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.

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.

[[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.

Subgroups in the strict sense come in two flavors: finite [[Wikipedia: Index of a subgroup|index]] and infinite index, where intuitively speaking the index measures the relative size of the subgroup within the entire ''p''-limit group. For example, the subgroups generated by 4 and 3, by 2 and 9, and by 4 and 6 all have index 2 in the full [[3-limit]] (Pythagorean) group. Half of the 3-limit intervals will belong to any one of them, and half will not, and all three groups are distinct. On the other hand, the group generated by 2, 3, and 7 is of infinite index in the full [[7-limit]] group, which is generated by 2, 3, 5 and 7. The index can be computed by taking the determinant of the matrix whose rows are the [[monzo]]s of the generators.

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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.
+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.
 [[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.
 Subgroups in the strict sense come in two flavors: finite [[Wikipedia: Index of a subgroup|index]] and infinite index, where intuitively speaking the index measures the relative size of the subgroup within the entire ''p''-limit group. For example, the subgroups generated by 4 and 3, by 2 and 9, and by 4 and 6 all have index 2 in the full [[3-limit]] (Pythagorean) group. Half of the 3-limit intervals will belong to any one of them, and half will not, and all three groups are distinct. On the other hand, the group generated by 2, 3, and 7 is of infinite index in the full [[7-limit]] group, which is generated by 2, 3, 5 and 7. The index can be computed by taking the determinant of the matrix whose rows are the [[monzo]]s of the generators.