Just intonation subgroup: Difference between revisions
m Cleanup |
No edit summary |
||
| (3 intermediate revisions by 3 users not shown) | |||
| Line 1: | Line 1: | ||
{{interwiki | {{interwiki | ||
| de = | | de = Untergruppe der reinen Stimmung | ||
| en = Just intonation subgroup | | en = Just intonation subgroup | ||
| es = | | es = | ||
| ja = | | ja = 純正律部分群 | ||
}} | }} | ||
A '''just intonation subgroup''' is a {{w|free abelian group|group}} generated by a finite set of positive rational numbers via arbitrary multiplications and divisions. Using subgroups implies a way to organize [[just intonation]] intervals such that they form a lattice. Therefore it is closely related to [[regular temperament theory]]. | A '''just intonation subgroup''' is a {{w|free abelian group|group}} generated by a finite set of positive rational numbers via arbitrary multiplications and divisions. Using subgroups implies a way to organize [[just intonation]] intervals such that they form a lattice. Therefore it is closely related to [[regular temperament theory]]. | ||
| Line 33: | Line 33: | ||
== Generalization == | == 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 | 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). | ||
== List of selected subgroups == | == List of selected subgroups == | ||
| Line 143: | Line 143: | ||
* [[2.17/13.19/13 subgroup]] | * [[2.17/13.19/13 subgroup]] | ||
; 5.7 | ; 8.9.5.7.11.13.17.23: | ||
* [[143ed11]] | * [[143ed11]] | ||