The wedgie: Difference between revisions

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== Basics ==

The ''[[Wedgies_and_Multivals|wedgie]]'' is a way of defining and working with an [[Abstract_regular_temperament|abstract regular temperament]]. If one takes r independent [[Vals|vals]] in a p-limit group of n primes, then the wedgie is defined by taking the [[Wedgies_and_Multivals|wedge product]] of the vals, and dividing out the greatest common divisior of the coefficients, to produce an r-multival. If the first non-zero coefficient of this multival is negative, it is then scalar multiplied by -1, changing the sign of the first non-zero coefficient to be positive. The result is the wedgie. Wedgies are in a one-to-one relationship with abstract regular temperaments; that is, regular temperaments where no tuning has been decided on.

===For beginners: How the period and generator falls out of a rank-2 wedgie===

This is both a procedure for finding a period and a generator for a rank-2 regular temperament given the temperament's wedgie, and a (hopefully enlightening) proof sketch of why the procedure always works.

'''The procedure:'''

Consider the entries of the wedgie W. The first few elements are: W(2, q_1), ..., W(2, q_n), assuming that your temperament is on the 2.q_1.(...).q_n JI subgroup.

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

To find the '''generator''': Treat W(2, v) as a linear map where you plug in a JI vector v, and use the [[w:extended Euclidean algorithm|extended Euclidean algorithm]] to find the 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.

For example, consider the wedgie for <<1 4 10 4 13 12||. We have W(2,3) = 1, W(2,5) = 4, W(2,7) = 10, 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 sketch:'''

This explanation assumes that:

* you can think of JI ratios as vectors living in the n-dimensional lattice of the "JI subgroup"

* you know what a "period" and a "generator" of a rank-2 temperament are

* 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. So rather than give the period and generator 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"; 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 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 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.

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.

Take for granted that we already 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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 == Basics ==
 The ''[[Wedgies_and_Multivals|wedgie]]'' is a way of defining and working with an [[Abstract_regular_temperament|abstract regular temperament]]. If one takes r independent [[Vals|vals]] in a p-limit group of n primes, then the wedgie is defined by taking the [[Wedgies_and_Multivals|wedge product]] of the vals, and dividing out the greatest common divisior of the coefficients, to produce an r-multival. If the first non-zero coefficient of this multival is negative, it is then scalar multiplied by -1, changing the sign of the first non-zero coefficient to be positive. The result is the wedgie. Wedgies are in a one-to-one relationship with abstract regular temperaments; that is, regular temperaments where no tuning has been decided on.
+===For beginners: How the period and generator falls out of a rank-2 wedgie===
+This is both a procedure for finding a period and a generator for a rank-2 regular temperament given the temperament's wedgie, and a (hopefully enlightening) proof sketch of why the procedure always works.
+'''The procedure:'''
+Consider the entries of the wedgie W. The first few elements are: W(2, q_1), ..., W(2, q_n), assuming that your temperament is on the 2.q_1.(...).q_n JI subgroup.
+To find the '''period''': let d = gcd(W(2, q_1), ..., W(2, q_n)). Then your period is 1\d.
+To find the '''generator''': Treat W(2, v) as a linear map where you plug in a JI vector v, and use the [[w:extended Euclidean algorithm|extended Euclidean algorithm]] to find the 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.
+For example, consider the wedgie for &lt;&lt;1 4 10 4 13 12||. We have W(2,3) = 1, W(2,5) = 4, W(2,7) = 10, 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 sketch:'''
+This explanation assumes that:
+* you can think of JI ratios as vectors living in the n-dimensional lattice of the "JI subgroup"
+* you know what a "period" and a "generator" of a rank-2 temperament are
+* 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. So rather than give the period and generator 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"; 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 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 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.
+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.
+Take for granted that we already 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 ==