Normal forms: Difference between revisions

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In the case of interval lists, the most common format they are presented in is as ratios, not vectors, e.g. [81/80, 64/63] rather than [{{vector|-4 4 1 0}} {{vector|-6 2 0 1}}]. So you may need to convert ratios to vectors and back when working with these forms.

=== Defactored Hermite form ===

=== Antitransposed Defactored Hermite form ===

Given a matrix of ''p''-limit intervals, we can find its defactored Hermite form in much the same way as a mapping. The only difference is that they must be put in an "antitranspose sandwich", or in other words, antitransposed once at the beginning, and then antitransposed again at the end.

Given a matrix of ''p''-limit intervals, we can find its antitransposed defactored Hermite form in much the same way as we find the defactored Hermite form of a mapping. The only difference is that they must be put in an "antitranspose sandwich", or in other words, antitransposed once at the beginning, and then antitransposed again at the end.

For example, suppose we have the comma basis for septimal meantone:

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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 list in normal form contains a minimal set of ratios, in an ordering of nondecreasing prime limit which is parsimonious in its use of higher limits. For example, if we put [81/80, 126/125] into normal form we obtain [80/81, 57344/59049]. 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.

Note that the defactored Hermite form of the comma list ~~requires~~ the list ~~to be~~ 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/is enfactored, so the former is beep's defactored Hermite form.

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/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.

=== Positive ratio form ===

Similar to [[Normal lists #Positive generator form|the situation with mappings]], even though by using the HNF the 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 [[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''.

For example, the 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 }}].

@@ Line 195: / Line 195: @@
 In the case of interval lists, the most common format they are presented in is as ratios, not vectors, e.g. [81/80, 64/63] rather than [{{vector|-4 4 1 0}} {{vector|-6 2 0 1}}]. So you may need to convert ratios to vectors and back when working with these forms.
-=== Defactored Hermite form ===
+=== Antitransposed Defactored Hermite form ===
-Given a matrix of ''p''-limit intervals, we can find its defactored Hermite form in much the same way as a mapping. The only difference is that they must be put in an "antitranspose sandwich", or in other words, antitransposed once at the beginning, and then antitransposed again at the end.
+Given a matrix of ''p''-limit intervals, we can find its antitransposed defactored Hermite form in much the same way as we find the defactored Hermite form of a mapping. The only difference is that they must be put in an "antitranspose sandwich", or in other words, antitransposed once at the beginning, and then antitransposed again at the end.
 For example, suppose we have the comma basis for septimal meantone:
@@ Line 270: / Line 270: @@
 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 list in normal form contains a minimal set of ratios, in an ordering of nondecreasing prime limit which is parsimonious in its use of higher limits. For example, if we put [81/80, 126/125] into normal form we obtain [80/81, 57344/59049]. 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.
-Note that the defactored Hermite form of the comma list requires the list to be 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/is enfactored, so the former is beep's defactored Hermite form.
+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/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.
 === Positive ratio form ===
-Similar to [[Normal lists #Positive generator form|the situation with mappings]], even though by using the HNF the 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 [[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''.
-For example, the 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 }}].