Wedgie/Archived version: Difference between revisions

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In geometric terms, given JI ratios ''u'' and ''v'', and a rank-2 temperament's wedgie ''W'', the number ''W''(''u'', ''v'') is the signed area of the parallelogram spanned by (tempered versions of) ''u'' and ''v''. This is the determinant of the tempered versions of ''u'' and ''v''. The musical interpretation of the parallelogram spanned by ''u'' and ''v'' is: If you want to consider intervals that are multiples of ''u'' apart the same note (for example, if you want an octave-equivalent scale), ''W''(''u'', ''v'') tells you how many generators of your rank-2 temperament it would take to get to ''v''. The reason that wedgies work as unique identifiers of temperaments is that the value ''W''(''u'', ''v'') only depends on what the temperament does to ''u'' and ''v'', and this dependence (in a sense) matches up exactly with what commas are tempered out by the temperament.

A wedgie is 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''1.[...].''q''''n'' (where the ''q''''i'' need not be prime). Then the entries of the wedgie ''W'' corresponding to the rank-2 temperament ''a''&''b'' of the JI subgroup ''q''1.[...].''q''''n'' are:

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''1.[...].''q''''n'' (where the ''q''''i'' need not be prime). Then the entries of the wedgie ''W'' corresponding to the rank-2 temperament ''a''&''b'' of the JI subgroup ''q''1.[...].''q''''n'' are:

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 In geometric terms, given JI ratios ''u'' and ''v'', and a rank-2 temperament's wedgie ''W'', the number ''W''(''u'', ''v'') is the signed area of the parallelogram spanned by (tempered versions of) ''u'' and ''v''. This is the determinant of the tempered versions of ''u'' and ''v''. The musical interpretation of the parallelogram spanned by ''u'' and ''v'' is: If you want to consider intervals that are multiples of ''u'' apart the same note (for example, if you want an octave-equivalent scale), ''W''(''u'', ''v'') tells you how many generators of your rank-2 temperament it would take to get to ''v''. The reason that wedgies work as unique identifiers of temperaments is that the value ''W''(''u'', ''v'') only depends on what the temperament does to ''u'' and ''v'', and this dependence (in a sense) matches up exactly with what commas are tempered out by the temperament.
-A wedgie is 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:
+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:
 <math>W(q_i, q_j) = a(q_i)b(q_j) - a(q_j)b(q_i) \text{ for } i < j.</math>