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i so want to remove "formal prime" here but it's actually used on the discord :sob:
 
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{{Todo|add introduction|comment=introduction needed that helps musicians/composers understand that this is relevant to them|inline=1}}
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 [[Wikipedia: Free abelian group|group]] generated by a finite set of positive rational numbers via arbitrary multiplications and divisions. Any such group will be contained in a [[Harmonic limit|''p''-limit]] group for some minimal choice of prime ''p'', which is the prime limit of the subgroup.
 
Just intonation subgroups can be described by listing their [[generator]]s with full stops between them; we use said convention below. In standard mathematical notation, let ''c''<sub>1</sub>, ..., ''c''<sub>''r''</sub> be positive reals, and suppose ''v''<sub>''k''</sub> is the musical interval of log<sub>2</sub>(''c''<sub>''k''</sub>) octaves. Then
 
<math>c_1.c_2.\cdots.c_r := \operatorname{span}_\mathbb{Z} \{v_1, ..., v_k\}.</math>


There are three categories of subgroups:
There are three categories of subgroups:


* Prime subgroups (e.g. 2.3.7) contain only primes
* '''Prime subgroups''' (e.g. 2.3.7) contain only primes
* Composite subgroups (e.g. 2.5.9) contain composite numbers and perhaps prime numbers too
* '''Composite subgroups''' (e.g. 2.9.5) contain composite numbers and perhaps prime numbers too
* Rational subgroups (e.g. 2.3.7/5) contain rational numbers and perhaps prime and/or composite numbers too
* '''Fractional subgroups''' (e.g. 2.3.7/5) contain fractional numbers and perhaps prime and/or composite numbers too
 
For composite and fractional subgroups, not all combinations of numbers are mathematically valid [[basis|bases]] for subgroups. For example, 2.3.9 has a redundant generator 9, and both 2.3.15 and 2.3.5/3 can be simplified to 2.3.5.
 
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.
 
== Normalization ==
A canonical naming system for just intonation subgroups is to give a [[Normal lists #Normal interval lists|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 (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). 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.


For composite and rational subgroups, not all combinations of numbers are mathematically valid ([[basis|bases]] for) subgroups. For example, 2.3.9 has a redundant generator 9, and both 2.3.15 and 2.3.5/3 can be simplified to 2.3.5.
== Index ==
{{See also| Wikipedia: Index of a subgroup }}


[[Inthar]] proposes the following simplifying terminology for pedagogical purposes: Given a subgroup written as generated by a fixed (non-redundant) set: ''a''.''b''.''c''.[...].''d'', call any member of this set a '''formal prime'''. (Mathematically, this is a synonym for an element of a fixed [[basis]].) For example, if the group is written 2.5/3.7/3, the formal primes are 2, 5/3 and 7/3. Formal primes may not be actual primes, but they behave similarly to primes in the ''p''-limit.
Intuitively speaking, the '''index''' measures the relative size of the subgroup within another subgroup, which is usually the ''p''-limit.  


A prime subgroup that doesn't 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 5-limit JI, 7-limit JI, etc. Thus a just intonation subgroup in the strict sense refers only to prime subgroups that do omit such primes, as well as the other two categories.
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 [[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.
== 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 350.978 cents, the square root of 3/2 (a neutral third which is exactly one half of 3/2). This is closely related to the [[3L 4s]] mos tuning with neutral third generator sqrt(3/2).


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.
== List of selected subgroups ==
__FORCETOC__
=== 7-limit subgroups ===
== 7-limit subgroups ==
{{See also| 2.3.7 subgroup }}


; 2.3.7:
; 2.3.7:
Line 52: Line 66:
; 2.9/5.9/7:
; 2.9/5.9/7:
* {{EDOs|legend=1| 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558 }}
* {{EDOs|legend=1| 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558 }}
* ''Terrain temperament'' subgroup, see [[Chromatic pairs #Terrain]]
* ''Terrain temperament'' subgroup, see [[Subgroup temperaments #Terrain]]
 
; 3.5.7:
* Does not have octaves, commonly used for non-octave [[EDT]]s


== 11-limit subgroups ==
=== 11-limit subgroups ===
{{See also| 2.3.7.11 subgroup }}
{{See also| Alpharabian tuning }}


; 2.3.11:
; 2.3.11:
Line 71: Line 90:
; 2.3.7.11:
; 2.3.7.11:
* {{EDOs|legend=1| 9, 17, 26, 31, 41, 46, 63, 72, 135 }}
* {{EDOs|legend=1| 9, 17, 26, 31, 41, 46, 63, 72, 135 }}
* The [[Chromatic pairs#Radon|Radon temperament]] subgroup, generated by the Ptolemy Intense Chromatic [22/21, 8/7, 4/3, 3/2, 11/7, 12/7, 2/1]
* The [[Gamelismic clan#Radon|Radon temperament]] subgroup, generated by the Ptolemy Intense Chromatic [22/21, 8/7, 4/3, 3/2, 11/7, 12/7, 2/1]
* See: [[Gallery of 2.3.7.11 Subgroup Scales]]
* See: [[Gallery of 2.3.7.11 Subgroup Scales]]


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; 2.5/3.7/3.11/3:
; 2.5/3.7/3.11/3:
* {{EDOs|legend=1| 33, 41, 49, 57, 106, 204, 253 }}
* {{EDOs|legend=1| 33, 41, 49, 57, 106, 204, 253 }}
* The [[Chromatic_pairs#Indium|Indium temperament]] subgroup.
* The [[Subgroup temperaments#Indium|Indium temperament]] subgroup.


== 13-limit subgroups ==
=== 13-limit subgroups ===


; 2.3.13:
; 2.3.13:
Line 89: Line 108:
; 2.3.5.13:
; 2.3.5.13:
* {{EDOs|legend=1| 15, 19, 34, 53, 87, 130, 140, 246, 270 }}
* {{EDOs|legend=1| 15, 19, 34, 53, 87, 130, 140, 246, 270 }}
* The [[Chromatic pairs#Cata|Cata]], [[The Archipelago#Trinidad|Trinidad]] and [[The Archipelago#Parizekmic|Parizekmic]] temperaments subgroup.
* The [[Kleismic family#Cata|Cata]], [[The Archipelago#Trinidad|Trinidad]] and [[The Archipelago#Parizekmic|Parizekmic]] temperaments subgroup.


; 2.3.7.13:
; 2.3.7.13:
Line 99: Line 118:
; 2.5.7.13:
; 2.5.7.13:
* {{EDOs|legend=1| 7, 10, 17, 27, 37, 84, 121, 400 }}
* {{EDOs|legend=1| 7, 10, 17, 27, 37, 84, 121, 400 }}
* The [[Chromatic_pairs#Huntington|Huntington temperament]] subgroup.
* The [[No-threes subgroup temperaments#Huntington|Huntington temperament]] subgroup.


; 2.5.7.11.13:
; 2.5.7.11.13:
* {{EDOs|legend=1| 6, 7, 13, 19, 25, 31, 37 }}
* {{EDOs|legend=1| 6, 7, 13, 19, 25, 31, 37 }}
* The [[Chromatic_pairs#Roulette|Roulette temperament]] subgroup
* The [[Hemimean clan#Roulette|Roulette temperament]] subgroup


; 2.3.13/5:
; 2.3.13/5:
Line 111: Line 130:
; 2.3.11/5.13/5:
; 2.3.11/5.13/5:
* {{EDOs|legend=1| 5, 9, 14, 19, 24, 29 }}
* {{EDOs|legend=1| 5, 9, 14, 19, 24, 29 }}
* The [[Chromatic pairs#Bridgetown|Bridgetown temperament]] subgroup.
* The [[Subgroup temperaments#Bridgetown|Bridgetown temperament]] subgroup.


; 2.3.11/7.13/7:
; 2.3.11/7.13/7:
* {{EDOs|legend=1| 5, 7, 12, 17, 29, 46, 75, 196, 271 }}
* {{EDOs|legend=1| 5, 7, 12, 17, 29, 46, 75, 196, 271 }}
* The [[Chromatic pairs#Pepperoni|Pepperoni temperament]] subgroup.
* The [[Subgroup temperaments#Pepperoni|Pepperoni temperament]] subgroup.


; 2.7/5.11/5.13/5:
; 2.7/5.11/5.13/5:
* {{EDOs|legend=1| 5, 8, 21, 29, 37, 66, 169, 235 }}
* {{EDOs|legend=1| 5, 8, 21, 29, 37, 66, 169, 235 }}
* The [[Chromatic pairs#Tridec|Tridec temperament]] subgroup.
* The [[Subgroup temperaments#Tridec|Tridec temperament]] subgroup.
 
=== Higher-limit subgroups ===
* [[2.11.13.17.19 subgroup]]
* [[2.17/13.19/13 subgroup]]
 
== See also ==
* [[Subgroup basis matrices]] – a formal discussion on matrix representations of subgroup bases
 
== Notes ==


[[Category:Subgroup| ]] <!-- main article -->
[[Category:Just intonation]]
[[Category:Just intonation]]
[[Category:Subgroup| ]] <!-- main article -->
[[Category:Theory]]
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