Wedgie/Archived version: Difference between revisions

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The '''wedgie''' is a way of defining and working with an [[abstract regular temperament]]. If one takes r independent [[vals]] in a p-limit group of n primes, then the wedgie is defined by taking the [https://en.wikipedia.org/wiki/Wedge_product wedge product] of the vals (called a '''multival'''), 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.

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

The following is a procedure for finding a period and a generator for a rank-2 regular temperament on the 2.q_1.(...).q_n [[Subgroup temperaments|JI subgroup]]~~, given the temperament's wedgie~~. We also give a (hopefully convincing and enlightening) proof of why the procedure always works. We'll assume that the equave (interval of equivalence) is the octave, but non-octave JI equaves can be substituted for the octave if needed, by substituting the appropriate JI ratio for 2/1.

The following is a procedure for finding a period and a generator for a rank-2 regular temperament on the 2.q_1.(...).q_n [[Subgroup temperaments|JI subgroup]]. We also give a (hopefully convincing and enlightening) proof of why the procedure always works. We'll assume that the equave (interval of equivalence) is the octave, but non-octave JI equaves can be substituted for the octave if needed, by substituting the appropriate JI ratio for 2/1.

The following 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 a [[val]] is

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

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

===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.)

Consider the entries of the wedgie W~~. The entries of W~~ are W(2, q_1), ..., W(2, q_n), and W(q_i, q_j) for i < j.

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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===Example===

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.

~~Consider the wedgie W~~ = ~~<<1 4 4|| for 2.3.5 meantone. 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===

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

~~===Proof===~~

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.

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 The '''wedgie''' is a way of defining and working with an [[abstract regular temperament]]. If one takes r independent [[vals]] in a p-limit group of n primes, then the wedgie is defined by taking the [https://en.wikipedia.org/wiki/Wedge_product wedge product] of the vals (called a '''multival'''), 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.
 ==How the period and generator falls out of a rank-2 wedgie==
-The following is a procedure for finding a period and a generator for a rank-2 regular temperament on the 2.q_1.(...).q_n [[Subgroup temperaments|JI subgroup]], given the temperament's wedgie. We also give a (hopefully convincing and enlightening) proof of why the procedure always works. We'll assume that the equave (interval of equivalence) is the octave, but non-octave JI equaves can be substituted for the octave if needed, by substituting the appropriate JI ratio for 2/1.
+The following is a procedure for finding a period and a generator for a rank-2 regular temperament on the 2.q_1.(...).q_n [[Subgroup temperaments|JI subgroup]]. We also give a (hopefully convincing and enlightening) proof of why the procedure always works. We'll assume that the equave (interval of equivalence) is the octave, but non-octave JI equaves can be substituted for the octave if needed, by substituting the appropriate JI ratio for 2/1.
 The following 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 a [[val]] is
+* you know what a [[val]] is.
-* you know what the words "basis", "linear map", and "determinant" mean.
 ===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.)
-Consider the entries of the wedgie W. The entries of W are W(2, q_1), ..., W(2, q_n), and W(q_i, q_j) for i < j.
 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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 ===Example===
+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.
-Consider the wedgie W = &lt;&lt;1 4 4|| for 2.3.5 meantone. 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===
+The following additionally assumes that you know what the words "basis", "linear map", and "determinant" mean.
-===Proof===
 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.