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''' 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.

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''', 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:

*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 2/1 is 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!

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!

== Gene Ward Smith's introduction ==

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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''' 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.
+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''', 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:
 *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 2/1 is 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!
+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!
 == Gene Ward Smith's introduction ==