Wedgie/Archived version: Difference between revisions

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Let ''d'' = gcd(W('''2''', '''q'''1), ..., W('''2''', '''q'''''n'')). This tells you that for any JI ratio '''v''' in your JI subgroup, W('''2''', '''v''') = 2''N''('''v''') for some number ''N''('''v''') [that depends linearly on '''v''']. This equation is also true when we replace 2/1 with any JI ratio '''u''' that is equated to 2/1. This tells us that for W('''p''', '''g''') = 1, we (up to some choices) need '''p''' to be a JI ratio such that ''d'''''p''' is equated to 2/1, i.e. '''p''' represents 1/''d'' of the octave.

Choose a basis '''e'''1, '''e'''2 for the temperament group and write (the image of) 2/1 as 2/1 = ''λ''1'''e'''1 + ''λ''2'''e'''2. Then:

Choose a basis '''e'''1, '''e'''2 for the temperament group and write (the image of) '''2''' as '''2''' = ''λ''1'''e'''1 + ''λ''2'''e'''2. Then:

*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.

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 Let ''d'' = gcd(W('''2''', '''q'''<sub>1</sub>), ..., W('''2''', '''q'''<sub>''n''</sub>)). This tells you that for any JI ratio '''v''' in your JI subgroup, W('''2''', '''v''') = 2''N''('''v''') for some number ''N''('''v''') [that depends linearly on '''v''']. This equation is also true when we replace 2/1 with any JI ratio '''u''' that is equated to 2/1. This tells us that for W('''p''', '''g''') = 1, we (up to some choices) need '''p''' to be a JI ratio such that ''d'''''p''' is equated to 2/1, i.e. '''p''' represents 1/''d'' of the octave.
-Choose a basis '''e'''<sub>1</sub>, '''e'''<sub>2</sub> for the temperament group and write (the image of) 2/1 as 2/1 = ''λ''<sub>1</sub>'''e'''<sub>1</sub> + ''λ''<sub>2</sub>'''e'''<sub>2</sub>. Then:
+Choose a basis '''e'''<sub>1</sub>, '''e'''<sub>2</sub> for the temperament group and write (the image of) '''2''' as '''2''' = ''λ''<sub>1</sub>'''e'''<sub>1</sub> + ''λ''<sub>2</sub>'''e'''<sub>2</sub>. Then:
 *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>.