Wedgie/Archived version: Difference between revisions

Line 52:

To find the '''period''': Let {{nowrap|''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 {{nowrap|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 {{nowrap|'''g''' {{=}} (the tempered version of) ''q''1''c''1 … ''q''''n''''c''''n''}} (written additively, a linear combination {{nowrap|'''g''' = ''c''1'''q'''1 + … + ''c''''n'''''q'''''n'')}}.

To find (a JI interpretation of) the '''generator''': Solve the equation {{nowrap|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 {{nowrap|'''g''' {{=}} (the tempered version of) ''q''1''c''1 … ''q''''n''''c''''n''}} (written additively, a linear combination {{nowrap|'''g''' {{=}} ''c''1'''q'''1 + … + ''c''''n'''''q'''''n'')}}.

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

@@ Line 52: / Line 52: @@
 To find the '''period''': Let {{nowrap|''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 {{nowrap|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 {{nowrap|'''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 {{nowrap|'''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 {{nowrap|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 {{nowrap|'''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 {{nowrap|'''g''' {{=}} ''c''<sub>1</sub>'''q'''<sub>1</sub> + … + ''c''<sub>''n''</sub>'''q'''<sub>''n''</sub>)}}.
 {{Proof|contents= The following additionally assumes that you know what the words "basis", "linear map", and "determinant" mean.