Just intonation subgroup: Difference between revisions

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== Normalization ==

A canonical ~~naming~~ system for just intonation subgroups is to give a [[normal forms #Normal forms for commas|normal form]] 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 ~~(the~~ [[Hermite normal form]] should be used here, not the [[canonical form]], because in the case of subgroups, [[enfactoring]] is ~~sometimes desirable~~, such as in the subgroup 2.9.7 which should not be reduced to 2.3.7 by subgroup canonicalization).

A canonical notation system for just intonation subgroups is to give a [[normal forms #Normal forms for commas|normal form]] for the generators of the group, which will also show the {{w|rank of an abelian group|rank}} of the group by the number of generators in the list. The [[Hermite normal form]] should be used here, not the [[canonical form]], because in the case of subgroups, [[enfactoring]] is usually desired, such as in the subgroup 2.9.7 which should not be reduced to 2.3.7 by subgroup canonicalization.

== Index ==

Intuitively speaking, the '''index''' measures the relative size of the subgroup within another subgroup, which is usually the ~~''p''-~~limit.

Intuitively speaking, the '''index''' measures the relative size of the subgroup within another subgroup, which is usually the minimal prime subgroup or the minimal prime limit.

Subgroups in the strict sense come in two flavors: finite index and infinite index. 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 [[subgroup basis matrix]], whose columns are the [[monzo]]s of the generators.

Subgroups in the strict sense come in two flavors: finite index and infinite index. 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 {{w|determinant}} of the [[subgroup basis matrix]], whose columns are the [[monzo]]s of the generators.

== Generalization ==

Non-JI intervals can also be used as basis elements, when the subgroup in question contains non-JI intervals. For example, 2.sqrt(3~~/2) (sometimes written 2.2ed3~~/2) is the group generated by [[2/1]] and [[sqrt(3/2)]] (a neutral third which is exactly one half of 3/2, 350.978 [[cent]]s). This is closely related to the [[3L 4s]] mos tuning with neutral third generator sqrt(3/2).

Non-JI intervals can also be used as basis elements, when the subgroup in question contains non-JI intervals. For example, 2.sqrt(3/2) is the group generated by [[2/1]] and [[sqrt(3/2)]] (a neutral third which is exactly one half of 3/2, 350.978 [[cent]]s). This is closely related to the [[3L 4s]] mos tuning with neutral third generator sqrt(3/2).

== List of selected subgroups ==

@@ Line 23: / Line 23: @@
 == Normalization ==
-A canonical naming system for just intonation subgroups is to give a [[normal forms #Normal forms for commas|normal form]] 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 (the [[Hermite normal form]] should be used here, not the [[canonical form]], because in the case of subgroups, [[enfactoring]] is sometimes desirable, such as in the subgroup 2.9.7 which should not be reduced to 2.3.7 by subgroup canonicalization).
+A canonical notation system for just intonation subgroups is to give a [[normal forms #Normal forms for commas|normal form]] for the generators of the group, which will also show the {{w|rank of an abelian group|rank}} of the group by the number of generators in the list. The [[Hermite normal form]] should be used here, not the [[canonical form]], because in the case of subgroups, [[enfactoring]] is usually desired, such as in the subgroup 2.9.7 which should not be reduced to 2.3.7 by subgroup canonicalization.
 == Index ==
-{{See also| Wikipedia: Index of a subgroup }}
+{{Wikipedia|Index of a subgroup}}
-Intuitively speaking, the '''index''' measures the relative size of the subgroup within another subgroup, which is usually the ''p''-limit.
+Intuitively speaking, the '''index''' measures the relative size of the subgroup within another subgroup, which is usually the minimal prime subgroup or the minimal prime limit.
-Subgroups in the strict sense come in two flavors: finite index and infinite index. 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 [[subgroup basis matrix]], whose columns are the [[monzo]]s of the generators.
+Subgroups in the strict sense come in two flavors: finite index and infinite index. 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 {{w|determinant}} of the [[subgroup basis matrix]], whose columns are the [[monzo]]s of the generators.
 == Generalization ==
-Non-JI intervals can also be used as basis elements, when the subgroup in question contains non-JI intervals. For example, 2.sqrt(3/2) (sometimes written 2.2ed3/2) is the group generated by [[2/1]] and [[sqrt(3/2)]] (a neutral third which is exactly one half of 3/2, 350.978 [[cent]]s). This is closely related to the [[3L 4s]] mos tuning with neutral third generator sqrt(3/2).
+Non-JI intervals can also be used as basis elements, when the subgroup in question contains non-JI intervals. For example, 2.sqrt(3/2) is the group generated by [[2/1]] and [[sqrt(3/2)]] (a neutral third which is exactly one half of 3/2, 350.978 [[cent]]s). This is closely related to the [[3L 4s]] mos tuning with neutral third generator sqrt(3/2).
 == List of selected subgroups ==