The wedgie: Difference between revisions

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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.

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

The following 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 convincing and enlightening) proof ~~sketch~~ of why the procedure always works. We'll assume that the [[interval of equivalence|equave]] is the octave, but non-octave equaves can be substituted for the octave if needed.

The following 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 convincing and enlightening) proof of why the procedure always works. We'll assume that the [[interval of equivalence|equave]] is the octave, but non-octave equaves can be substituted for the octave if needed.

'''The procedure:'''

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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~~:'''

'''Proof:'''

This explanation assumes that:

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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.

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. Using the fact that W = a&b where a and b are two edos (properly, rank-1 [[val]]s), you 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.

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 (*).

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 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.
 ===How the period and generator falls out of a rank-2 wedgie===
-The following 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 convincing and enlightening) proof sketch of why the procedure always works. We'll assume that the [[interval of equivalence|equave]] is the octave, but non-octave equaves can be substituted for the octave if needed.
+The following 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 convincing and enlightening) proof of why the procedure always works. We'll assume that the [[interval of equivalence|equave]] is the octave, but non-octave equaves can be substituted for the octave if needed.
 '''The procedure:'''
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 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:'''
+'''Proof:'''
 This explanation assumes that:
@@ Line 26: / Line 26: @@
 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.
+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. Using the fact that W = a&b where a and b are two edos (properly, rank-1 [[val]]s), you 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.
 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 (*).