Normal forms: Difference between revisions

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=== Minimal form ===

The '''minimal form''', or '''reduced form''' shows the simplest comma list sufficient to define the temperament, where the commas are rated by a complexity measure such as the [[Tenney height]], in which case it is called the ''Tenney-minimal form''. It is obvious why this form is useful, but unfortunately, it is not easy to find since it is equivalent to the ''shortest basis problem'', an NP-hard problem. The [[LLL reduction]] algorithm gives a good approximation. Otherwise, it is typically solved heuristically by hand.

The ''minimal form'', or ''reduced form'' shows the simplest comma list sufficient to define the temperament, where the commas are rated by a complexity measure such as the [[Tenney height]], in which case it is called the ''Tenney-minimal form''. It is obvious why this form is useful, but unfortunately, it is not easy to find since it is equivalent to the ''shortest basis problem'', an NP-hard problem. The [[LLL reduction]] algorithm gives a good approximation. Otherwise, it is typically solved heuristically by hand.

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

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Note that the antitransposed defactored Hermite form of the comma list involves the list being defactored (e.g. torsion to be removed). For example, both [25/27, 35/36] and [25/27, 49/48] characterize beep. But the latter has torsion (i.e. is enfactored), so the former is beep's antitransposed defactored Hermite 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 [[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.

=== Positive ratio form ===

=== Positive-ratio form ===

Similar to [[~~Normal lists~~ #Positive generator form|the situation with mappings]], even though by using the HNF the antitransposed defactored Hermite form ensures that the first nonzero entry of each comma's [[prime-count vector]] is a positive ''number'', this does not necessarily mean that the corresponding commas are all positive ''in pitch''.

Similar to [[#Positive-generator form|the situation with mappings]], even though by using the HNF the antitransposed defactored Hermite form ensures that the first nonzero entry of each comma's [[prime-count vector]] is a positive ''number'', this does not necessarily mean that the corresponding commas are all positive ''in pitch''.

For example, the antitransposed defactored Hermite form of meantone's comma basis is [{{vector| 4 -4 1 }}]. HNF has ensured that the first number is positive. But the vector {{vector|4 -4 1}} represents the ratio 80/81, which is less than unity; it is still the meantone comma, but it is the meantone comma ''downward'' in pitch.

For example, the antitransposed defactored Hermite form of meantone's comma basis is [{{vector| 4 -4 1 }}]. HNF has ensured that the first number is positive. But the vector {{vector| 4 -4 1 }} represents the ratio 80/81, which is less than unity; it is still the meantone comma, but it is the meantone comma ''downward'' in pitch.

Having negative commas like this can be surprising or confusing, and so the '''positive ratio form''' addresses this concern. To correct any negative comma, simply replace it with its reciprocal. In vector form, this can be done by changing the signs on every number; meantone can be put into positive ratio form as [{{vector| -4 4 -1 }}].

Having negative commas like this can be surprising or confusing, and so the ''positive-ratio form'' addresses this concern. To correct any negative comma, simply replace it with its reciprocal. In vector form, this can be done by changing the signs on every number; meantone can be put into positive ratio form as [{{vector| -4 4 -1 }}].

The positive ratio form of septimal meantone is [81/80, 59049/57344].

=== 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 ~~[[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 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]].

@@ Line 190: / Line 190: @@
 === Minimal form ===
-The '''minimal form''', or '''reduced form''' shows the simplest comma list sufficient to define the temperament, where the commas are rated by a complexity measure such as the [[Tenney height]], in which case it is called the ''Tenney-minimal form''. It is obvious why this form is useful, but unfortunately, it is not easy to find since it is equivalent to the ''shortest basis problem'', an NP-hard problem. The [[LLL reduction]] algorithm gives a good approximation. Otherwise, it is typically solved heuristically by hand.
+The ''minimal form'', or ''reduced form'' shows the simplest comma list sufficient to define the temperament, where the commas are rated by a complexity measure such as the [[Tenney height]], in which case it is called the ''Tenney-minimal form''. It is obvious why this form is useful, but unfortunately, it is not easy to find since it is equivalent to the ''shortest basis problem'', an NP-hard problem. The [[LLL reduction]] algorithm gives a good approximation. Otherwise, it is typically solved heuristically by hand.
 '''Note''': This is the form shown in the "comma lists" of each temperament on this wiki.
@@ Line 271: / Line 271: @@
 Note that the antitransposed defactored Hermite form of the comma list involves the list being defactored (e.g. torsion to be removed). For example, both [25/27, 35/36] and [25/27, 49/48] characterize beep. But the latter has torsion (i.e. is enfactored), so the former is beep's antitransposed defactored Hermite 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 [[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.
-=== Positive ratio form ===
+=== Positive-ratio form ===
-Similar to [[Normal lists #Positive generator form|the situation with mappings]], even though by using the HNF the antitransposed defactored Hermite form ensures that the first nonzero entry of each comma's [[prime-count vector]] is a positive ''number'', this does not necessarily mean that the corresponding commas are all positive ''in pitch''.
+Similar to [[#Positive-generator form|the situation with mappings]], even though by using the HNF the antitransposed defactored Hermite form ensures that the first nonzero entry of each comma's [[prime-count vector]] is a positive ''number'', this does not necessarily mean that the corresponding commas are all positive ''in pitch''.
-For example, the antitransposed defactored Hermite form of meantone's comma basis is [{{vector| 4 -4 1 }}]. HNF has ensured that the first number is positive. But the vector {{vector|4 -4 1}} represents the ratio 80/81, which is less than unity; it is still the meantone comma, but it is the meantone comma ''downward'' in pitch.
+For example, the antitransposed defactored Hermite form of meantone's comma basis is [{{vector| 4 -4 1 }}]. HNF has ensured that the first number is positive. But the vector {{vector| 4 -4 1 }} represents the ratio 80/81, which is less than unity; it is still the meantone comma, but it is the meantone comma ''downward'' in pitch.
-Having negative commas like this can be surprising or confusing, and so the '''positive ratio form''' addresses this concern. To correct any negative comma, simply replace it with its reciprocal. In vector form, this can be done by changing the signs on every number; meantone can be put into positive ratio form as [{{vector| -4 4 -1 }}].
+Having negative commas like this can be surprising or confusing, and so the ''positive-ratio form'' addresses this concern. To correct any negative comma, simply replace it with its reciprocal. In vector form, this can be done by changing the signs on every number; meantone can be put into positive ratio form as [{{vector| -4 4 -1 }}].
 The positive ratio form of septimal meantone is [81/80, 59049/57344].
 === 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 [[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 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]].