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]] q1.[...].qn (where the qi need not be prime). Then the entries of the wedgie W corresponding to the rank-2 temperament a&b of the JI subgroup q1.[...].qn are:

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:

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(Note that by the alternating property, W(qi, qi) = 0 for all i.)

For the p_n-prime limit, the entries of W are conventionally listed in the order

For the p_n-prime limit, the entries of ''W'' are conventionally listed in the order

<math>\langle\langle W(2, 3) \ ... \ W(2, p_n) \ W(3, 5) \ ... \ W(3, p_n) \ ... W(p_{n-2}, p_{n-1}) \ W(p_{n-2}, p_n)\ W(p_{n-1}, p_n)]].</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.

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]] q1.[...].qn, then the wedgie for the rank-r temperament ''V''1& ...&''V''''r'' is defined by:

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:

# taking the [https://en.wikipedia.org/wiki/Wedge_product wedge product] of the vals (called a '''multival'''), and

# dividing out the greatest common divisior of the coefficients, to produce an ''r''-multival.

# If the first non-zero coefficient of this multival is negative, the multival is then scalar multiplied by -1, changing the sign of the first non-zero coefficient to be positive.

The result is the wedgie of the rank-n temperament V1&...&Vr, whose entries are

The result is the wedgie of the rank-n temperament ''V''1&...&''V''r, whose entries are

where <math>[V_i(q_{k_j})]_{i,j}</math> denotes the ''n''×''n'' matrix whose (''i'',''j'') entry is <math>V_i(q_{k_j})</math>.

where <math>[V_i(q_{k_j})]_{i,j}</math> denotes the ''n''×''n'' matrix whose (''i'', ''j'') entry is <math>V_i(q_{k_j})</math>.

==How the period and generator falls out of a rank-2 wedgie==

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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:
+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:
 <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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 (Note that by the alternating property, W(q<sub>i</sub>, q<sub>i</sub>) = 0 for all i.)
-For the p_n-prime limit, the entries of W are conventionally listed in the order
+For the p_n-prime limit, the entries of ''W'' are conventionally listed in the order
 <math>\langle\langle W(2, 3) \ ... \ W(2, p_n) \ W(3, 5) \ ... \ W(3, p_n) \ ... W(p_{n-2}, p_{n-1}) \ W(p_{n-2}, p_n)\ W(p_{n-1}, p_n)]].</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.
+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:
+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:
 # taking the [https://en.wikipedia.org/wiki/Wedge_product wedge product] of the vals (called a '''multival'''), and
 # dividing out the greatest common divisior of the coefficients, to produce an ''r''-multival.
 # If the first non-zero coefficient of this multival is negative, the multival is then scalar multiplied by -1, changing the sign of the first non-zero coefficient to be positive.
-The result is the wedgie of the rank-n temperament V1&...&Vr, whose entries are
+The result is the wedgie of the rank-n temperament ''V''<sub>1</sub>&...&''V''<sub>r</sub>, whose entries are
 <math>W(q_{k_1}, ..., q_{k_n}) = \det[V_i(q_{k_j})]_{i,j},</math>
-where <math>[V_i(q_{k_j})]_{i,j}</math> denotes the ''n''×''n'' matrix whose (''i'',''j'') entry is <math>V_i(q_{k_j})</math>.
+where <math>[V_i(q_{k_j})]_{i,j}</math> denotes the ''n''×''n'' matrix whose (''i'', ''j'') entry is <math>V_i(q_{k_j})</math>.
 ==How the period and generator falls out of a rank-2 wedgie==