals that are multiples of u apart the same note (for example, if you want an octave-equivalent scale), W(u, v) tells you how many generators it take to get to v.

The musical interpretation of the parallelogram spanned by u and v is: If you want to consider intervals that are multiples of u apart the same note (for example, if you want an octave-equivalent scale), W(u, v) tells you how many generators it take to get to v. Let's label this (*).

The key fact about the determinant we use here is that two integer vectors v_1, v_2 form a basis for the rank-2 integer lattice '''Z'''<sup>2</sup> iff det(v_1, v_2) = ±1. So in order to find a period and generator for our tempearment, we need a pair of vectors {p, g} such that W(p, g) = 1 and p is 1\d for some integer d.

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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 ~~believe me~~ 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 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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 In the language of linear algebra, the wedgie is an "alternating bilinear form"; this means that it acts like the operation of finding the determinant of two vectors on the space of intervals of your rank-2 temperament. In geometric terms, given JI ratios u and v, and wedgie W, the number W(u,v) is the signed area of the parallelogram spanned by (tempered versions of) u and v. The entries of the wedgie give the values of the wedgie on the basis elements of the JI subgroup that the temperament is on. By the alternating property [i.e. W(u, v) = -W(v, u)] and bilinearity [W is linear in each argument separately], specifying the values on basis elements of the JI subgroup is enough to define W as an alternating bilinear form on all of the JI subgroup.
-The musical interpretation of the parallelogram spanned by u and v is: If you want to consider intervals that are multiples of u apart the same note (for example, if you want an octave-equivalent scale), W(u, v) tells you how many generators it take to get to v.
+The musical interpretation of the parallelogram spanned by u and v is: If you want to consider intervals that are multiples of u apart the same note (for example, if you want an octave-equivalent scale), W(u, v) tells you how many generators it take to get to v. Let's label this (*).
 The key fact about the determinant we use here is that two integer vectors v_1, v_2 form a basis for the rank-2 integer lattice '''Z'''<sup>2</sup> iff det(v_1, v_2) = ±1. So in order to find a period and generator for our tempearment, we need a pair of vectors {p, g} such that W(p, g) = 1 and p is 1\d for some integer d.
@@ Line 34: / Line 34: @@
 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 believe me 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 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 ==