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}(\mathbf{q}_i, \mathbf{q}_j) = a(\mathbf{q}_i)b(\mathbf{q}_j) - a(\mathbf{q}_j)b(\mathbf{q}_i) \text{ for } i < j,</math> | <math>\mathrm{W}(\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> | ||
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. | ||