The wedgie: Difference between revisions

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===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 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 procedure:'''

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'''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. 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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 ===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 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 procedure:'''
@@ Line 18: / Line 23: @@
 '''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. 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.