Normal forms: Difference between revisions
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An integral matrix is a matrix whose entries are all integers. An important normalized form for integral matrices is [[Wikipedia: Hermite normal form|Hermite normal form]], and by using Hermite normal form we may define normalized forms of lists of | == Introduction to Hermite normal form == | ||
An integral matrix is a matrix whose entries are all integers. An important normalized form for integral matrices is [[Wikipedia: Hermite normal form|Hermite normal form]], and by using Hermite normal form we may define normalized forms of lists of intervals or [[Monzos and interval space|monzos]] (aka comma lists) or lists of [[Vals and tuning space|vals]] (aka [[mapping]]s), the '''normal interval/monzo list''' and the '''normal val list'''. | |||
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Another important normalized form for integral matrices is what [[Kite Giedraitis]] has dubbed the IRREF, the '''integral reduced row echelon form'''. It is the [[Wikipedia: Row echelon form|reduced row echelon form]] made integral 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 Hermite normal form. | Another important normalized form for integral matrices is what [[Kite Giedraitis]] has dubbed the IRREF, the '''integral reduced row echelon form'''. It is the [[Wikipedia: Row echelon form|reduced row echelon form]] made integral 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 Hermite normal form. | ||
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There are slightly different definitions of Hermite normal form 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 Hermite normal form 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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== Normal interval list == | == Normal interval list == | ||
=== Hermite normal form === | |||
Given a list of ''p''-limit intervals, we can convert it to a normal list by the following procedure: | Given a list of ''p''-limit intervals, we can convert it to a normal list by the following procedure: | ||
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The result is a normal interval list. The set of elements of the original list generates a finitely generated free abelian subgroup of the positive rationals under multiplication, and therefore of any ''p''-limit group it lives inside. The normalized list contains a minimal set of generators, each greater than zero, in an ordering of nondecreasing prime limit which is parsimonious in its use of higher limits. For example, if we normalize [81/80, 126/125] we obtain [81/80, 59049/57344]. The first interval is 5-limit, which is as small as possible. The second is 7-limit, which must be the case because the group these two generate is 7-limit. However, it uses only 2, 3 and 7 in its prime factorization, parsimoniously rejecting 5 as the next highest prime limit. Because [[abstract regular temperament]]s, where the prime mappings are known but not the specific tuning of the generators, are fully characterized by their kernel, the group of intervals they map to the unison, they can also be characterized by the regular interval list of a set of generators (called commas or unison vectors) for the kernel. The above normal interval list, for example, characterizes septimal meantone, defining the normal comma sequence of septimal meantone. | The result is a normal interval list. The set of elements of the original list generates a finitely generated free abelian subgroup of the positive rationals under multiplication, and therefore of any ''p''-limit group it lives inside. The normalized list contains a minimal set of generators, each greater than zero, in an ordering of nondecreasing prime limit which is parsimonious in its use of higher limits. For example, if we normalize [81/80, 126/125] we obtain [81/80, 59049/57344]. The first interval is 5-limit, which is as small as possible. The second is 7-limit, which must be the case because the group these two generate is 7-limit. However, it uses only 2, 3 and 7 in its prime factorization, parsimoniously rejecting 5 as the next highest prime limit. Because [[abstract regular temperament]]s, where the prime mappings are known but not the specific tuning of the generators, are fully characterized by their kernel, the group of intervals they map to the unison, they can also be characterized by the regular interval list of a set of generators (called commas or unison vectors) for the kernel. The above normal interval list, for example, characterizes septimal meantone, defining the normal comma sequence of septimal meantone. | ||
There is only one normal comma sequence that characterizes septimal meantone. But sometimes a temperament can be characterized by multiple normal comma sequences | There is only one normal comma sequence that characterizes septimal meantone. But sometimes a temperament can be characterized by multiple normal comma sequences due to [[enfactoring]], leading to the following section of canonical form. | ||
Normal interval lists can also be used to characterize the [[just intonation subgroups]] on which subgroup temperaments are defined and using which subgroup scales may be constructed. On the pages [[chromatic pairs]], [[subgroup temperaments]] and [[just intonation subgroup]]s can be found many examples; the subgroup lists are given in a form where generators of the subgroup are separated by periods so as to flag the fact that the list defines a subgroup. An example would be the Barbados subgroup, 2.3.13/5. | Normal interval lists can also be used to characterize the [[just intonation subgroups]] on which subgroup temperaments are defined and using which subgroup scales may be constructed. On the pages [[chromatic pairs]], [[subgroup temperaments]] and [[just intonation subgroup]]s can be found many examples; the subgroup lists are given in a form where generators of the subgroup are separated by periods so as to flag the fact that the list defines a subgroup. An example would be the Barbados subgroup, 2.3.13/5. | ||
In each temperament page on this wiki, however, the "comma lists" are not normal interval lists as defined above. Instead, it shows ratio-wise the simplest comma sequence sufficient to define the temperament. | === Canonical form === | ||
{{See also| Canonical form }} | |||
The canonical form of the comma list, however, requires that the normal comma sequence be defactored. There is only one such form of any temperament. For example, both [27/25, 36/35] and [27/25, 49/48] are normal, and they both characterize Beep. But the latter is enfactored, so the former is Beep's canonical form. | |||
=== Tenney-minimal form === | |||
In each temperament page on this wiki, however, the "comma lists" are not normal interval lists as defined above. Instead, it shows ratio-wise the simplest comma sequence sufficient to define the temperament, which may be called the Tenney-minimal form. | |||
== Normal val list == | == Normal val list == | ||
=== Hermite normal form === | |||
If L is a list of ''n'' vals, we may write it as an ''n''×''m'' matrix, where the rows of the matrix are the vals, and ''m'' = π (''p''), where ''p'' is the prime limit. To get the normal val list, we do the following: | If L is a list of ''n'' vals, we may write it as an ''n''×''m'' matrix, where the rows of the matrix are the vals, and ''m'' = π (''p''), where ''p'' is the prime limit. To get the normal val list, we do the following: | ||
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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). | 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). | ||
=== Canonical form === | |||
{{See also| Canonical form }} | |||
The canonical form of the val list requires that the normal val list be defactored, therefore uniquely characterizing each temperament. | |||
== Maple code == | == Maple code == | ||
Below is [[Wikipedia: Maple (software)|Maple]] code for finding the normal interval and val list, given an interval list or a val list. | Below is [[Wikipedia: Maple (software)|Maple]] code for finding the normal interval and val list, given an interval list or a val list. | ||
{{Databox| Code | | |||
<syntaxhighlight lang="mupad"> | <syntaxhighlight lang="mupad"> | ||
log2 := proc(x) evalf(ln(x) / ln(2)) end: | log2 := proc(x) evalf(ln(x) / ln(2)) end: | ||
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end: | end: | ||
</syntaxhighlight> | </syntaxhighlight> | ||
}} | |||
[[Category: | [[Category:Regular temperament theory]] | ||
[[Category: | [[Category:Math]] | ||
{{Todo| add introduction }} | |||