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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]].


{{Todo|add introduction|comment=introduction needed that helps musicians/composers understand that this is relevant to them|inline=1}}
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]].


== Definition ==
In standard mathematical notation, let ''r''<sub>1</sub>, …, ''r''<sub>''n''</sub> be positive rationals, and suppose ''s''<sub>''i''</sub> is the musical interval of log<sub>2</sub>(''r''<sub>''i''</sub>) octaves. Then
A just intonation ''subgroup'' is a [http://en.wikipedia.org/wiki/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.


It is only when the group in question is not the entire p-limit group that we have a just intonation subgroup in the strict sense. Such subgroups come in two flavors: finite [http://en.wikipedia.org/wiki/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 [[monzos]] of the generators.
$$ r_1.r_2.\cdots.r_n := \operatorname{span}_\mathbb{Z} \{v_1, \cdots, v_n\}. $$


A canonical naming system for just intonation subgroups is to give a [[Normal lists|normal interval list]] for the generators of the group, which will also show the [http://en.wikipedia.org/wiki/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.
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''<sub>1</sub>.''r''<sub>2</sub>.''r''<sub>3</sub>.[…].''r''<sub>''n''</sub>, 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>


== 7-limit subgroups ==
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.


; 2.3.7:
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.
* {{EDOs|legend=1| 5, 17, 31, 36, 135, 571 }}
* Archytas Diatonic [8/7, 32/27, 4/3, 3/2, 12/7, 16/9, 2/1]
* Safi al-Din Septimal [8/7, 9/7, 4/3, 32/21, 12/7, 16/9, 2/1]


; 2.5.7:
== Normalization ==
* {{EDOs|legend=1| 6, 25, 31, 35, 47, 171, 239, 379, 410, 789 }}
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.


; 2.3.7/5:
== Index ==
* {{EDOs|legend=1| 10, 29, 31, 41, 70, 171, 241, 412 }}
{{Wikipedia|Index of a subgroup}}


; 2.5/3.7:
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.
* {{EDOs|legend=1| 12, 15, 42, 57, 270, 327 }}


; 2.5.7/3:
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.
* {{EDOs|legend=1| 9, 31, 40, 50, 81, 90, 171, 261 }}


; 2.5/3.7/3:
== Generalization ==
* {{EDOs|legend=1| 27, 68, 72, 99, 171, 517 }}
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).


; 2.27/25.7/3:
== List of selected subgroups ==
* {{EDOs|legend=1| 9 }}
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.
* In effect, equivalent to 9EDO, which has a 7-limit version given by [27/25, 7/6, 63/50, 49/36, 72/49, 100/63, 12/7, 50/27, 2]


; 2.9/5.9/7:
=== 7-limit subgroups ===
* {{EDOs|legend=1| 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558 }}
* [[2.3.7 subgroup]]
* ''Terrain temperament'' subgroup, see [[Chromatic pairs #Terrain]]
* [[2.5.7 subgroup]]
* [[3.5.7 subgroup]]


== 11-limit subgroups ==
Others:
* 2.3.7/5
** {{EDOs|legend=1| 10, 29, 31, 41, 70, 171, 241, 412 }}
* 2.5/3.7
** {{EDOs|legend=1| 12, 15, 42, 57, 270, 327 }}
* 2.5.7/3
** {{EDOs|legend=1| 9, 31, 40, 50, 81, 90, 171, 261 }}
* 2.5/3.7/3
** {{EDOs|legend=1| 27, 68, 72, 99, 171, 517 }}
* 2.27/25.7/3
** {{EDOs|legend=1| 9 }}
** In effect, equivalent to 9edo, which has a 7-limit version given by [27/25, 7/6, 63/50, 49/36, 72/49, 100/63, 12/7, 50/27, 2]
* 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 }}
** The [[terrain]] temperament subgroup


; 2.3.11:
=== 11-limit subgroups ===
* {{EDOs|legend=1| 7, 15, 17, 24, 159, 494, 518, 653 }}
* [[2.3.11 subgroup]]
* Zalzal, al-Farabi's version [9/8, 27/22, 4/3, 3/2, 18/11, 16/9, 2/1]
* [[2.3.5.11 subgroup]]
* [[2.3.7.11 subgroup]]


; 2.5.11:
Others:
* {{EDOs|legend=1| 6, 7, 9, 13, 15, 22, 37, 87, 320 }}
* 2.5.11
** {{EDOs|legend=1| 6, 7, 9, 13, 15, 22, 37, 87, 320 }}
* 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.


; 2.7.11:
=== 13-limit subgroups ===
* {{EDOs|legend=1| 6, 9, 11, 20, 26, 135, 161, 296 }}
* [[2.3.5.13 subgroup]]
* [[2.3.5.7.13 subgroup]]
* [[2.3.7.11.13 subgroup]]


; 2.3.5.11:
Others:
* {{EDOs|legend=1| 7, 15, 22, 31, 65, 72, 87, 270, 342, 407, 494 }}
* 2.3.13
** {{EDOs|legend=1| 7, 10, 17, 60, 70, 130, 147, 277, 424 }}
** Mustaqim mode, Ibn Sina [9/8, 39/32, 4/3, 3/2, 13/8, 16/9, 2/1]
* 2.3.5.13
** {{EDOs|legend=1| 15, 19, 34, 53, 87, 130, 140, 246, 270 }}
** The [[cata]], [[trinidad]] and [[parizekmic]] temperaments subgroup
* 2.3.7.13
** {{EDOs|legend=1| 10, 26, 27, 36, 77, 94, 104, 130, 234 }}
** 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]
** Ibn Sina tuning [14/13, 7/6, 4/3, 3/2, 21/13, 7/4, 2]
* 2.3.5.11.13
* 2.5.7.13
** {{EDOs|legend=1| 7, 10, 17, 27, 37, 84, 121, 400 }}
** The [[huntington]] temperament subgroup
* 2.5.7.11.13
** {{EDOs|legend=1| 6, 7, 13, 19, 25, 31, 37 }}
** The [[roulette]] temperament subgroup
* 2.3.13/5
** {{EDOs|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}
** The [[barbados]] temperament subgroup.
* 2.3.11/5.13/5
** {{EDOs|legend=1| 5, 9, 14, 19, 24, 29 }}
** The [[bridgetown]] temperament subgroup
* 2.3.11/7.13/7
** {{EDOs|legend=1| 5, 7, 12, 17, 29, 46, 75, 196, 271 }}
** The [[pepperoni]] temperament subgroup.
* 2.7/5.11/5.13/5
** {{EDOs|legend=1| 5, 8, 21, 29, 37, 66, 169, 235 }}
** The [[tridec]] temperament subgroup.


; 2.3.7.11:
=== Higher-limit subgroups ===
* {{EDOs|legend=1| 9, 17, 26, 31, 41, 46, 63, 72, 135 }}
* [[2.3.5.7.11.13.19 subgroup]]
* 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]
* [[2.3.5.7.11.13.19.29 subgroup]]
* See: [[Gallery of 2.3.7.11 Subgroup Scales]]


; 2.5.7.11:
=== Irrational subgroups ===
* {{EDOs|legend=1| 6, 15, 31, 35, 37, 109, 618, 960 }}
* [[Hemipyth]] (√2.√3 subgroup)
* [[Hemipent]] (√2.√3.√5 subgroup)


; 2.5/3.7/3.11/3:
== See also ==
* {{EDOs|legend=1| 33, 41, 49, 57, 106, 204, 253 }}
* [[Subgroup basis matrix]] – a formal discussion on matrix representations of subgroup bases
* The [[Chromatic_pairs#Indium|Indium temperament]] subgroup.


== 13-limit subgroups ==
== Notes ==
<references group="note"/>


; 2.3.13:
* {{EDOs|legend=1| 7, 10, 17, 60, 70, 130, 147, 277, 424 }}
* Mustaqim mode, Ibn Sina [9/8, 39/32, 4/3, 3/2, 13/8, 16/9, 2/1]
; 2.3.5.13:
* {{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.
; 2.3.7.13:
* {{EDOs|legend=1| 10, 26, 27, 36, 77, 94, 104, 130, 234 }}
* 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]
* Ibn Sina tuning [14/13, 7/6, 4/3, 3/2, 21/13, 7/4, 2]
; 2.5.7.13:
* {{EDOs|legend=1| 7, 10, 17, 27, 37, 84, 121, 400 }}
* The [[Chromatic_pairs#Huntington|Huntington temperament]] subgroup.
; 2.5.7.11.13:
* {{EDOs|legend=1| 6, 7, 13, 19, 25, 31, 37 }}
* The [[Chromatic_pairs#Roulette|Roulette temperament]] subgroup
; 2.3.13/5:
* {{EDOs|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}
* The [[The Archipelago#Barbados|Barbados temperament]] subgroup.
; 2.3.11/5.13/5:
* {{EDOs|legend=1| 5, 9, 14, 19, 24, 29 }}
* The [[Chromatic pairs#Bridgetown|Bridgetown temperament]] subgroup.
; 2.3.11/7.13/7:
* {{EDOs|legend=1| 5, 7, 12, 17, 29, 46, 75, 196, 271 }}
* The [[Chromatic pairs#Pepperoni|Pepperoni temperament]] subgroup.
; 2.7/5.11/5.13/5:
* {{EDOs|legend=1| 5, 8, 21, 29, 37, 66, 169, 235 }}
* The [[Chromatic pairs#Tridec|Tridec temperament]] subgroup.
[[Category:Just]]
[[Category:Subgroup| ]] <!-- main article -->
[[Category:Subgroup| ]] <!-- main article -->
[[Category:Theory]]
[[Category:Just intonation]]

Latest revision as of 05:11, 22 August 2026

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 subgroup refers to its mathematical structure with respect to JI – a subset of a 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 generators in frequency ratios 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, 3, and 7.

In standard mathematical notation, let r1, …, rn be positive rationals, and suppose si is the musical interval of log2(ri) 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: r1.r2.r3.[…].rn, each member of this set is called a basis element, structural prime, or "formal prime".[note 1]

Subgroups have been categorized as follows (after 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 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.

Normalization

A canonical notation system for just intonation subgroups is to give a normal form for the generators of the group, which will also show the 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

English Wikipedia has an article on:

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 monzos 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) is the group generated by 2/1 and sqrt(3/2) (a neutral third which is exactly one half of 3/2, 350.978 cents). This is closely related to the 3L 4s mos tuning with neutral third generator sqrt(3/2).

List of selected subgroups

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.

7-limit subgroups

Others:

11-limit subgroups

Others:

13-limit subgroups

Others:

Higher-limit subgroups

Irrational subgroups

See also

Notes

  1. ↑ 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.
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