Normal forms: Difference between revisions
Cmloegcmluin (talk | contribs) link for mapping to lattice |
Cmloegcmluin (talk | contribs) extract RREF from Mathematical theory of regular temperaments#Characterizing a regular temperament, because it is a subtopic of the Normal list characterization listed there |
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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. | [[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 | HNF by itself is not used as a normal form for regular temperament theory. However, it is part of the definition of most normal forms here, 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. | 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 problem also precludes the possibility of a definitive maximum generator which is still less than half of the previous generator. | This problem also precludes the possibility of a definitive maximum generator which is still less than half of the previous generator. | ||
=== Reduced row echelon form (RREF) === | |||
{{main| Wikipedia: Row echelon form }} | |||
If you have a routine which will compute the reduced row echelon form of a matrix with rows consisting of vals using rational number arithmetic, this can be computed and any rows consisting of the zero val stripped off. The result is a unique identifier for the abstract temperament which is closely related to the normal val list. The intervals of the abstract temperament may be found in the same way, by applying the mappings (which are fractional vals) to monzos; or put in another way, by matrix multiplication of the monzos by the reduced row echelon form matrix. | |||
For example, if we feed [{{val| 22 35 51 62 }}, {{val| 31 49 72 87 }}, {{val| 84 133 195 236 }}] into a reduced row echelon form routine, we obtain [{{val| 1 0 3 1 }}, {{val| 0 1 -3/7 8/7 }}, {{val| 0 0 0 0 }}]. Stripping off the zero val in the final row, we get E = [{{val| 1 0 3 1 }}, {{val| 0 1 -3/7 8/7 }}]. The monzo for 7/6 is {{monzo| -1 -1 0 1 }}, and E{{monzo| -1 -1 0 1 }} = [0 1/7]. Multiply by {{monzo| 1 0 0 0 }}, the monzo for 2, and the result is E{{monzo| 1 0 0 0 }}, which is [1 0]. We have in fact 7-limit orwell temperament, with period 2 and generator approximately 7/6. | |||
For more information, see [[RREF]]. | |||
== Normal interval lists == | == Normal interval lists == | ||
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This is the form shown in the "comma lists" of each temperament on this wiki. | This is the form shown in the "comma lists" of each temperament on this wiki. | ||
=== IRREF === | === Integer reduced row echelon form (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. | 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. | ||
Sometimes the IRREF is identical to the HNF. For more information, see [[IRREF]]. | |||
== Maple code == | == Maple code == | ||