Wedgie/Archived version: Difference between revisions
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A wedgie is written as a list of entries that give the values of the wedgie on the [[basis]] elements of the [[JI subgroup]] that the temperament is on. By the alternating property [i.e. W('''u''', '''v''') = −W('''v''', '''u''')] and bilinearity [W is linear in each argument separately], specifying the values on basis elements of the JI subgroup is enough to define W as an alternating bilinear form on all of the JI subgroup. The simplest example is rank-2 wedgies: Let a and b be (non-[[contorted]]) vals on a [[JI subgroup]] ''q''<sub>1</sub>.[…].''q''<sub>''n''</sub> (where the ''q''<sub>''i''</sub> need not be prime). Then the entries of the wedgie W corresponding to the rank-2 temperament a&b of the JI subgroup ''q''<sub>1</sub>.[…].''q''<sub>''n''</sub> are (ignoring sign and normalization): | A wedgie is written as a list of entries that give the values of the wedgie on the [[basis]] elements of the [[JI subgroup]] that the temperament is on. By the alternating property [i.e. W('''u''', '''v''') = −W('''v''', '''u''')] and bilinearity [W is linear in each argument separately], specifying the values on basis elements of the JI subgroup is enough to define W as an alternating bilinear form on all of the JI subgroup. The simplest example is rank-2 wedgies: Let a and b be (non-[[contorted]]) vals on a [[JI subgroup]] ''q''<sub>1</sub>.[…].''q''<sub>''n''</sub> (where the ''q''<sub>''i''</sub> need not be prime). Then the entries of the wedgie W corresponding to the rank-2 temperament a&b of the JI subgroup ''q''<sub>1</sub>.[…].''q''<sub>''n''</sub> are (ignoring sign and normalization): | ||
<math>\mathrm{W}\left(\mathbf{q}_i, \mathbf{q}_j) = \mathrm{a}(\mathbf{q}_i)\mathrm{b}(\mathbf{q}_j) - \mathrm{a}(\mathbf{q}_j)\mathrm{b}(\mathbf{q}_i) \text{ for } i < j,</math> | <math>\mathrm{W}\left(\mathbf{q}_i, \mathbf{q}_j\right) = \mathrm{a}\left(\mathbf{q}_i\right)\mathrm{b}\left(\mathbf{q}_j\right) - \mathrm{a}\left(\mathbf{q}_j\right)\mathrm{b}\left(\mathbf{q}_i\right) \text{ for } i < j,</math> | ||
where bolded variables and numbers represent the ordinary numbers written in [[monzo]] form. | where bolded variables and numbers represent the ordinary numbers written in [[monzo]] form. | ||
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For the ''p''<sub>''n''</sub>-prime limit, the entries of W are conventionally listed in the order | For the ''p''<sub>''n''</sub>-prime limit, the entries of W are conventionally listed in the order | ||
<math>\wedgie{\mathrm{W}(\mathbf{2}, \mathbf{3}) \ \ldots \ \mathrm{W}(\mathbf{2}, \mathbf{p}_n) & \mathrm{W}(\mathbf{3}, \mathbf{5}) \ \ldots \ \mathrm{W}(\mathbf{3}, \mathbf{p}_n) \ldots \mathrm{W}(\mathbf{p}_{n-2}, \mathbf{p}_{n-1}) & \mathrm{W}(\mathbf{p}_{n-2}, \mathbf{p}_n) & \mathrm{W}(\mathbf{p}_{n-1}, \mathbf{p}_n)}.</math> | <math>\wedgie{\mathrm{W}\left(\mathbf{2}, \mathbf{3}\right) \ \ldots \ \mathrm{W}\left(\mathbf{2}, \mathbf{p}_n\right) & \mathrm{W}\left(\mathbf{3}, \mathbf{5}\right) \ \ldots \ \mathrm{W}\left(\mathbf{3}, \mathbf{p}_n\right) \ldots \mathrm{W}\left(\mathbf{p}_{n-2}, \mathbf{p}_{n-1}\right) & \mathrm{W}\left(\mathbf{p}_{n-2}, \mathbf{p}_n\right) & \mathrm{W}\left(\mathbf{p}_{n-1}, \mathbf{p}_n\right)}.</math> | ||
For example, a 5-limit wedgie is of the form | For example, a 5-limit wedgie is of the form | ||
<math>\wedgie{\mathrm{W}(\mathbf{2}, \mathbf{3}) \ \mathrm{W}(\mathbf{2},\mathbf{5}) \ \mathrm{W}(\mathbf{3}, \mathbf{5})},</math> | <math>\wedgie{\mathrm{W}\left(\mathbf{2}, \mathbf{3}\right) \ \mathrm{W}\left(\mathbf{2},\mathbf{5}\right) \ \mathrm{W}\left(\mathbf{3}, \mathbf{5}\right)},</math> | ||
and a 7-limit wedgie is of the form | and a 7-limit wedgie is of the form | ||
<math>\wedgie{\mathrm{W}(\mathbf{2}, \mathbf{3}) & \mathrm{W}(\mathbf{2},\mathbf{5}) & \mathrm{W}(\mathbf{2}, \mathbf{7}) & \mathrm{W}(\mathbf{3}, \mathbf{5}) & \mathrm{W}(\mathbf{3}, \mathbf{7}) & \mathrm{W}(\mathbf{5}, \mathbf{7})}.</math> | <math>\wedgie{\mathrm{W}\left(\mathbf{2}, \mathbf{3}\right) & \mathrm{W}\left(\mathbf{2},\mathbf{5}\right) & \mathrm{W}\left(\mathbf{2}, \mathbf{7}\right) & \mathrm{W}\left(\mathbf{3}, \mathbf{5}\right) & \mathrm{W}\left(\mathbf{3}, \mathbf{7}\right) & \mathrm{W}\left(\mathbf{5}, \mathbf{7}\right)}.</math> | ||
More generally, if one takes ''r'' independent [[vals]] V<sub>1</sub>, …, V<sub>''r''</sub> in a rank-''n'' [[JI subgroup]] ''q''<sub>1</sub>.[…].''q''<sub>''n''</sub>, then the wedgie for the rank-''r'' temperament V<sub>1</sub>&…&V<sub>''r''</sub> is defined by: | More generally, if one takes ''r'' independent [[vals]] V<sub>1</sub>, …, V<sub>''r''</sub> in a rank-''n'' [[JI subgroup]] ''q''<sub>1</sub>.[…].''q''<sub>''n''</sub>, then the wedgie for the rank-''r'' temperament V<sub>1</sub>&…&V<sub>''r''</sub> is defined by: | ||
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The result is the wedgie of the rank-''r'' temperament V<sub>1</sub>&…&V<sub>''r''</sub>, whose entries are (ignoring steps 2 and 3): | The result is the wedgie of the rank-''r'' temperament V<sub>1</sub>&…&V<sub>''r''</sub>, whose entries are (ignoring steps 2 and 3): | ||
<math>\mathrm{W}(\mathbf{q}_{k_1}, \ldots, \mathbf{q}_{k_r}) = \det[\mathrm{V}_i(\mathbf{q}_{k_j})]_{i,j}, \ \text{for} \ 1 < k_j < n, </math> | <math>\mathrm{W}\left(\mathbf{q}_{k_1}, \ldots, \mathbf{q}_{k_r}\right) = \det[\mathrm{V}_i\left(\mathbf{q}_{k_j}\right)]_{i,j}, \ \text{for} \ 1 < k_j < n, </math> | ||
where <math>[\mathrm{V}_i(\mathbf{q}_{k_j})]_{i,j}</math> denotes the ''r''×''r'' matrix whose (''i'', ''j'') entry is <math>\mathrm{V}_i(\mathbf{q}_{k_j})</math>. These are ''r''-dimensional quantities, the volumes of the ''r''-dimensional parallelograms spanned by '''q'''<sub>''k''<sub>''1''</sub></sub>, ..., '''q'''<sub>''k''<sub>''r''</sub></sub> in the temperament's lattice. | where <math>[\mathrm{V}_i\left((\mathbf{q}_{k_j}\right)]_{i,j}</math> denotes the ''r''×''r'' matrix whose (''i'', ''j'') entry is <math>\mathrm{V}_i\left((\mathbf{q}_{k_j}\right)</math>. These are ''r''-dimensional quantities, the volumes of the ''r''-dimensional parallelograms spanned by '''q'''<sub>''k''<sub>''1''</sub></sub>, ..., '''q'''<sub>''k''<sub>''r''</sub></sub> in the temperament's lattice. | ||
== How the period and generator falls out of a rank-2 wedgie == | == How the period and generator falls out of a rank-2 wedgie == | ||