The wedgie: Difference between revisions

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* you know what the words "basis", "linear map", and "determinant" mean.

The period ''p'' (fraction of octave) and generator ''g'' form a basis for all the intervals of a rank-2 temperament. For example, p = 2/1 and g = 3/2 form a basis for meantone. But from a linear algebra perspective, there's nothing special about the basis {p, g}; I could have chosen the basis p' = 3/1 and g' = 2/1. What makes the wedgie a unique identifier for a temperament is that rather than specify ~~the period and generator~~ directly, the wedgie acts more like a set of constraints that any basis for the temperament must satisfy.

The period ''p'' (fraction of octave) and generator ''g'' form a basis for all the intervals of a rank-2 temperament. For example, p = 2/1 and g = 3/2 form a basis for meantone. But from a linear algebra perspective, there's nothing special about the basis {p, g}; I could have chosen the basis p' = 3/1 and g' = 2/1. What makes the wedgie a unique identifier for a temperament is that rather than specify a basis directly, the wedgie acts more like a set of constraints that any basis for the temperament must satisfy.

In the language of linear algebra, the wedgie is an "alternating bilinear form" on the appropriate JI group M; this means that it acts like the operation of finding the determinant of two vectors on the appropriate quotient module M' = M/K of M, where K is the kernel of the biliear form W. One can verify that K is exactly the kernel of the 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. This is the determinant of the tempered versions of u and v.

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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 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!

Choose a basis e1, e2 for the module and write 2/1 = a1 e1 + a2 e2. Then:

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

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

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).

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 ==

@@ Line 24: / Line 24: @@
 * you know what the words "basis", "linear map", and "determinant" mean.
-The period ''p'' (fraction of octave) and generator ''g'' form a basis for all the intervals of a rank-2 temperament. For example, p = 2/1 and g = 3/2 form a basis for meantone. But from a linear algebra perspective, there's nothing special about the basis {p, g}; I could have chosen the basis p' = 3/1 and g' = 2/1. What makes the wedgie a unique identifier for a temperament is that rather than specify the period and generator directly, the wedgie acts more like a set of constraints that any basis for the temperament must satisfy.
+The period ''p'' (fraction of octave) and generator ''g'' form a basis for all the intervals of a rank-2 temperament. For example, p = 2/1 and g = 3/2 form a basis for meantone. But from a linear algebra perspective, there's nothing special about the basis {p, g}; I could have chosen the basis p' = 3/1 and g' = 2/1. What makes the wedgie a unique identifier for a temperament is that rather than specify a basis directly, the wedgie acts more like a set of constraints that any basis for the temperament must satisfy.
 In the language of linear algebra, the wedgie is an "alternating bilinear form" on the appropriate JI group M; this means that it acts like the operation of finding the determinant of two vectors on the appropriate quotient module M' = M/K of M, where K is the kernel of the biliear form W. One can verify that K is exactly the kernel of the 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. This is the determinant of the tempered versions of u and v.
@@ 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 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!
+Choose a basis e1, e2 for the module and write 2/1 = a1 e1 + a2 e2. Then:
+*W(2/1, e1) = W(a2 e2, e1) = -a2 W(e1, e2) = -a2
+*W(2/1,e2) = W(a1e1, e2) = a1 W(e1, e2) = a1.
+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).
+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 ==