Just intonation subgroup: Difference between revisions

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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 ''r''₁, …, ''r''_''n'' be positive rationals, and suppose ''s''_''i'' is the musical interval of log₂(''r''_''i'') octaves. Then

$$ r_1.r_2.\cdots.r_n := \operatorname{span}_\mathbb{Z} \{v_1, \cdots, v_n\}. $$

If any redundant generators are eliminated, the set of generators is a [[basis]]. In general, given a subgroup written as generated by such a set: ''r''₁.''r''₂.''r''₃.[…].''r''_''n'', each member of this set is called 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>

Subgroups have been categorized as follows (after [[#Normalization|normalization]]):  

* ''Prime subgroups'' (e.g. 2.3.7) contain only prime basis elements;

* ''Composite subgroups'' (e.g. 2.9.5) contain composite and perhaps prime basis elements too;

* ''Fractional subgroups'' (e.g. 2.3.7/5) contain fractional numbers and perhaps prime and/or composite numbers too.  

A prime subgroup that does not omit any primes less than ''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 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.  

{{Wikipedia|Index of a subgroup}}

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 {{w|determinant}} of the [[subgroup basis matrix]], whose columns are the [[monzo]]s of the generators.

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).

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.

* [[2.5.7 subgroup]]

* [[3.5.7 subgroup]]

* 2.5.7/3

** {{EDOs|legend=1| 9, 31, 40, 50, 81, 90, 171, 261 }}

* [[2.3.11 subgroup]]

* [[2.3.5.11 subgroup]]

* [[2.3.7.11 subgroup]]

* 2.5.11

* 2.7.11

** {{EDOs|legend=1| 6, 9, 11, 20, 26, 135, 161, 296 }}

* 2.5.7.11

** {{EDOs|legend=1| 6, 15, 31, 35, 37, 109, 618, 960 }}

* 2.5/3.7/3.11/3

** {{EDOs|legend=1| 33, 41, 49, 57, 106, 204, 253 }}

** The [[indium]] temperament subgroup.

* [[2.3.5.13 subgroup]]

* [[2.3.5.7.13 subgroup]]

* [[2.3.7.11.13 subgroup]]

* 2.3.5.13

* 2.3.7.13

** Buzurg [14/13, 16/13, 4/3, 56/39, 3/2]

** Safi al-Din tuning [8/7, 16/13, 4/3, 32/21, 64/39, 16/9, 2/1]

** The [[pepperoni]] temperament subgroup.

=== Higher-limit subgroups ===

* [[Hemipyth]] (√2.√3 subgroup)

* [[Hemipent]] (√2.√3.√5 subgroup)

* [[Subgroup basis matrix]] – a formal discussion on matrix representations of subgroup bases

== Notes ==

<references group="note"/>