Wedgie/Archived version: Difference between revisions

Line 44:

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'') = a1 W(2, ''q''1) + … a''n'' W(2, q''n'') = ''d'' for the coefficinets a1, ..., an (using some algorithm such as the [[Wikipedia: Extended Euclidean algorithm|extended Euclidean algorithm]]). Then one valid generator for the temperament is (the tempered version of) ''g'' = ''q''1''a''1 … q''n''''a''''n'' (equivalently, a linear combination g = ''a''1''q''1 + … + ''a''''n''''q''''n'').

To find (a JI interpretation of) the '''generator''': Solve the equation W(2, ''g'') = ''a''1 W(2, ''q''1) + … ''a''''n'' W(2, q''n'') = ''d'' for the coefficinets ''a''1, ..., ''a''n (using some algorithm such as the [[Wikipedia: Extended Euclidean algorithm|extended Euclidean algorithm]]). Then one valid generator for the temperament is (the tempered version of) ''g'' = ''q''1''a''1 … q''n''''a''''n'' (equivalently, a linear combination g = ''a''1''q''1 + … + ''a''''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 with linear algebra (read: a program such as the x31eq temperament finder). For example, the [[TE tuning|TE]] and [[POTE tuning|POTE]] tunings are based on minimizing [[TE error]].

@@ Line 44: / Line 44: @@
 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'') = a<sub>1</sub> W(2, ''q''<sub>1</sub>) + … a<sub>''n''</sub> W(2, q<sub>''n''</sub>) = ''d'' for the coefficinets a<sub>1</sub>, ..., a<sub>n</sub> (using some algorithm such as the [[Wikipedia: Extended Euclidean algorithm|extended Euclidean algorithm]]). Then one valid generator for the temperament is (the tempered version of) ''g'' = ''q''<sub>1</sub><sup>''a''<sub>1</sub></sup> … q<sub>''n''</sub><sup>''a''<sub>''n''</sub></sup> (equivalently, a linear combination g = ''a''<sub>1</sub>''q''<sub>1</sub> + … + ''a''<sub>''n''</sub>''q''<sub>''n''</sub>).
+To find (a JI interpretation of) the '''generator''': Solve the equation W(2, ''g'') = ''a''<sub>1</sub> W(2, ''q''<sub>1</sub>) + … ''a''<sub>''n''</sub> W(2, q<sub>''n''</sub>) = ''d'' for the coefficinets ''a''<sub>1</sub>, ..., ''a''<sub>n</sub> (using some algorithm such as the [[Wikipedia: Extended Euclidean algorithm|extended Euclidean algorithm]]). Then one valid generator for the temperament is (the tempered version of) ''g'' = ''q''<sub>1</sub><sup>''a''<sub>1</sub></sup> … q<sub>''n''</sub><sup>''a''<sub>''n''</sub></sup> (equivalently, a linear combination g = ''a''<sub>1</sub>''q''<sub>1</sub> + … + ''a''<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 with linear algebra (read: a program such as the x31eq temperament finder). For example, the [[TE tuning|TE]] and [[POTE tuning|POTE]] tunings are based on minimizing [[TE error]].