Just intonation subgroup: Difference between revisions

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* Rational subgroups (e.g. 2.3.7/5) contain rational numbers and perhaps prime and/or composite numbers too

For composite and rational subgroups, not all combinations of numbers are mathematically valid subgroups. For example, 2.3.9 has a redundant generator 9, and both 2.3.15 and 2.3.5/3 can be simplified to 2.3.5.

For composite and rational subgroups, not all combinations of numbers are mathematically valid ([[basis|bases]] for) subgroups. For example, 2.3.9 has a redundant generator 9, and both 2.3.15 and 2.3.5/3 can be simplified to 2.3.5.

[[Inthar]] proposes the following simplifying terminology for pedagogical purposes: Given a subgroup written as generated by a fixed (non-redundant) set: ''a''.''b''.''c''.[...].''d'', call any member of this set a ''formal prime'' (Mathematically, this is a synonym for an element of a fixed [[basis]]).

A prime subgroup that doesn't omit any primes < ''p'' (e.g. 2.3.5, 2.3.5.7, 2.3.5.7.11, etc. but not 2.3.7 or 3.5.7) is simply called 5-limit JI, 7-limit JI, etc. Thus a just intonation subgroup in the strict sense refers only to prime subgroups that do omit such primes, as well as the other two categories.

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 * Rational subgroups (e.g. 2.3.7/5) contain rational numbers and perhaps prime and/or composite numbers too
-For composite and rational subgroups, not all combinations of numbers are mathematically valid subgroups. For example, 2.3.9 has a redundant generator 9, and both 2.3.15 and 2.3.5/3 can be simplified to 2.3.5.
+For composite and rational subgroups, not all combinations of numbers are mathematically valid ([[basis|bases]] for) subgroups. For example, 2.3.9 has a redundant generator 9, and both 2.3.15 and 2.3.5/3 can be simplified to 2.3.5.
+[[Inthar]] proposes the following simplifying terminology for pedagogical purposes: Given a subgroup written as generated by a fixed (non-redundant) set: ''a''.''b''.''c''.[...].''d'', call any member of this set a ''formal prime'' (Mathematically, this is a synonym for an element of a fixed [[basis]]).
 A prime subgroup that doesn't omit any primes < ''p'' (e.g. 2.3.5, 2.3.5.7, 2.3.5.7.11, etc. but not 2.3.7 or 3.5.7) is simply called 5-limit JI, 7-limit JI, etc. Thus a just intonation subgroup in the strict sense refers only to prime subgroups that do omit such primes, as well as the other two categories.