Just intonation subgroup: Difference between revisions
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| en = Just intonation subgroup | |||
| de = Untergruppe der reinen Stimmung | | de = Untergruppe der reinen Stimmung | ||
| es = | | es = | ||
| ja = 純正律部分群 | | ja = 純正律部分群 | ||
}} | }} | ||
A '''just intonation subgroup''' | A '''just intonation subgroup''' consists of all [[just intonation]] intervals formed by arbitrarily [[stacking]] a set of intervals and their inverses finitely many times. The term ''{{w|subgroup}}'' refers to its mathematical structure with respect to JI – a subset of a {{w|group (mathematics)|group}} that is also a group. Just intonation subgroups organize intervals consistently, with each subgroup corresponding to a [[lattice]]; the use of these as an approach to JI is closely related to [[regular temperament theory]]. | ||
Just intonation subgroups can be described by listing their [[generator]]s in [[frequency ratio]]s with full stops between them; we use said convention below. For example, the [[2.3.7 subgroup]] is a subgroup consisting of intervals that are combinations of [[2/1|2]], [[3/1|3]], and [[7/1|7]]. | |||
In standard mathematical notation, let ''c''<sub>1</sub>, …, ''c''<sub>''r''</sub> be positive rationals, and suppose ''v''<sub>''k''</sub> is the musical interval of log<sub>2</sub>(''c''<sub>''k''</sub>) octaves. Then | |||
$$ c_1.c_2.\cdots.c_r := \operatorname{span}_\mathbb{Z} \{v_1, \cdots, v_k\}. $$ | |||
There are three categories of subgroups: | There are three categories of subgroups: | ||
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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 [[harmonic limit|''p''-limit JI]]. 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. | 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 [[harmonic limit|''p''-limit JI]]. 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. | ||
The following terminology has been proposed 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 '''basis element''', '''structural prime''', or "'''formal prime'''".<ref>The meaning of "formal" this term is using is "of external form or structure, rather than nature or content", which is to say that a formal prime is not necessarily ''actually'' a prime, but we treat them as if they were. The original coiner of this term, [[Inthar]], has recommended its disuse, in favor of the mathematically accurate and generic "basis element", or possibly something else which indicates the co-uniqueness of the elements.</ref> For example, if the group is written 2.5/3.7/3, the basis elements are 2, 5/3 and 7/3. | The following terminology has been proposed 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 '''basis element''', '''structural prime''', or "'''formal prime'''".<ref group="note">The meaning of "formal" this term is using is "of external form or structure, rather than nature or content", which is to say that a formal prime is not necessarily ''actually'' a prime, but we treat them as if they were. The original coiner of this term, [[Inthar]], has recommended its disuse, in favor of the mathematically accurate and generic "basis element", or possibly something else which indicates the co-uniqueness of the elements.</ref> For example, if the group is written 2.5/3.7/3, the basis elements are 2, 5/3 and 7/3. | ||
== Normalization == | == Normalization == | ||
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== Notes == | == Notes == | ||
<references group="note"/> | |||
[[Category:Subgroup| ]] <!-- main article --> | [[Category:Subgroup| ]] <!-- main article --> | ||
[[Category:Just intonation]] | [[Category:Just intonation]] | ||