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{{interwiki
{{interwiki
| de =  
| de = Untergruppe der reinen Stimmung
| en = Just intonation subgroup
| en = Just intonation subgroup
| es =  
| es =  
| ja = 純正律サブグループ
| 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]].  


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
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>
<math>c_1.c_2.\cdots.c_r := \operatorname{span}_\mathbb{Z} \{v_1, \cdots, 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. 2.9.5) contain composite numbers and perhaps prime numbers too
* '''Composite subgroups''' (e.g. 2.9.5) contain composite numbers and perhaps prime numbers too
* ''Fractional subgroups'' (e.g. 2.3.7/5) contain fractional 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.
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.
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>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 ==
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.
A canonical naming 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 [[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.


== Index ==
== Index ==
Line 34: 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 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).
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 ==
=== 7-limit subgroups ===
=== 7-limit subgroups ===
{{See also| 2.3.7 subgroup }}
* [[2.3.7 subgroup]]
 
* [[2.5.7 subgroup]]
; 2.3.7:
* [[3.5.7 subgroup]]
* {{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:
* {{EDOs|legend=1| 6, 25, 31, 35, 47, 171, 239, 379, 410, 789 }}
 
; 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 }}
* ''Terrain temperament'' subgroup, see [[Subgroup temperaments #Terrain]]


; 3.5.7:
Others:
* Does not have octaves, commonly used for non-octave [[EDT]]s
* 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


=== 11-limit subgroups ===
=== 11-limit subgroups ===
{{See also| 2.3.7.11 subgroup }}
* [[2.3.11 subgroup]]
{{See also| Alpharabian tuning }}
* [[2.3.5.11 subgroup]]
 
* [[2.3.7.11 subgroup]]
; 2.3.11:
* {{EDOs|legend=1| 7, 15, 17, 24, 159, 494, 518, 653 }}
* Zalzal, al-Farabi's version [9/8, 27/22, 4/3, 3/2, 18/11, 16/9, 2/1]
 
; 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 }}
 
; 2.3.5.11:
* {{EDOs|legend=1| 7, 15, 22, 31, 65, 72, 87, 270, 342, 407, 494 }}
 
; 2.3.7.11:
* {{EDOs|legend=1| 9, 17, 26, 31, 41, 46, 63, 72, 135 }}
* 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]]
 
; 2.5.7.11:
* {{EDOs|legend=1| 6, 15, 31, 35, 37, 109, 618, 960 }}


; 2.5/3.7/3.11/3:
Others:
* {{EDOs|legend=1| 33, 41, 49, 57, 106, 204, 253 }}
* 2.5.11
* The [[Subgroup temperaments#Indium|Indium temperament]] subgroup.
** {{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.


=== 13-limit subgroups ===
=== 13-limit subgroups ===
* [[2.3.5.13 subgroup]]
* [[2.3.5.7.13 subgroup]]
* [[2.3.7.11.13 subgroup]]


; 2.3.13:
Others:
* {{EDOs|legend=1| 7, 10, 17, 60, 70, 130, 147, 277, 424 }}
* 2.3.13
* Mustaqim mode, Ibn Sina [9/8, 39/32, 4/3, 3/2, 13/8, 16/9, 2/1]
** {{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:
* 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 [[Kleismic family#Cata|Cata]], [[The Archipelago#Trinidad|Trinidad]] and [[The Archipelago#Parizekmic|Parizekmic]] temperaments subgroup.
** The [[cata]], [[trinidad]] and [[parizekmic]] temperaments subgroup
 
* 2.3.7.13
; 2.3.7.13:
** {{EDOs|legend=1| 10, 26, 27, 36, 77, 94, 104, 130, 234 }}
* {{EDOs|legend=1| 10, 26, 27, 36, 77, 94, 104, 130, 234 }}
** Buzurg [14/13, 16/13, 4/3, 56/39, 3/2]
* 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]
* 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]
* 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
; 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 [[huntington]] temperament subgroup
* The [[No-threes subgroup temperaments#Huntington|Huntington temperament]] subgroup.
* 2.5.7.11.13
 
** {{EDOs|legend=1| 6, 7, 13, 19, 25, 31, 37 }}
; 2.5.7.11.13:
** The [[roulette]] temperament subgroup
* {{EDOs|legend=1| 6, 7, 13, 19, 25, 31, 37 }}
* 2.3.13/5
* The [[Hemimean clan#Roulette|Roulette temperament]] subgroup
** {{EDOs|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}
 
** The [[barbados]] temperament subgroup.
; 2.3.13/5:
* 2.3.11/5.13/5
* {{EDOs|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}
** {{EDOs|legend=1| 5, 9, 14, 19, 24, 29 }}
* The [[The Archipelago#Barbados|Barbados temperament]] subgroup.
** The [[bridgetown]] temperament subgroup
 
* 2.3.11/7.13/7
; 2.3.11/5.13/5:
** {{EDOs|legend=1| 5, 7, 12, 17, 29, 46, 75, 196, 271 }}
* {{EDOs|legend=1| 5, 9, 14, 19, 24, 29 }}
** The [[pepperoni]] temperament subgroup.
* The [[Subgroup temperaments#Bridgetown|Bridgetown temperament]] subgroup.
* 2.7/5.11/5.13/5
 
** {{EDOs|legend=1| 5, 8, 21, 29, 37, 66, 169, 235 }}
; 2.3.11/7.13/7:
** The [[tridec]] temperament subgroup.
* {{EDOs|legend=1| 5, 7, 12, 17, 29, 46, 75, 196, 271 }}
* The [[Subgroup temperaments#Pepperoni|Pepperoni temperament]] subgroup.
 
; 2.7/5.11/5.13/5:
* {{EDOs|legend=1| 5, 8, 21, 29, 37, 66, 169, 235 }}
* The [[Subgroup temperaments#Tridec|Tridec temperament]] subgroup.


=== Higher-limit subgroups ===
=== Higher-limit subgroups ===
* [[2.11.13.17.19 subgroup]]
* [[2.3.5.7.11.13.19 subgroup]]
* [[2.17/13.19/13 subgroup]]
* [[2.3.5.7.11.13.19.29 subgroup]]
 
; 5.7.8.9.11.13.17.23:
* [[143ed11]]


=== Irrational subgroups ===
=== Irrational subgroups ===
Line 152: Line 119:


== See also ==
== See also ==
* [[Subgroup basis matrices]] – a formal discussion on matrix representations of subgroup bases
* [[Subgroup basis matrix]] – a formal discussion on matrix representations of subgroup bases


== Notes ==
== Notes ==
Retrieved from "https://en.xen.wiki/w/Just_intonation_subgroup"