Wedgie/Archived version: Difference between revisions

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To find the '''period''': Let ''d'' = gcd(W('''2''', '''q'''1), …, W('''2''', '''q'''''n'')). Then your period is 1\''d''.

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'').

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 [[POTE tuning|POTE]] tunings are based on minimizing [[TE error]], and those tunings can be found using the x31eq temperament finder.

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 To find the '''period''': Let ''d'' = gcd(W('''2''', '''q'''<sub>1</sub>), …, W('''2''', '''q'''<sub>''n''</sub>)). Then your period is 1\''d''.
-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>).
+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 [[POTE tuning|POTE]] tunings are based on minimizing [[TE error]], and those tunings can be found using the x31eq temperament finder.