Normal forms: Difference between revisions

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# Hermite reduce the matrix for L

# Throw away all rows which consist of nothing but zeros, resulting in a ''k''×''m'' matrix

# Find the [[Wikipedia: Moore–Penrose pseudoinverse|Moore–Penrose pseudoinverse]] of the ''k''×''m'' matrix, and multiply this from the left by the row vector 1200·[1 log23 log25 … log2''p'']. If the ''i''-th entry in the result is negative, multiply the corresponding val by -1. Return the result ~~as the normalized val list~~.

# Find the [[Wikipedia: Moore–Penrose pseudoinverse|Moore–Penrose pseudoinverse]] of the ''k''×''m'' matrix, and multiply this from the left by the row vector 1200·[1 log23 log25 … log2''p'']. If the ''i''-th entry in the result is negative, multiply the corresponding val by -1. Return the result.

The point of steps two and three 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. The "~~Maps~~" (though not the "Map to lattice") listed on temperament pages of this wiki are all normal val lists; an example would be [{{val| 1 0 -4 -13 }}, {{val| 0 1 4 10 }}], the normal val list for septimal meantone. Using this as input, the output is 1199.25¢, 1899.45¢ (tempered 2/1, tempered 3/1).

We may call this the '''Smith normal form'''.

The point of steps two and three 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. The "mapping" (though not the "Map to lattice") listed on temperament pages of this wiki are all normal val lists; an example would be [{{val| 1 0 -4 -13 }}, {{val| 0 1 4 10 }}], the normal val list for septimal meantone. Using this as input, the output is 1199.25¢, 1899.45¢ (tempered 2/1, tempered 3/1).

== Maple code ==

@@ Line 43: / Line 43: @@
 # Hermite reduce the matrix for L
 # Throw away all rows which consist of nothing but zeros, resulting in a ''k''×''m'' matrix
-# Find the [[Wikipedia: Moore–Penrose pseudoinverse|Moore–Penrose pseudoinverse]] of the ''k''×''m'' matrix, and multiply this from the left by the row vector 1200·[1 log<sub>2</sub>3 log<sub>2</sub>5 … log<sub>2</sub>''p'']. If the ''i''-th entry in the result is negative, multiply the corresponding val by -1. Return the result as the normalized val list.
+# Find the [[Wikipedia: Moore–Penrose pseudoinverse|Moore–Penrose pseudoinverse]] of the ''k''×''m'' matrix, and multiply this from the left by the row vector 1200·[1 log<sub>2</sub>3 log<sub>2</sub>5 … log<sub>2</sub>''p'']. If the ''i''-th entry in the result is negative, multiply the corresponding val by -1. Return the result.
-The point of steps two and three 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. The "Maps" (though not the "Map to lattice") listed on temperament pages of this wiki are all normal val lists; an example would be [{{val| 1 0 -4 -13 }}, {{val| 0 1 4 10 }}], the normal val list for septimal meantone. Using this as input, the output is 1199.25¢, 1899.45¢ (tempered 2/1, tempered 3/1).
+We may call this the '''Smith normal form'''.
+The point of steps two and three 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. The "mapping" (though not the "Map to lattice") listed on temperament pages of this wiki are all normal val lists; an example would be [{{val| 1 0 -4 -13 }}, {{val| 0 1 4 10 }}], the normal val list for septimal meantone. Using this as input, the output is 1199.25¢, 1899.45¢ (tempered 2/1, tempered 3/1).
 == Maple code ==