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 [[Harmonic limit|''p''-limit]] musical intervals (or [[Monzos and Interval Space|monzos]]) or lists of [[Vals and Tuning Space|vals]], the '''normal interval/monzo list''' and the '''normal val list'''.

An integral matrix is a matrix whose entries are all integers. An important normalized form for integral matrices is [http://en.wikipedia.org/wiki/Hermite_normal_form Hermite normal form], and by using Hermite normal form we may define normalized forms of lists of [[Harmonic_Limit|p-limit]] musical intervals (or [[Monzos_and_Interval_Space|monzos]]) or lists of [[Vals_and_Tuning_Space|vals]], the normal interval (or monzo) list and the normal val list.

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 [[~~KiteGiedraitis|~~Kite Giedraitis]] has dubbed the IRREF, the integral reduced row echelon form. It is the [[~~wikipedia~~:~~Row_echelon_form~~|~~RREF~~]] made integral by multiplying each row of the matrix by the ~~LCM~~ 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.

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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+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 [[Harmonic limit|''p''-limit]] musical intervals (or [[Monzos and Interval Space|monzos]]) or lists of [[Vals and Tuning Space|vals]], the '''normal interval/monzo list''' and the '''normal val list'''.
-An integral matrix is a matrix whose entries are all integers. An important normalized form for integral matrices is [http://en.wikipedia.org/wiki/Hermite_normal_form Hermite normal form], and by using Hermite normal form we may define normalized forms of lists of [[Harmonic_Limit|p-limit]] musical intervals (or [[Monzos_and_Interval_Space|monzos]]) or lists of [[Vals_and_Tuning_Space|vals]], the normal interval (or monzo) list and the normal val list.
+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 [[KiteGiedraitis|Kite Giedraitis]] has dubbed the IRREF, the integral reduced row echelon form. It is the [[wikipedia:Row_echelon_form|RREF]] made integral by multiplying each row of the matrix by the LCM 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.
 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.