Wedgie/Archived version: Difference between revisions

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The period '''p''' (fraction of octave) and generator '''g''' form a basis for all the intervals of a rank-2 temperament. For example, '''p''' = 2/1 and '''g''' = 3/2 form a basis for meantone. But from a purely linear-algebra perspective, there's nothing special about the basis {'''p''', '''g'''}; I could have chosen another basis, for example '''p'''' = 3/1 for my "period" and '''g'''' = 2/1 for my "generator". What makes the wedgie a unique identifier for a temperament is that rather than specify a basis directly, the wedgie specifies a ''constraint'' that any basis for the temperament must satisfy: namely, that a basis '''e'''1, '''e'''2 must satisfy W('''e'''1, '''e'''2) = ±1.

In the language of linear algebra, the wedgie is an "alternating bilinear form" on the appropriate JI group ''M''; this means that (ignoring sign) it acts like the operation of finding the determinant of two vectors on the appropriate quotient group ''M' '' = ''M''/''K'' of ''M'', where ''K'' is the kernel of the bilinear form W. Using the fact that W = a&b where a and b are two edos (properly, rank-1 [[val]]s), you can verify that K is exactly the kernel of the rank-2 temperament, as follows. (Hence ''M''/''K' '' is a rank-2 lattice on which W is an alternating non-degenerate bilinear form, which justifies the intuition of viewing W as a determinant-like function.)

In the language of linear algebra, the wedgie is an "alternating bilinear form" on the appropriate JI group ''M''; this means that (ignoring sign) it acts like the operation of finding the determinant of two vectors on the appropriate quotient group ''M' '' = ''M''/''λ'' of ''M'', where ''λ'' is the kernel of the bilinear form W. Using the fact that W = a&b where a and b are two edos (properly, rank-1 [[val]]s), you can verify that K is exactly the kernel of the rank-2 temperament, as follows. (Hence ''M''/''K' '' is a rank-2 lattice on which W is an alternating non-degenerate bilinear form, which justifies the intuition of viewing W as a determinant-like function.)

Let ''K''1 = the kernel of the temperament (i.e. the set of commas tempered out by the temperament), and ''K''2 = ker W = {'''v''' ∈ ''M'' : W('''v''', '''w''') = 0 ∀'''w''' ∈ ''M''}. If '''v''' ∈ ''K''1, then '''v''' is tempered out by both a and b, so W('''v''', '''w''') = a('''v''')b('''w''') − a('''w''')b('''v''') = 0, and '''v''' ∈ ''K''2. Conversely, if '''v''' ∈ ''K''2, then W('''v''', '''w''') = a('''v''')b('''w''') − a('''w''')b('''v''') = 0 for all w, which implies a('''v''')b('''w''') = a('''w''')b('''v''') (*) for all w. Since a and b both have rank 1 but a&b has rank 2, a and b are linearly independent in ''M*'' (the dual '''Z'''-module of M); so we can choose '''w''' such that a('''w''') = 0 but b('''w''') ≠ 0. Then (*) shows a('''v''') = 0. By the same argument, b('''v''') = 0. So '''v''' is in ''K''1 and ''K''1 = ''K''2; the kernel of the temperament is exactly the intervals that the wedgie "treats as zero".

Let ''λ''1 = the kernel of the temperament (i.e. the set of commas tempered out by the temperament), and ''λ''2 = ker W = {'''v''' ∈ ''M'' : W('''v''', '''w''') = 0 ∀'''w''' ∈ ''M''}. If '''v''' ∈ ''λ''1, then '''v''' is tempered out by both a and b, so W('''v''', '''w''') = a('''v''')b('''w''') − a('''w''')b('''v''') = 0, and '''v''' ∈ ''λ''2. Conversely, if '''v''' ∈ ''λ''2, then W('''v''', '''w''') = a('''v''')b('''w''') − a('''w''')b('''v''') = 0 for all w, which implies a('''v''')b('''w''') = a('''w''')b('''v''') (*) for all w. Since a and b both have rank 1 but a&b has rank 2, a and b are linearly independent in ''M*'' (the dual '''Z'''-module of M); so we can choose '''w''' such that a('''w''') = 0 but b('''w''') ≠ 0. Then (*) shows a('''v''') = 0. By the same argument, b('''v''') = 0. So '''v''' is in ''λ''1 and ''λ''1 = ''λ''2; the kernel of the temperament is exactly the intervals that the wedgie "treats as zero".

By the First Isomorphism Theorem it follows that ''M' '' is the group of intervals in the rank-2 temperament in question.

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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 = ''k''1'''e'''1 + ''k''2'''e'''2. Then:

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:

*W('''2''', '''e'''1) = W(''k''2'''e'''2, '''e'''1) = −''k''2W('''e'''1, '''e'''2) = −''k''2

*W('''2''', '''e'''1) = W(''λ''2'''e'''2, '''e'''1) = −''λ''2W('''e'''1, '''e'''2) = −''λ''2

*W('''2''', '''e'''2) = W(''k''1'''e'''1, '''e'''2) = ''k''1W('''e'''1, '''e'''2) = ''k''1.

*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 ''k''1 and ''k''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 2/1 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!

== Technical introduction ==

@@ Line 58: / Line 58: @@
 The period '''p''' (fraction of octave) and generator '''g''' form a basis for all the intervals of a rank-2 temperament. For example, '''p''' = 2/1 and '''g''' = 3/2 form a basis for meantone. But from a purely linear-algebra perspective, there's nothing special about the basis {'''p''', '''g'''}; I could have chosen another basis, for example '''p'''' = 3/1 for my "period" and '''g'''' = 2/1 for my "generator". What makes the wedgie a unique identifier for a temperament is that rather than specify a basis directly, the wedgie specifies a ''constraint'' that any basis for the temperament must satisfy: namely, that a basis '''e'''<sub>1</sub>, '''e'''<sub>2</sub> must satisfy W('''e'''<sub>1</sub>, '''e'''<sub>2</sub>) = ±1.
-In the language of linear algebra, the wedgie is an "alternating bilinear form" on the appropriate JI group ''M''; this means that (ignoring sign) it acts like the operation of finding the determinant of two vectors on the appropriate quotient group ''M' '' = ''M''/''K'' of ''M'', where ''K'' is the kernel of the bilinear form W. Using the fact that W = a&b where a and b are two edos (properly, rank-1 [[val]]s), you can verify that K is exactly the kernel of the rank-2 temperament, as follows. (Hence ''M''/''K' '' is a rank-2 lattice on which W is an alternating non-degenerate bilinear form, which justifies the intuition of viewing W as a determinant-like function.)
+In the language of linear algebra, the wedgie is an "alternating bilinear form" on the appropriate JI group ''M''; this means that (ignoring sign) it acts like the operation of finding the determinant of two vectors on the appropriate quotient group ''M' '' = ''M''/''λ'' of ''M'', where ''λ'' is the kernel of the bilinear form W. Using the fact that W = a&b where a and b are two edos (properly, rank-1 [[val]]s), you can verify that K is exactly the kernel of the rank-2 temperament, as follows. (Hence ''M''/''K' '' is a rank-2 lattice on which W is an alternating non-degenerate bilinear form, which justifies the intuition of viewing W as a determinant-like function.)
-Let ''K''<sub>1</sub> = the kernel of the temperament (i.e. the set of commas tempered out by the temperament), and ''K''<sub>2</sub> = ker W = {'''v''' ∈ ''M'' : W('''v''', '''w''') = 0 ∀'''w''' ∈ ''M''}. If '''v''' ∈ ''K''<sub>1</sub>, then '''v''' is tempered out by both a and b, so W('''v''', '''w''') = a('''v''')b('''w''') &minus; a('''w''')b('''v''') = 0, and '''v''' ∈ ''K''<sub>2</sub>. Conversely, if '''v''' ∈ ''K''<sub>2</sub>, then W('''v''', '''w''') = a('''v''')b('''w''') &minus; a('''w''')b('''v''') = 0 for all w, which implies a('''v''')b('''w''') = a('''w''')b('''v''') (*) for all w. Since a and b both have rank 1 but a&b has rank 2, a and b are linearly independent in ''M*'' (the dual '''Z'''-module of M); so we can choose '''w''' such that a('''w''') = 0 but b('''w''') ≠ 0. Then (*) shows a('''v''') = 0. By the same argument, b('''v''') = 0. So '''v''' is in ''K''<sub>1</sub> and ''K''<sub>1</sub> = ''K''<sub>2</sub>; the kernel of the temperament is exactly the intervals that the wedgie "treats as zero".
+Let ''λ''<sub>1</sub> = the kernel of the temperament (i.e. the set of commas tempered out by the temperament), and ''λ''<sub>2</sub> = ker W = {'''v''' ∈ ''M'' : W('''v''', '''w''') = 0 ∀'''w''' ∈ ''M''}. If '''v''' ∈ ''λ''<sub>1</sub>, then '''v''' is tempered out by both a and b, so W('''v''', '''w''') = a('''v''')b('''w''') &minus; a('''w''')b('''v''') = 0, and '''v''' ∈ ''λ''<sub>2</sub>. Conversely, if '''v''' ∈ ''λ''<sub>2</sub>, then W('''v''', '''w''') = a('''v''')b('''w''') &minus; a('''w''')b('''v''') = 0 for all w, which implies a('''v''')b('''w''') = a('''w''')b('''v''') (*) for all w. Since a and b both have rank 1 but a&b has rank 2, a and b are linearly independent in ''M*'' (the dual '''Z'''-module of M); so we can choose '''w''' such that a('''w''') = 0 but b('''w''') ≠ 0. Then (*) shows a('''v''') = 0. By the same argument, b('''v''') = 0. So '''v''' is in ''λ''<sub>1</sub> and ''λ''<sub>1</sub> = ''λ''<sub>2</sub>; the kernel of the temperament is exactly the intervals that the wedgie "treats as zero".
 By the First Isomorphism Theorem it follows that ''M' '' is the group of intervals in the rank-2 temperament in question.
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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 = ''k''<sub>1</sub>'''e'''<sub>1</sub> + ''k''<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) 2/1 as 2/1 = ''λ''<sub>1</sub>'''e'''<sub>1</sub> + ''λ''<sub>2</sub>'''e'''<sub>2</sub>. Then:
-*W('''2''', '''e'''<sub>1</sub>) = W(''k''<sub>2</sub>'''e'''<sub>2</sub>, '''e'''<sub>1</sub>) = &minus;''k''<sub>2</sub>W('''e'''<sub>1</sub>, '''e'''<sub>2</sub>) = &minus;''k''<sub>2</sub>
+*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(''k''<sub>1</sub>'''e'''<sub>1</sub>, '''e'''<sub>2</sub>) = ''k''<sub>1</sub>W('''e'''<sub>1</sub>, '''e'''<sub>2</sub>) = ''k''<sub>1</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 ''k''<sub>1</sub> and ''k''<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 2/1 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!
 == Technical introduction ==