Wedgie/Archived version: Difference between revisions

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* you know what a [[val]] is.

===The procedure===

Consider the rank-2 temperament a&b, where a and b are two [[val]]s. Then the entries of the wedgie W corresponding to a&b are W(2, q_1), ..., W(2, q_n), and W(q_i, q_j) for i < j, and the entry W(p,q) is given by a(p)b(q) - a(q)b(p). (This is how the wedge product of two 1-forms a and b works.)

[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. This is the determinant of the tempered versions of u and 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.]

To find the '''period''': Let d = gcd(W(2, q_1), ..., W(2, q_n)). Then your period is 1\d.

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Consider the wedgie W = <<1 4 4|| for 2.3.5 meantone (the 12&19 temperament). We have W(2,3) = 1 and W(2,5) = 4, 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.

===Proof===

===Proof (a bit technical)===

The following additionally assumes that you know what the words "basis", "linear map", and "determinant" mean.

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By the First Isomorphism Theorem it follows that M' is the group of intervals in the rank-2 temperament in question.

[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. This is the determinant of the tempered versions of u and 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.]

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 9: / Line 9: @@
 * you know what a [[val]] is.
 ===The procedure===
 Consider the rank-2 temperament a&b, where a and b are two [[val]]s. Then the entries of the wedgie W corresponding to a&b are W(2, q_1), ..., W(2, q_n), and W(q_i, q_j) for i < j, and the entry W(p,q) is given by a(p)b(q) - a(q)b(p). (This is how the wedge product of two 1-forms a and b works.)
+[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. This is the determinant of the tempered versions of u and 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.]
 To find the '''period''': Let d = gcd(W(2, q_1), ..., W(2, q_n)). Then your period is 1\d.
@@ Line 18: / Line 20: @@
 Consider the wedgie W = &lt;&lt;1 4 4|| for 2.3.5 meantone (the 12&19 temperament). We have W(2,3) = 1 and W(2,5) = 4, 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.
-===Proof===
+===Proof (a bit technical)===
 The following additionally assumes that you know what the words "basis", "linear map", and "determinant" mean.
@@ Line 28: / Line 30: @@
 By the First Isomorphism Theorem it follows that M' is the group of intervals in the rank-2 temperament in question.
-[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. This is the determinant of the tempered versions of u and 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.]
 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.