Wedgie/Archived version: Difference between revisions

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To find (a JI interpretation of) the '''generator''': Solve the equation W('''2''', '''g''') = ''c''1W('''2''', '''q'''1) + … + ''c''''n''W('''2''', '''q'''''n'') = ''d'' for the coefficients ''c''1, ..., ''c''''n'' (using some algorithm such as the [[Wikipedia: Extended Euclidean algorithm|extended Euclidean algorithm]]). Then one valid generator for the temperament is '''g''' = (the tempered version of) ''q''1''c''1 … ''q''''n''''c''''n'' (written additively, a linear combination '''g''' = ''c''1'''q'''1 + … + ''c''''n'''''q'''''n'').

Now choosing an optimal tuning for the temperament is a matter of choosing a way to measure error from JI and minimizing the error. For example, the [[TE tuning|TE]] and [[CTE tuning|CTE]] tunings are based on minimizing [[TE error]], and those tunings can be found using [https://sintel.pythonanywhere.com/ Sintel's temperament finder].

{{Proof|contents= The following additionally assumes that you know what the words "basis", "linear map", and "determinant" mean.

~~=== Example ===~~

~~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. 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) ==~~=

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

Consider the 2.''q''1.(…).q''n'' [[JI subgroup]], with basis '''2''', '''q'''1, ..., '''q'''''n''.

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*W('''2''', '''e'''1) = W(''λ''2'''e'''2, '''e'''1) = −''λ''2W('''e'''1, '''e'''2) = −''λ''2

*W('''2''', '''e'''2) = W(''λ''1'''e'''1, '''e'''2) = ''λ''1W('''e'''1, '''e'''2) = ''λ''1.

Divisibility by ''d'' and the fact that '''e'''1 and '''e'''2 represent JI ratios in the 2.''q''1.[...].''q''''n'' subgroup imply that ''λ''1 and ''λ''2 are both divisible by ''d'', and hence '''2''' is mapped to a ''d''th power in ''M' '' (the temperament space). Since gcd(W('''2''', '''q'''1), ..., W('''2''', '''q'''''n'')) = ''d'', we can always find a linear combination ''g'' = ''c''1'''q'''1 + ... + ''c''''n'''''q'''''n'' such that W('''2''', '''g''') = ''c''1W('''2''', '''q'''1) + ... + ''c''''n''W('''2''', '''q'''''n'') = ''d'' using the extended Euclidean algorithm. Then since W('''2''', '''g''') = W(''d'''''p''', '''g''') = ''d''W('''p''', '''g''') = ''d'', we have W('''p''', '''g''') = 1. {{~~qed~~}}

Divisibility by ''d'' and the fact that '''e'''1 and '''e'''2 represent JI ratios in the 2.''q''1.[...].''q''''n'' subgroup imply that ''λ''1 and ''λ''2 are both divisible by ''d'', and hence '''2''' is mapped to a ''d''th power in ''M' '' (the temperament space). Since gcd(W('''2''', '''q'''1), ..., W('''2''', '''q'''''n'')) = ''d'', we can always find a linear combination ''g'' = ''c''1'''q'''1 + ... + ''c''''n'''''q'''''n'' such that W('''2''', '''g''') = ''c''1W('''2''', '''q'''1) + ... + ''c''''n''W('''2''', '''q'''''n'') = ''d'' using the extended Euclidean algorithm. Then since W('''2''', '''g''') = W(''d'''''p''', '''g''') = ''d''W('''p''', '''g''') = ''d'', we have W('''p''', '''g''') = 1. Ta-da!}}

Now choosing an optimal tuning for the temperament is a matter of choosing a way to measure error from JI and minimizing the error. For example, the [[TE tuning|TE]] and [[CTE tuning|CTE]] tunings are based on minimizing [[TE error]], and those tunings can be found using [https://sintel.pythonanywhere.com/ Sintel's temperament finder].

=== Example ===

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. 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.

== Gene Ward Smith's introduction ==

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 To find (a JI interpretation of) the '''generator''': Solve the equation W('''2''', '''g''') = ''c''<sub>1</sub>W('''2''', '''q'''<sub>1</sub>) + … + ''c''<sub>''n''</sub>W('''2''', '''q'''<sub>''n''</sub>) = ''d'' for the coefficients ''c''<sub>1</sub>, ..., ''c''<sub>''n''</sub> (using some algorithm such as the [[Wikipedia: Extended Euclidean algorithm|extended Euclidean algorithm]]). Then one valid generator for the temperament is '''g''' = (the tempered version of) ''q''<sub>1</sub><sup>''c''<sub>1</sub></sup> … ''q''<sub>''n''</sub><sup>''c''<sub>''n''</sub></sup> (written additively, a linear combination '''g''' = ''c''<sub>1</sub>'''q'''<sub>1</sub> + … + ''c''<sub>''n''</sub>'''q'''<sub>''n''</sub>).
-Now choosing an optimal tuning for the temperament is a matter of choosing a way to measure error from JI and minimizing the error. For example, the [[TE tuning|TE]] and [[CTE tuning|CTE]] tunings are based on minimizing [[TE error]], and those tunings can be found using [https://sintel.pythonanywhere.com/ Sintel's temperament finder].
+{{Proof|contents= The following additionally assumes that you know what the words "basis", "linear map", and "determinant" mean.
-=== Example ===
-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. 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) ===
-The following additionally assumes that you know what the words "basis", "linear map", and "determinant" mean.
 Consider the 2.''q''<sub>1</sub>.(…).q<sub>''n''</sub> [[JI subgroup]], with basis '''2''', '''q'''<sub>1</sub>, ..., '''q'''<sub>''n''</sub>.
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 *W('''2''', '''e'''<sub>1</sub>) = W(''λ''<sub>2</sub>'''e'''<sub>2</sub>, '''e'''<sub>1</sub>) = &minus;''λ''<sub>2</sub>W('''e'''<sub>1</sub>, '''e'''<sub>2</sub>) = &minus;''λ''<sub>2</sub>
 *W('''2''', '''e'''<sub>2</sub>) = W(''λ''<sub>1</sub>'''e'''<sub>1</sub>, '''e'''<sub>2</sub>) = ''λ''<sub>1</sub>W('''e'''<sub>1</sub>, '''e'''<sub>2</sub>) = ''λ''<sub>1</sub>.
-Divisibility by ''d'' and the fact that '''e'''<sub>1</sub> and '''e'''<sub>2</sub> represent JI ratios in the 2.''q''<sub>1</sub>.[...].''q''<sub>''n''</sub> subgroup imply that ''λ''<sub>1</sub> and ''λ''<sub>2</sub> are both divisible by ''d'', and hence '''2''' is mapped to a ''d''th power in ''M' '' (the temperament space). Since gcd(W('''2''', '''q'''<sub>1</sub>), ..., W('''2''', '''q'''<sub>''n''</sub>)) = ''d'', we can always find a linear combination ''g'' = ''c''<sub>1</sub>'''q'''<sub>1</sub> + ... + ''c''<sub>''n''</sub>'''q'''<sub>''n''</sub> such that W('''2''', '''g''') = ''c''<sub>1</sub>W('''2''', '''q'''<sub>1</sub>) + ... + ''c''<sub>''n''</sub>W('''2''', '''q'''<sub>''n''</sub>) = ''d'' using the extended Euclidean algorithm. Then since W('''2''', '''g''') = W(''d'''''p''', '''g''') = ''d''W('''p''', '''g''') = ''d'', we have W('''p''', '''g''') = 1. {{qed}}
+Divisibility by ''d'' and the fact that '''e'''<sub>1</sub> and '''e'''<sub>2</sub> represent JI ratios in the 2.''q''<sub>1</sub>.[...].''q''<sub>''n''</sub> subgroup imply that ''λ''<sub>1</sub> and ''λ''<sub>2</sub> are both divisible by ''d'', and hence '''2''' is mapped to a ''d''th power in ''M' '' (the temperament space). Since gcd(W('''2''', '''q'''<sub>1</sub>), ..., W('''2''', '''q'''<sub>''n''</sub>)) = ''d'', we can always find a linear combination ''g'' = ''c''<sub>1</sub>'''q'''<sub>1</sub> + ... + ''c''<sub>''n''</sub>'''q'''<sub>''n''</sub> such that W('''2''', '''g''') = ''c''<sub>1</sub>W('''2''', '''q'''<sub>1</sub>) + ... + ''c''<sub>''n''</sub>W('''2''', '''q'''<sub>''n''</sub>) = ''d'' using the extended Euclidean algorithm. Then since W('''2''', '''g''') = W(''d'''''p''', '''g''') = ''d''W('''p''', '''g''') = ''d'', we have W('''p''', '''g''') = 1. Ta-da!}}
+Now choosing an optimal tuning for the temperament is a matter of choosing a way to measure error from JI and minimizing the error. For example, the [[TE tuning|TE]] and [[CTE tuning|CTE]] tunings are based on minimizing [[TE error]], and those tunings can be found using [https://sintel.pythonanywhere.com/ Sintel's temperament finder].
+=== Example ===
+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. 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.
 == Gene Ward Smith's introduction ==