Just intonation subgroup: Difference between revisions

Line 22:

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.

A canonical naming system for just intonation subgroups is to give a [[Normal lists #Normal interval list|normal interval list]] for the generators of the group, which will also show the [[Wikipedia: Rank of an abelian group|rank]] of the group by the number of ~~generators~~ in the list. Below we give some of the more interesting subgroup systems. If a scale is given with the system, it means the subgroup is generated by the notes of the scale. Just intonation subgroups can be described by listing their ~~generators~~ with dots between them; the purpose of using dots is to flag the fact that it is a subgroup which is being referred to. This naming convention is employed below.

A canonical naming system for just intonation subgroups is to give a [[Normal lists #Normal interval list|normal interval list]] for the formal primes (linearly independent generators) of the group, which will also show the [[Wikipedia: Rank of an abelian group|rank]] of the group by the number of formal primes in the list. Below we give some of the more interesting subgroup systems. If a scale is given with the system, it means the subgroup is generated by the notes of the scale. Just intonation subgroups can be described by listing their formal primes with dots between them; the purpose of using dots is to flag the fact that it is a subgroup which is being referred to. This naming convention is employed below.

Non-JI intervals can also be formal primes. For example, 2.sqrt(3/2) (sometimes written 2.2ed3/2) is the group generated by 2/1 and the square root of 3/2 (a neutral third which is exactly one half of 3/2).

__FORCETOC__

== 7-limit subgroups ==

@@ Line 22: / Line 22: @@
 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.
-A canonical naming system for just intonation subgroups is to give a [[Normal lists #Normal interval list|normal interval list]] for the generators of the group, which will also show the [[Wikipedia: Rank of an abelian group|rank]] of the group by the number of generators in the list. Below we give some of the more interesting subgroup systems. If a scale is given with the system, it means the subgroup is generated by the notes of the scale. Just intonation subgroups can be described by listing their generators with dots between them; the purpose of using dots is to flag the fact that it is a subgroup which is being referred to. This naming convention is employed below.
+A canonical naming system for just intonation subgroups is to give a [[Normal lists #Normal interval list|normal interval list]] for the formal primes (linearly independent generators) of the group, which will also show the [[Wikipedia: Rank of an abelian group|rank]] of the group by the number of formal primes in the list. Below we give some of the more interesting subgroup systems. If a scale is given with the system, it means the subgroup is generated by the notes of the scale. Just intonation subgroups can be described by listing their formal primes with dots between them; the purpose of using dots is to flag the fact that it is a subgroup which is being referred to. This naming convention is employed below.
+Non-JI intervals can also be formal primes. For example, 2.sqrt(3/2) (sometimes written 2.2ed3/2) is the group generated by 2/1 and the square root of 3/2 (a neutral third which is exactly one half of 3/2).
 __FORCETOC__
 == 7-limit subgroups ==