Wedgie/Archived version: Difference between revisions

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=== Example ===

Consider the wedgie <math>W = \bival{1 & 4 & 4}</math> for 5-limit meantone (the 12&19 temperament). We have {{nowrap|W('''2''', '''3''') {{=}} 1}} and {{nowrap|W('''2''', '''5''') {{=}} 4, so {{nowrap|''d'' {{=}} 1}}, and our period is 1\1. Further, we have that {{nowrap|1*W(~~'''~~2~~'''~~, ~~'''~~3~~'''~~) + 0*W(~~'''~~2~~'''~~, ~~'''~~5~~'''~~) {{=}} 1}}, so {{nowrap|''c''1 {{=}} 1}}, {{nowrap|''c''2 {{=}} 0}} is one solution, and we can use {{nowrap|3150 {{=}} 3/1}} as our generator.

Consider the wedgie <math>W = \bival{1 & 4 & 4}</math> for 5-limit meantone (the 12&19 temperament). We have {{nowrap|W('''2''', '''3''') {{=}} 1}} and {{nowrap|W('''2''', '''5''') {{=}} 4}}, so {{nowrap|''d'' {{=}} 1}}, and our period is 1\1. Further, we have that <math>W\left(\mathbf{2}, \mathbf{3}) + 0*W(\mathbf{2}, \mathbf{5}\right) = 1</math>, so {{nowrap|''c''1 {{=}} 1}}, {{nowrap|''c''2 {{=}} 0}} is one solution, and we can use {{nowrap|3150 {{=}} 3/1}} as our generator.

Note that <math>-3\,W\left(\textbf{2}, \textbf{3}\right) + W\left(\textbf{2}, \textbf{5}\right) = -3 * 1 + 1 * 4 = 1</math>, so {{nowrap|''c''1 {{=}} −3}}, {{nowrap|''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.

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 === Example ===
-Consider the wedgie <math>W = \bival{1 & 4 & 4}</math> for 5-limit meantone (the 12&19 temperament). We have {{nowrap|W('''2''', '''3''') {{=}} 1}} and {{nowrap|W('''2''', '''5''') {{=}} 4, so {{nowrap|''d'' {{=}} 1}}, and our period is 1\1. Further, we have that {{nowrap|1*W('''2''', '''3''') + 0*W('''2''', '''5''') {{=}} 1}}, so {{nowrap|''c''<sub>1</sub> {{=}} 1}}, {{nowrap|''c''<sub>2</sub> {{=}} 0}} is one solution, and we can use {{nowrap|3<sup>1</sup>5<sup>0</sup> {{=}} 3/1}} as our generator.
+Consider the wedgie <math>W = \bival{1 & 4 & 4}</math> for 5-limit meantone (the 12&19 temperament). We have {{nowrap|W('''2''', '''3''') {{=}} 1}} and {{nowrap|W('''2''', '''5''') {{=}} 4}}, so {{nowrap|''d'' {{=}} 1}}, and our period is 1\1. Further, we have that <math>W\left(\mathbf{2}, \mathbf{3}) + 0*W(\mathbf{2}, \mathbf{5}\right) = 1</math>, so {{nowrap|''c''<sub>1</sub> {{=}} 1}}, {{nowrap|''c''<sub>2</sub> {{=}} 0}} is one solution, and we can use {{nowrap|3<sup>1</sup>5<sup>0</sup> {{=}} 3/1}} as our generator.
 Note that <math>-3\,W\left(\textbf{2}, \textbf{3}\right) + W\left(\textbf{2}, \textbf{5}\right) = -3 * 1 + 1 * 4 = 1</math>, so {{nowrap|''c''<sub>1</sub> {{=}} &minus;3}}, {{nowrap|''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.