Normal forms: Difference between revisions

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And there is 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.

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. Such comma list that ordered in ascending prime limits was called [http://lumma.org/tuning/gws/commaseq.htm comma sequence].

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.

@@ Line 267: / Line 267: @@
 And there is 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.
+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. Such comma list that ordered in ascending prime limits was called [http://lumma.org/tuning/gws/commaseq.htm comma sequence].
 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.