Wedgie/Archived version: Difference between revisions

Line 77:

*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''''2'p''', '''g''') = ''d''W('''p''', '''g''') = ''d'', we have W('''p''', '''g''') = 1. Ta-da!

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!

== Gene Ward Smith's introduction ==

@@ Line 77: / Line 77: @@
 *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''''2'p''', '''g''') = ''d''W('''p''', '''g''') = ''d'', we have W('''p''', '''g''') = 1. Ta-da!
+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!
 == Gene Ward Smith's introduction ==