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 ([[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.

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.

A prime subgroup that does not 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. It is customary of just intonation subgroups to refer only to prime subgroups that do omit such primes, as well as the other two categories.

[[Inthar]] proposes the following terminology for streamlining pedagogy: 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.

[[Inthar]] proposes the following terminology for streamlining pedagogy: 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.

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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 * 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 ([[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.
+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.
+A prime subgroup that does not 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. It is customary of just intonation subgroups to refer only to prime subgroups that do omit such primes, as well as the other two categories.
-[[Inthar]] proposes the following terminology for streamlining pedagogy: 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.
+[[Inthar]] proposes the following terminology for streamlining pedagogy: 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.
 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.