Wedgie/Archived version: Difference between revisions

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The key fact about the determinant we use here is that two integer vectors '''v'''1, '''v'''2 form a basis for the rank-2 integer lattice '''Z'''2 iff det('''v'''1, '''v'''2) = ±1. So in order to find a period and generator for our temperament, we need a pair of vectors {'''p''', '''g'''} such that W('''p''', '''g''') = 1 and '''p''' is 1\''d'' for some integer ''d''.

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.

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''' with any JI ratio '''u''' that is equated to '''2'''. 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''' as '''2''' = ''λ''1'''e'''1 + ''λ''2'''e'''2. Then:

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 The key fact about the determinant we use here is that two integer vectors '''v'''<sub>1</sub>, '''v'''<sub>2</sub> form a basis for the rank-2 integer lattice '''Z'''<sup>2</sup> iff det('''v'''<sub>1</sub>, '''v'''<sub>2</sub>) = ±1. So in order to find a period and generator for our temperament, we need a pair of vectors {'''p''', '''g'''} such that W('''p''', '''g''') = 1 and '''p''' is 1\''d'' for some integer ''d''.
-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.
+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''' with any JI ratio '''u''' that is equated to '''2'''. 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''' as '''2''' = ''λ''<sub>1</sub>'''e'''<sub>1</sub> + ''λ''<sub>2</sub>'''e'''<sub>2</sub>. Then: