Wedgie/Archived version: Difference between revisions

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The '''wedgie''' is a way of defining and working with an [[abstract regular temperament]] (that is, a regular temperament where no tuning has been decided on). Wedgies are in a one-to-one relationship with abstract regular temperaments.

A wedgie is ~~defined~~ by ~~its~~ values on the basis of the temperament's 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:

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 entries of the wedgie 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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<math>\langle \langle W(2, 3) \ W(2,5) \ W(2, 7) \ W(3, 5) \ W(3, 7) \ W(5, 7)]].</math>

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. The entries of the wedgie 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. 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''.

More generally, if one takes ''r'' independent [[vals]] ''V''1, ..., ''V''''r'' in a rank-''n'' [[JI subgroup]] ''q''1.[...].''q''''n'', then the wedgie for the rank-r temperament ''V''1& ...&''V''''r'' is defined by:

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 The '''wedgie''' is a way of defining and working with an [[abstract regular temperament]] (that is, a regular temperament where no tuning has been decided on). Wedgies are in a one-to-one relationship with abstract regular temperaments.
-A wedgie is defined by its values on the basis of the temperament's 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:
+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 entries of the wedgie 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>
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 <math>\langle \langle W(2, 3) \ W(2,5) \ W(2, 7) \ W(3, 5) \ W(3, 7) \ W(5, 7)]].</math>
-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. The entries of the wedgie 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. 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''.
 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: