Wedgie/Archived version: Difference between revisions

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

For the p_n-limit, the entries of W are 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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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]] V1, ..., Vr in a rank-n [[JI subgroup]] q1.[...].qn, then the wedgie for the rank-r temperament V1& ...&Vr is defined by:

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:

# 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.

# 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

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where <math>[V_i(q_{k_j})]_{i,j}</math> denotes the 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==

@@ Line 8: / Line 8: @@
 (Note that by the alternating property, W(q<sub>i</sub>, q<sub>i</sub>) = 0 for all i.)
-For the p_n-limit, the entries of W are 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>
@@ Line 22: / Line 22: @@
 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]] V1, ..., Vr in a rank-n [[JI subgroup]] q<sub>1</sub>.[...].q<sub>n</sub>, then the wedgie for the rank-r temperament V1& ...&Vr 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.
+# 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
@@ Line 30: / Line 30: @@
 <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 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==