Normal forms: Difference between revisions

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(9) For any number q < 1 on this list, replace q with 1/q

The result is a normal interval list. 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 generators, each greater than zero, 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 [81/80, 59049/57344]. 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 [[Abstract_regular_temperament|abstract regular temperaments]], 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 be 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_sequences|~~comma sequence]] of septimal meantone.

The result is a normal interval list. 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 generators, each greater than zero, 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 [81/80, 59049/57344]. 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 [[Abstract_regular_temperament|abstract regular temperaments]], 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 be 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 sequence of septimal meantone.

There is only one normal comma sequence that characterizes septimal meantone. But sometimes a temperament can be characterized by multiple normal comma sequences. However, if a requirement is added that the normal comma sequence be torsion-free, then there is only one characteristic normal comma sequence, and we can speak of <u>the</u> normal comma sequence of any temperament. For example, both [27/25, 21/20] and [27/25, 49/48] are normal, and they both characterize Beep. But the latter has torsion, so Beep's normal comma sequence is the former.

Normal interval lists can also be used to characterize the [[Just_intonation_subgroups|just intonation subgroups]] on which subgroup temperaments are defined and using which subgroup scales may be constructed. On the pages [[Chromatic_pairs|Chromatic pairs]], [[Subgroup_temperaments|Subgroup temperaments]] and [[Just_intonation_subgroups|Just intonation subgroups]] 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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 (9) For any number q &lt; 1 on this list, replace q with 1/q
-The result is a normal interval list. 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 generators, each greater than zero, 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 [81/80, 59049/57344]. 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 [[Abstract_regular_temperament|abstract regular temperaments]], 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 be 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_sequences|comma sequence]] of septimal meantone.
+The result is a normal interval list. 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 generators, each greater than zero, 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 [81/80, 59049/57344]. 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 [[Abstract_regular_temperament|abstract regular temperaments]], 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 be 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 sequence of septimal meantone.
+There is only one normal comma sequence that characterizes septimal meantone. But sometimes a temperament can be characterized by multiple normal comma sequences. However, if a requirement is added that the normal comma sequence be torsion-free, then there is only one characteristic normal comma sequence, and we can speak of <u>the</u> normal comma sequence of any temperament. For example, both [27/25, 21/20] and [27/25, 49/48] are normal, and they both characterize Beep. But the latter has torsion, so Beep's normal comma sequence is the former.
 Normal interval lists can also be used to characterize the [[Just_intonation_subgroups|just intonation subgroups]] on which subgroup temperaments are defined and using which subgroup scales may be constructed. On the pages [[Chromatic_pairs|Chromatic pairs]], [[Subgroup_temperaments|Subgroup temperaments]] and [[Just_intonation_subgroups|Just intonation subgroups]] 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.