Normal forms: Difference between revisions

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== Normal interval lists ==

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.

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. 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.

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And there's our canonical comma basis.

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. 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.

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 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.

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 185: / Line 185: @@
 == Normal interval lists ==
-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.
+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. 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.
@@ Line 262: / Line 262: @@
 And there's our canonical comma basis.
-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. 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.
+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 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.
 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.