Normal forms: Difference between revisions

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-22 03:16:05 UTC</tt>.<br>

: The original revision id was <tt>143888077</tt>.<br>

(2) Throw away all rows which consist of nothing but zeros

 

(3) In the resulting kxm matrix, take the first k rows and columns, and test if the matrix is singular

(4) If it is, return the rows of the kxm matrix as the normalized list (in practice you would need to be doing something unusual for this to happen, and at that point you should probably work matters out for yourself.) If it is nonsingular, invert the matrix

and multiply the inverse matrix from the left by the row vector

[1 log2(3) log2(5) ... log2(p)]. If the ith entry in the result is negative, multiply the corresponding val by -1. Return the result as the normalized val list.

The point of steps three and four is that now the vals on the list correspond to a list of generators which are all positive (written additively) or equivalently greater than 1 (written multiplicatively.) Just as a normal comma list can be used to classify an abstract regular temperament, so can a normal val list. The val list is what on [[Graham Breed]]'s [[http://x31eq.com/temper/|web site]]  is called a "mapping", put into a canonical form.

with(padic, ordp):

with(LinearAlgebra):

log2 := proc(x) evalf(ln(x)/ln(2)) end:

plim := proc(q)

  # prime limit of rational number q

r := r*p^ordp(q, p) od;

local M, N, u, v, i, j, a;

j := Rank(Matrix(u));

for i from 1 to j do

M := Matrix(v);

N := SubMatrix(M, 1..j, 1..j);

N := N^(-1);

a := Vector[row](pril(j));

a := a.N;

for i from 1 to j do

(2) Throw away all rows which consist of nothing but zeros&lt;br /&gt;

(3) In the resulting kxm matrix, take the first k rows and columns, and test if the matrix is singular&lt;br /&gt;

(4) If it is, return the rows of the kxm matrix as the normalized list (in practice you would need to be doing something unusual for this to happen, and at that point you should probably work matters out for yourself.) If it is nonsingular, invert the matrix&lt;br /&gt;

and multiply the inverse matrix from the left by the row vector&lt;br /&gt;

[1 log2(3) log2(5) ... log2(p)]. If the ith entry in the result is negative, multiply the corresponding val by -1. Return the result as the normalized val list.&lt;br /&gt;

The point of steps three and four is that now the vals on the list correspond to a list of generators which are all positive (written additively) or equivalently greater than 1 (written multiplicatively.) Just as a normal comma list can be used to classify an abstract regular temperament, so can a normal val list. The val list is what on &lt;a class="wiki_link" href="/Graham%20Breed"&gt;Graham Breed&lt;/a&gt;'s &lt;a class="wiki_link_ext" href="http://x31eq.com/temper/" rel="nofollow"&gt;web site&lt;/a&gt;  is called a &amp;quot;mapping&amp;quot;, put into a canonical form.&lt;br /&gt;

with(padic, ordp):&lt;br /&gt;

with(LinearAlgebra):&lt;br /&gt;

log2 := proc(x) evalf(ln(x)/ln(2)) end:&lt;br /&gt;

plim := proc(q)&lt;br /&gt;

  # prime limit of rational number q&lt;br /&gt;

r := r*p^ordp(q, p) od;&lt;br /&gt;

local M, N, u, v, i, j, a;&lt;br /&gt;

j := Rank(Matrix(u));&lt;br /&gt;

for i from 1 to j do&lt;br /&gt;

M := Matrix(v);&lt;br /&gt;

N := SubMatrix(M, 1..j, 1..j);&lt;br /&gt;

N := N^(-1);&lt;br /&gt;

a := Vector[row](pril(j));&lt;br /&gt;

a := a.N;&lt;br /&gt;

for i from 1 to j do&lt;br /&gt;