Normal forms: Difference between revisions

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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 ''the'' 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]] on which subgroup temperaments are defined and using which subgroup scales may be constructed. On the pages [[chromatic pairs]], [[subgroup temperaments]] and [[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.

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.

In each temperament page on this wiki, however, the "comma lists" are not normal interval lists as defined above. Instead, it shows ratio-wise the simplest comma sequence sufficient to define the temperament.

== Normal val list ==

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 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 ''the'' 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]] on which subgroup temperaments are defined and using which subgroup scales may be constructed. On the pages [[chromatic pairs]], [[subgroup temperaments]] and [[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.
+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.
+In each temperament page on this wiki, however, the "comma lists" are not normal interval lists as defined above. Instead, it shows ratio-wise the simplest comma sequence sufficient to define the temperament.
 == Normal val list ==