Normal forms: Difference between revisions

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=== Defactored Hermite Form ===

This is Dave Keenan and Douglas Blumeyer's "~~Canonical~~ form" for a temperament, formed from [[defactoring]] the matrix (aka removing [[contorsion]]) prior to putting it into Hermite form.

This is Dave Keenan and Douglas Blumeyer's "canonical form" for a temperament, formed from [[defactoring]] the matrix (aka removing [[contorsion]]) prior to putting it into Hermite form.

We may write a list of <math>k</math> vals as an <math>k×d</math> matrix, where the rows of the matrix are the vals, and <math>d</math> is the ''dimensionality'' of the system<ref>Calling the [[wikipedia: Prime-counting function|prime-counting function]], written π(x), on the prime limit will give us this number. For examples, π(2) = 1, π(3) = 2, π(5) = 3, π(7) = 4, π(11) = 5, etc.</ref>. To get the '''Defactored Hermite form''', we do the following:

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=== Positive generator form ===

Even though by using the HNF the ~~canonical form~~ ensures that the pivot (first nonzero entry) of each mapping row is a positive ''number'', this does not necessarily mean that the corresponding generators are all positive ''in pitch''. For example, the ~~canonical form~~ of porcupine is the matrix {{ket|{{map| 1 2 3 }} {{map| 0 3 5 }}}}. The second column of this matrix tells us it takes 2 of the first generator and 3 of the second generator to reach its approximation of 3/1. But as we can tell from the first column of this matrix, it takes only 1 of the first generator and nothing else to reach its approximation of 2/1. Therefore, if we move by 2 of the first generator, we are already at this temperament's approximation of 4/1, and so if we still need to move by 3 of the second generator to reach its approximation of 3/1, then the second generator must be negative. Indeed, it is about 163 cents ''downward'' in pitch. Negative generators like this can be surprising and confusing, and so the '''positive generator form''' was developed to address this concern.

Even though by using the HNF the Defactored Hermite Form ensures that the pivot (first nonzero entry) of each mapping row is a positive ''number'', this does not necessarily mean that the corresponding generators are all positive ''in pitch''. For example, the Defactored Hermite Form of porcupine is the matrix {{ket|{{map| 1 2 3 }} {{map| 0 3 5 }}}}. The second column of this matrix tells us it takes 2 of the first generator and 3 of the second generator to reach its approximation of 3/1. But as we can tell from the first column of this matrix, it takes only 1 of the first generator and nothing else to reach its approximation of 2/1. Therefore, if we move by 2 of the first generator, we are already at this temperament's approximation of 4/1, and so if we still need to move by 3 of the second generator to reach its approximation of 3/1, then the second generator must be negative. Indeed, it is about 163 cents ''downward'' in pitch. Negative generators like this can be surprising and confusing, and so the '''positive generator form''' was developed to address this concern.

To obtain this form, we first need to know whether each generator is positive or negative in pitch. Many methods are available for finding this information, but the one which is the easiest (and therefore the one this form is defined as using) is to find the [[Frobenius generator]]s of the temperament. To break this down, we find the [[Wikipedia: Moore–Penrose pseudoinverse|Moore–Penrose pseudoinverse]] of the ''k''×''m'' matrix, A+, and multiply this from the left by the row vector of [[JIP]], J0 = [1 log23 log25 … log2''p''].

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If the ''i''-th entry in the result is negative, change the signs of every entry in the corresponding row of the mapping.

The "mapping" (though not the "map to lattice") listed on temperament pages of this wiki are in this form. The generators in ~~canonical form~~ of septimal meantone is positive already, so its positive generator form is the same as its ~~canonical form~~, [{{val| 1 0 -4 -13 }}, {{val| 0 1 4 10 }}], corresponding to generators of ~2/1 and ~3/1. An example of positive generator form that is different from the ~~canonical form~~ is the porcupine temperament, elaborated below.

The "mapping" (though not the "map to lattice") listed on temperament pages of this wiki are in this form. The generators in Defactored Hermite Form of septimal meantone is positive already, so its positive generator form is the same as its Defactored Hermite Form, [{{val| 1 0 -4 -13 }}, {{val| 0 1 4 10 }}], corresponding to generators of ~2/1 and ~3/1. An example of positive generator form that is different from the Defactored Hermite Form is the porcupine temperament, elaborated below.

The ~~canonical~~ mapping for porcupine is

The Defactored Hermite Form of the mapping for porcupine is

<math>

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== Normal interval lists ==

A similar set of normal forms are defined for interval lists. The ~~canonical~~ and positive forms parallel those for vals, however, the normal form defined for intervals which has "minimal" in its name is quite different conceptually than the normal form defined for vals which has "minimal" in its name. Also, there is no notion of an equave-reduced form for intervals.

A similar set of normal forms are defined for interval lists. The Defactored Hermite Form and positive forms parallel those for vals, however, the normal form defined for intervals which has "minimal" in its name is quite different conceptually than the normal form defined for vals which has "minimal" in its name. Also, there is no notion of an equave-reduced form for intervals.

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 {{bra|{{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.

=== ~~Canonical form~~ ===

=== Defactored Hermite Form ===

Given a matrix of ''p''-limit intervals, we can find its ~~canonical form~~ in much the same way as a mapping. The only difference is that the matrix must be antitransposed once at the beginning of the operation and once again at the end.

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 the matrix must be antitransposed once at the beginning of the operation and once again at the end. For an example, see: [[Defactoring#canonical comma-bases]]

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 ratios, 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 [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. Because [[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 list of septimal meantone.

Note that the ~~canonical form~~ of the comma list requires the list to be defactored. For example, both [25/27, 35/36] and [25/27, 49/48] characterize Beep. But the latter is enfactored, so the former is Beep's ~~canonical form~~.

Note that the Defactored Hermite Form of the comma list requires the list to be defactored. For example, both [25/27, 35/36] and [25/27, 49/48] characterize Beep. But the latter is enfactored, so the former is Beep's 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.

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=== Positive ratio form ===

Similar to [[Normal lists #Positive generator form|the situation with mappings]], even though by using the HNF the ~~canonical 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 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 ~~canonical form~~ of meantone's comma-basis is {{bra|{{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 Defactored Hermite Form of meantone's comma-basis is {{bra|{{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 {{bra|{{vector| -4 4 -1 }}}}.

@@ Line 31: / Line 31: @@
 === Defactored Hermite Form ===
-This is Dave Keenan and Douglas Blumeyer's "Canonical form" for a temperament, formed from [[defactoring]] the matrix (aka removing [[contorsion]]) prior to putting it into Hermite form.
+This is Dave Keenan and Douglas Blumeyer's "canonical form" for a temperament, formed from [[defactoring]] the matrix (aka removing [[contorsion]]) prior to putting it into Hermite form.
 We may write a list of <span><math>k</math></span> vals as an <span><math>k×d</math></span> matrix, where the rows of the matrix are the vals, and <span><math>d</math></span> is the ''dimensionality'' of the system<ref>Calling the [[wikipedia: Prime-counting function|prime-counting function]], written π(x), on the prime limit will give us this number. For examples, π(2) = 1, π(3) = 2, π(5) = 3, π(7) = 4, π(11) = 5, etc.</ref>. To get the '''Defactored Hermite form''', we do the following:
@@ Line 48: / Line 48: @@
 === Positive generator form ===
-Even though by using the HNF the canonical form ensures that the pivot (first nonzero entry) of each mapping row is a positive ''number'', this does not necessarily mean that the corresponding generators are all positive ''in pitch''. For example, the canonical form of porcupine is the matrix {{ket|{{map| 1 2 3 }} {{map| 0 3 5 }}}}. The second column of this matrix tells us it takes 2 of the first generator and 3 of the second generator to reach its approximation of 3/1. But as we can tell from the first column of this matrix, it takes only 1 of the first generator and nothing else to reach its approximation of 2/1. Therefore, if we move by 2 of the first generator, we are already at this temperament's approximation of 4/1, and so if we still need to move by 3 of the second generator to reach its approximation of 3/1, then the second generator must be negative. Indeed, it is about 163 cents ''downward'' in pitch. Negative generators like this can be surprising and confusing, and so the '''positive generator form''' was developed to address this concern.
+Even though by using the HNF the Defactored Hermite Form ensures that the pivot (first nonzero entry) of each mapping row is a positive ''number'', this does not necessarily mean that the corresponding generators are all positive ''in pitch''. For example, the Defactored Hermite Form of porcupine is the matrix {{ket|{{map| 1 2 3 }} {{map| 0 3 5 }}}}. The second column of this matrix tells us it takes 2 of the first generator and 3 of the second generator to reach its approximation of 3/1. But as we can tell from the first column of this matrix, it takes only 1 of the first generator and nothing else to reach its approximation of 2/1. Therefore, if we move by 2 of the first generator, we are already at this temperament's approximation of 4/1, and so if we still need to move by 3 of the second generator to reach its approximation of 3/1, then the second generator must be negative. Indeed, it is about 163 cents ''downward'' in pitch. Negative generators like this can be surprising and confusing, and so the '''positive generator form''' was developed to address this concern.
 To obtain this form, we first need to know whether each generator is positive or negative in pitch. Many methods are available for finding this information, but the one which is the easiest (and therefore the one this form is defined as using) is to find the [[Frobenius generator]]s of the temperament. To break this down, we find the [[Wikipedia: Moore–Penrose pseudoinverse|Moore–Penrose pseudoinverse]] of the ''k''×''m'' matrix, A<sup>+</sup>, and multiply this from the left by the row vector of [[JIP]], J<sub>0</sub> = [1 log<sub>2</sub>3 log<sub>2</sub>5 … log<sub>2</sub>''p''].
@@ Line 56: / Line 56: @@
 If the ''i''-th entry in the result is negative, change the signs of every entry in the corresponding row of the mapping.
-The "mapping" (though not the "map to lattice") listed on temperament pages of this wiki are in this form. The generators in canonical form of septimal meantone is positive already, so its positive generator form is the same as its canonical form, [{{val| 1 0 -4 -13 }}, {{val| 0 1 4 10 }}], corresponding to generators of ~2/1 and ~3/1. An example of positive generator form that is different from the canonical form is the porcupine temperament, elaborated below.
+The "mapping" (though not the "map to lattice") listed on temperament pages of this wiki are in this form. The generators in Defactored Hermite Form of septimal meantone is positive already, so its positive generator form is the same as its Defactored Hermite Form, [{{val| 1 0 -4 -13 }}, {{val| 0 1 4 10 }}], corresponding to generators of ~2/1 and ~3/1. An example of positive generator form that is different from the Defactored Hermite Form is the porcupine temperament, elaborated below.
-The canonical mapping for porcupine is
+The Defactored Hermite Form of the mapping for porcupine is
 <math>
@@ Line 179: / Line 179: @@
 == Normal interval lists ==
-A similar set of normal forms are defined for interval lists. The canonical and positive forms parallel those for vals, however, the normal form defined for intervals which has "minimal" in its name is quite different conceptually than the normal form defined for vals which has "minimal" in its name. Also, there is no notion of an equave-reduced form for intervals.
+A similar set of normal forms are defined for interval lists. The Defactored Hermite Form and positive forms parallel those for vals, however, the normal form defined for intervals which has "minimal" in its name is quite different conceptually than the normal form defined for vals which has "minimal" in its name. Also, there is no notion of an equave-reduced form for intervals.
 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 {{bra|{{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.
-=== Canonical form ===
+=== Defactored Hermite Form ===
-Given a matrix of ''p''-limit intervals, we can find its canonical form in much the same way as a mapping. The only difference is that the matrix must be antitransposed once at the beginning of the operation and once again at the end.
+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 the matrix must be antitransposed once at the beginning of the operation and once again at the end. For an example, see: [[Defactoring#canonical comma-bases]]
 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 ratios, 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 [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. Because [[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 list of septimal meantone.
-Note that the canonical form of the comma list requires the list to be defactored. For example, both [25/27, 35/36] and [25/27, 49/48] characterize Beep. But the latter is enfactored, so the former is Beep's canonical form.
+Note that the Defactored Hermite Form of the comma list requires the list to be defactored. For example, both [25/27, 35/36] and [25/27, 49/48] characterize Beep. But the latter is enfactored, so the former is Beep's 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.
@@ Line 195: / Line 196: @@
 === Positive ratio form ===
-Similar to [[Normal lists #Positive generator form|the situation with mappings]], even though by using the HNF the canonical 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 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 canonical form of meantone's comma-basis is {{bra|{{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 Defactored Hermite Form of meantone's comma-basis is {{bra|{{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 {{bra|{{vector| -4 4 -1 }}}}.