Wedgie/Archived version: Difference between revisions

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Consider the wedgie W = {{wedgie|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. Further, we have that 1*W('''2''', '''3''') + 0*W('''2''', '''5''') = 1, so ''c''1 = 1, ''c''2 = 0 is one solution, and we can use 3150 = 3/1 as our generator.

Note that -3*W('''2''', '''3''') + 1*W('''2''', '''5''') = −3*1 + 1*4 = 1, so ''c''1 = −3, ''c''2 = 1 is another solution to the equation. Thus 5/27 is also a valid generator. This octave reduces to the [[40/27]] grave fifth, which is equated to 3/2 in meantone.

Note that -3*W('''2''', '''3''') + 1*W('''2''', '''5''') = −3*1 + 1*4 = 1, so ''c''1 = −3, ''c''2 = 1 is another solution to the equation. Thus 5/27 is also a valid generator. This octave reduces to the [[40/27]] grave fifth, which is equated to 3/2 in meantone. This is an example of how any solution of the equation corresponds to a valid generator, and when two solutions correspond to the "same" generator on the nose, the difference between the solutions corresponds to a comma that is tempered out by the temperament.

=== Proof (a bit technical) ===

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 Consider the wedgie W = {{wedgie|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. Further, we have that 1*W('''2''', '''3''') + 0*W('''2''', '''5''') = 1, so ''c''<sub>1</sub> = 1, ''c''<sub>2</sub> = 0 is one solution, and we can use 3<sup>1</sup>5<sup>0</sup> = 3/1 as our generator.
-Note that -3*W('''2''', '''3''') + 1*W('''2''', '''5''') = &minus;3*1 + 1*4 = 1, so ''c''<sub>1</sub> = &minus;3, ''c''<sub>2</sub> = 1 is another solution to the equation. Thus 5/27 is also a valid generator. This octave reduces to the [[40/27]] grave fifth, which is equated to 3/2 in meantone.
+Note that -3*W('''2''', '''3''') + 1*W('''2''', '''5''') = &minus;3*1 + 1*4 = 1, so ''c''<sub>1</sub> = &minus;3, ''c''<sub>2</sub> = 1 is another solution to the equation. Thus 5/27 is also a valid generator. This octave reduces to the [[40/27]] grave fifth, which is equated to 3/2 in meantone. This is an example of how any solution of the equation corresponds to a valid generator, and when two solutions correspond to the "same" generator on the nose, the difference between the solutions corresponds to a comma that is tempered out by the temperament.
 === Proof (a bit technical) ===