Normal forms: Difference between revisions

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The ''minimal-generator form'' (or '''mingen form''') is a form specific to rank-2 temperaments, where the generator is positive and no greater than half the period.<ref group="note">This is somewhat like octave reduction combined with octave inversion, because you can't just add or subtract half octaves until it's between 0 and 600 cents; you have to add or subtract octaves until it's between −600 and +600 cents, then multiply by −1 if it's negative.</ref><ref group="note">You could always find a smaller and smaller generator by going negative, so this assumes positive generators.</ref>

[[Graham Breed]]'s [http://x31eq.com/temper/ temperament finder] uses this form for all rank-2 temperaments. Septimal meantone in minimal-generator form is {{mapping| 1 2 4 7 | 0 -1 -4 -10 }}, corresponding to generators of ~2/1 and ~4/3.

[[Graham Breed]]'s [http://x31eq.com/temper/ temperament finder] uses this form for all rank-2 temperaments. Septimal meantone in minimal-generator form is {{mapping| 1 2 4 7 | 0 -1 -4 -10 }}, corresponding to generators of ~2/1 and ~4/3. [[Kite Giedraitis]]'s [[pergen]] also uses this form, except for when the generator is a perfect fifth.

Beyond rank-2, the mingen form of a temperament is no longer unique. You can always get smaller and smaller generators. This is why on Graham Breed's temperament finding tool, beyond rank-2 he simply uses the Hermite Normal Form.

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=== Integer reduced row echelon form (IRREF) ===

Another important normal form for integer matrices is what [[Kite Giedraitis]] has dubbed the IRREF, the ''integer reduced row echelon form''. It is the {{w|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 normal form for integer matrices is what Kite Giedraitis has dubbed the IRREF, the ''integer reduced row echelon form''. It is the {{w|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]].

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 The ''minimal-generator form'' (or '''mingen form''') is a form specific to rank-2 temperaments, where the generator is positive and no greater than half the period.<ref group="note">This is somewhat like octave reduction combined with octave inversion, because you can't just add or subtract half octaves until it's between 0 and 600 cents; you have to add or subtract octaves until it's between −600 and +600 cents, then multiply by −1 if it's negative.</ref><ref group="note">You could always find a smaller and smaller generator by going negative, so this assumes positive generators.</ref>
-[[Graham Breed]]'s [http://x31eq.com/temper/ temperament finder] uses this form for all rank-2 temperaments. Septimal meantone in minimal-generator form is {{mapping| 1 2 4 7 | 0 -1 -4 -10 }}, corresponding to generators of ~2/1 and ~4/3.
+[[Graham Breed]]'s [http://x31eq.com/temper/ temperament finder] uses this form for all rank-2 temperaments. Septimal meantone in minimal-generator form is {{mapping| 1 2 4 7 | 0 -1 -4 -10 }}, corresponding to generators of ~2/1 and ~4/3. [[Kite Giedraitis]]'s [[pergen]] also uses this form, except for when the generator is a perfect fifth.
 Beyond rank-2, the mingen form of a temperament is no longer unique. You can always get smaller and smaller generators. This is why on Graham Breed's temperament finding tool, beyond rank-2 he simply uses the Hermite Normal Form.
@@ Line 283: / Line 283: @@
 === Integer reduced row echelon form (IRREF) ===
-Another important normal form for integer matrices is what [[Kite Giedraitis]] has dubbed the IRREF, the ''integer reduced row echelon form''. It is the {{w|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 normal form for integer matrices is what Kite Giedraitis has dubbed the IRREF, the ''integer reduced row echelon form''. It is the {{w|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]].