Normal forms: Difference between revisions

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[[Wikipedia: Hermite normal form|Hermite normal form]], or HNF for short, is an important normal form that was defined for integer matrices in the mathematical field of linear algebra. An integer matrix is simply a 2D array of integers, and so we can easily think of lists of vals or commas as integer matrices and therefore leverage HNF in regular temperament theory.

HNF by itself is not used as a normal form for regular temperament theory. However, it is part of the definition of every normal form here, so it is important to have a basic understanding of it.

HNF by itself is not used as a normal form for regular temperament theory. However, it is part of the definition of almost every normal form here (except the IRREF), so it is important to have a basic understanding of it.

There are slightly different definitions of HNF in use, and if you are using a computer program to compute it, you should take care that the same normal monzo or val list is finally achieved. The definition used by the Wikipedia article on Hermite form, probably the most standard, works as follows.

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This is the form shown in the "comma lists" of each temperament on this wiki.

=== IRREF ===

Another important normalized form for integer matrices is what [[Kite Giedraitis]] has dubbed the IRREF, the '''integer reduced row echelon form'''. It is the [[Wikipedia: Row echelon form|reduced row echelon form]] but with integer entries, found by multiplying each row of the matrix by the least common multiple of all denominators in that row. It differs from the Hermite normal form in that each pivot is the only nonzero entry in its column. For a monzo list, it has the advantage of limiting the appearance of the ''N'' highest primes to only one comma each (where ''N'' is the codimension), isolating each prime's effect on the [[pergen]], but has the disadvantage that the commas tend to have high odd limits, and the comma list may have torsion.

This is the only form on this page which does not use the HNF, although sometimes the IRREF is identical to the HNF. For more information, see [[IRREF]].

== Maple code ==

@@ Line 6: / Line 6: @@
 [[Wikipedia: Hermite normal form|Hermite normal form]], or HNF for short, is an important normal form that was defined for integer matrices in the mathematical field of linear algebra. An integer matrix is simply a 2D array of integers, and so we can easily think of lists of vals or commas as integer matrices and therefore leverage HNF in regular temperament theory.
-HNF by itself is not used as a normal form for regular temperament theory. However, it is part of the definition of every normal form here, so it is important to have a basic understanding of it.
+HNF by itself is not used as a normal form for regular temperament theory. However, it is part of the definition of almost every normal form here (except the IRREF), so it is important to have a basic understanding of it.
 There are slightly different definitions of HNF in use, and if you are using a computer program to compute it, you should take care that the same normal monzo or val list is finally achieved. The definition used by the Wikipedia article on Hermite form, probably the most standard, works as follows.
@@ Line 202: / Line 202: @@
 This is the form shown in the "comma lists" of each temperament on this wiki.
+=== IRREF ===
+Another important normalized form for integer matrices is what [[Kite Giedraitis]] has dubbed the IRREF, the '''integer reduced row echelon form'''. It is the [[Wikipedia: Row echelon form|reduced row echelon form]] but with integer entries, found by multiplying each row of the matrix by the least common multiple of all denominators in that row. It differs from the Hermite normal form in that each pivot is the only nonzero entry in its column. For a monzo list, it has the advantage of limiting the appearance of the ''N'' highest primes to only one comma each (where ''N'' is the codimension), isolating each prime's effect on the [[pergen]], but has the disadvantage that the commas tend to have high odd limits, and the comma list may have torsion.
+This is the only form on this page which does not use the HNF, although sometimes the IRREF is identical to the HNF. For more information, see [[IRREF]].
 == Maple code ==