Just intonation subgroup: Difference between revisions

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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]].) For example, if the group is written 2.5/3.7/3, the formal primes are 2, 5/3 and 7/3. Formal primes may not be actual primes, but they behave similarly to primes in the ''p''-limit.

[[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]].) For example, if the group is written 2.5/3.7/3, the formal primes are 2, 5/3 and 7/3. Formal primes may not necessarily be actual primes, but they behave similarly to primes in the ''p''-limit.

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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 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]].) For example, if the group is written 2.5/3.7/3, the formal primes are 2, 5/3 and 7/3. Formal primes may not be actual primes, but they behave similarly to primes in the ''p''-limit.
+[[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]].) For example, if the group is written 2.5/3.7/3, the formal primes are 2, 5/3 and 7/3. Formal primes may not necessarily be actual primes, but they behave similarly to primes in the ''p''-limit.
 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.