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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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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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<h4>Original Wikitext content:</h4>
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]].  
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">By a just intonation subgroup is meant 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.
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


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.
<math>c_1.c_2.\cdots.c_r := \operatorname{span}_\mathbb{Z} \{v_1, ..., v_k\}.</math>


===7-limit subgroups===
There are three categories of subgroups:


2.3.7
* '''Prime subgroups''' (e.g. 2.3.7) contain only primes
Ets: 5, 31, 36, 135, 571
* '''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


Archytas Diatonic  [8/7, 32/27, 4/3, 3/2, 12/7, 16/9, 2/1]
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.
Safi al-Din Septimal [8/7, 9/7, 4/3, 32/21, 12/7, 16/9, 2/1]


2.5.7
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.
Ets: 6, 25, 31, 171, 239, 379, 410, 789


2.5.7/5
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.
Ets: 10, 29, 31, 41, 70, 171, 241, 412


2.5/3.7
== Normalization ==
Ets: 12, 15, 42, 57, 270, 327
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.


2.5.7/3
== Index ==
Ets: 9, 31, 40, 50, 81, 90, 171, 261
{{See also| Wikipedia: Index of a subgroup }}


2.5/3.7/3
Intuitively speaking, the '''index''' measures the relative size of the subgroup within another subgroup, which is usually the ''p''-limit.
Ets: 27, 68, 72, 99, 171, 517


2.27/25.7/3
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 determinant of the [[subgroup basis matrix]], whose columns are the [[monzo]]s of the generators.
Ets: 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]
== 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).


===11-limit subgroups===
== List of selected subgroups ==
=== 7-limit subgroups ===
{{See also| 2.3.7 subgroup }}


2.3.11
; 2.3.7:
Ets: 7, 15, 17, 24, 159, 494, 518, 653
* {{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]


Zalzal, al-Farabi's version [9/8, 27/22, 4/3, 3/2, 18/11, 16/9, 2/1]
; 2.5.7:
* {{EDOs|legend=1| 6, 25, 31, 35, 47, 171, 239, 379, 410, 789 }}


2.5.11
; 2.3.7/5:
Ets: 6, 7, 9, 13, 15, 22, 37, 87, 320
* {{EDOs|legend=1| 10, 29, 31, 41, 70, 171, 241, 412 }}


2.7.11
; 2.5/3.7:
Ets: 6, 9, 11, 20, 26, 135, 161, 296
* {{EDOs|legend=1| 12, 15, 42, 57, 270, 327 }}


2.3.5.11
; 2.5.7/3:
Ets: 7, 15, 22, 31, 65, 72, 87, 270, 342, 407, 494
* {{EDOs|legend=1| 9, 31, 40, 50, 81, 90, 171, 261 }}


2.3.7.11
; 2.5/3.7/3:
Ets: 9, 17, 26, 31, 41, 46, 63, 72, 135
* {{EDOs|legend=1| 27, 68, 72, 99, 171, 517 }}


Ptolemy Intense Chromatic [22/21, 8/7, 4/3, 3/2, 11/7, 12/7, 2/1]
; 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]


2.5.7.11
; 2.9/5.9/7:
Ets: 6, 15, 31, 35, 37, 109, 618, 960
* {{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]]


===13-limit subgroups
; 3.5.7:
* Does not have octaves, commonly used for non-octave [[EDT]]s


2.3.13
=== 11-limit subgroups ===
Ets: 7, 10, 17, 60, 70, 130, 147, 277, 424
{{See also| 2.3.7.11 subgroup }}
{{See also| Alpharabian tuning }}


Mustaqim mode, Ibn Sina [9/8, 39/32, 4/3, 3/2, 13/8, 16/9, 2/1]
; 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]


2.3.7.13
; 2.5.11:
Ets: 10, 26, 27, 36, 77, 94, 104, 130, 234
* {{EDOs|legend=1| 6, 7, 9, 13, 15, 22, 37, 87, 320 }}


Buzurg [14/13, 16/13, 4/3, 56/39, 3/2]
; 2.7.11:
Safi al-Din tuning [8/7, 16/13, 4/3, 32/21, 64/39, 16/9, 2/1]
* {{EDOs|legend=1| 6, 9, 11, 20, 26, 135, 161, 296 }}
Ibn Sina tuning [14/13, 7/6, 4/3, 3/2, 21/13, 7/4, 2]</pre></div>
 
<h4>Original HTML content:</h4>
; 2.3.5.11:
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Just intonation subgroups&lt;/title&gt;&lt;/head&gt;&lt;body&gt;By a just intonation subgroup is meant a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Free_abelian_group" rel="nofollow"&gt;group&lt;/a&gt; generated by a finite set of positive rational numbers via arbitrary multiplications and divisions. Any such group will be contained in a &lt;a class="wiki_link" href="/Harmonic%20Limit"&gt;p-limit&lt;/a&gt; group for some minimal choice of prime p, which is the prime limit of the subgroup. &lt;br /&gt;
* {{EDOs|legend=1| 7, 15, 22, 31, 65, 72, 87, 270, 342, 407, 494 }}
&lt;br /&gt;
 
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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Index_of_a_subgroup" rel="nofollow"&gt;index&lt;/a&gt; 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 &lt;a class="wiki_link" href="/3-limit"&gt;3-limit&lt;/a&gt; (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.&lt;br /&gt;
; 2.3.7.11:
&lt;br /&gt;
* {{EDOs|legend=1| 9, 17, 26, 31, 41, 46, 63, 72, 135 }}
A canonical naming system for just intonation subgroups is to give a &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal interval list&lt;/a&gt; for the generators of the group, which will also show the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rank_of_an_abelian_group" rel="nofollow"&gt;rank&lt;/a&gt; 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.&lt;br /&gt;
* 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]
&lt;br /&gt;
* See: [[Gallery of 2.3.7.11 Subgroup Scales]]
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc0"&gt;&lt;a name="x--7-limit subgroups"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;7-limit subgroups&lt;/h3&gt;
 
&lt;br /&gt;
; 2.5.7.11:
2.3.7&lt;br /&gt;
* {{EDOs|legend=1| 6, 15, 31, 35, 37, 109, 618, 960 }}
Ets: 5, 31, 36, 135, 571&lt;br /&gt;
 
&lt;br /&gt;
; 2.5/3.7/3.11/3:
Archytas Diatonic  [8/7, 32/27, 4/3, 3/2, 12/7, 16/9, 2/1]&lt;br /&gt;
* {{EDOs|legend=1| 33, 41, 49, 57, 106, 204, 253 }}
Safi al-Din Septimal [8/7, 9/7, 4/3, 32/21, 12/7, 16/9, 2/1]&lt;br /&gt;
* The [[Subgroup temperaments#Indium|Indium temperament]] subgroup.
&lt;br /&gt;
 
2.5.7&lt;br /&gt;
=== 13-limit subgroups ===
Ets: 6, 25, 31, 171, 239, 379, 410, 789&lt;br /&gt;
 
&lt;br /&gt;
; 2.3.13:
2.5.7/5&lt;br /&gt;
* {{EDOs|legend=1| 7, 10, 17, 60, 70, 130, 147, 277, 424 }}
Ets: 10, 29, 31, 41, 70, 171, 241, 412&lt;br /&gt;
* Mustaqim mode, Ibn Sina [9/8, 39/32, 4/3, 3/2, 13/8, 16/9, 2/1]
&lt;br /&gt;
 
2.5/3.7&lt;br /&gt;
; 2.3.5.13:
Ets: 12, 15, 42, 57, 270, 327&lt;br /&gt;
* {{EDOs|legend=1| 15, 19, 34, 53, 87, 130, 140, 246, 270 }}
&lt;br /&gt;
* The [[Kleismic family#Cata|Cata]], [[The Archipelago#Trinidad|Trinidad]] and [[The Archipelago#Parizekmic|Parizekmic]] temperaments subgroup.
2.5.7/3&lt;br /&gt;
 
Ets: 9, 31, 40, 50, 81, 90, 171, 261&lt;br /&gt;
; 2.3.7.13:
&lt;br /&gt;
* {{EDOs|legend=1| 10, 26, 27, 36, 77, 94, 104, 130, 234 }}
2.5/3.7/3&lt;br /&gt;
* Buzurg [14/13, 16/13, 4/3, 56/39, 3/2]
Ets: 27, 68, 72, 99, 171, 517&lt;br /&gt;
* Safi al-Din tuning [8/7, 16/13, 4/3, 32/21, 64/39, 16/9, 2/1]
&lt;br /&gt;
* Ibn Sina tuning [14/13, 7/6, 4/3, 3/2, 21/13, 7/4, 2]
2.27/25.7/3&lt;br /&gt;
 
Ets: 9&lt;br /&gt;
; 2.5.7.13:
&lt;br /&gt;
* {{EDOs|legend=1| 7, 10, 17, 27, 37, 84, 121, 400 }}
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]&lt;br /&gt;
* The [[No-threes subgroup temperaments#Huntington|Huntington temperament]] subgroup.
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x--11-limit subgroups"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11-limit subgroups&lt;/h3&gt;
; 2.5.7.11.13:
&lt;br /&gt;
* {{EDOs|legend=1| 6, 7, 13, 19, 25, 31, 37 }}
2.3.11&lt;br /&gt;
* The [[Hemimean clan#Roulette|Roulette temperament]] subgroup
Ets: 7, 15, 17, 24, 159, 494, 518, 653&lt;br /&gt;
 
&lt;br /&gt;
; 2.3.13/5:
Zalzal, al-Farabi's version [9/8, 27/22, 4/3, 3/2, 18/11, 16/9, 2/1]&lt;br /&gt;
* {{EDOs|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}
&lt;br /&gt;
* The [[The Archipelago#Barbados|Barbados temperament]] subgroup.
2.5.11&lt;br /&gt;
 
Ets: 6, 7, 9, 13, 15, 22, 37, 87, 320&lt;br /&gt;
; 2.3.11/5.13/5:
&lt;br /&gt;
* {{EDOs|legend=1| 5, 9, 14, 19, 24, 29 }}
2.7.11&lt;br /&gt;
* The [[Subgroup temperaments#Bridgetown|Bridgetown temperament]] subgroup.
Ets: 6, 9, 11, 20, 26, 135, 161, 296&lt;br /&gt;
 
&lt;br /&gt;
; 2.3.11/7.13/7:
2.3.5.11&lt;br /&gt;
* {{EDOs|legend=1| 5, 7, 12, 17, 29, 46, 75, 196, 271 }}
Ets: 7, 15, 22, 31, 65, 72, 87, 270, 342, 407, 494&lt;br /&gt;
* The [[Subgroup temperaments#Pepperoni|Pepperoni temperament]] subgroup.
&lt;br /&gt;
 
2.3.7.11&lt;br /&gt;
; 2.7/5.11/5.13/5:
Ets: 9, 17, 26, 31, 41, 46, 63, 72, 135&lt;br /&gt;
* {{EDOs|legend=1| 5, 8, 21, 29, 37, 66, 169, 235 }}
&lt;br /&gt;
* The [[Subgroup temperaments#Tridec|Tridec temperament]] subgroup.
Ptolemy Intense Chromatic [22/21, 8/7, 4/3, 3/2, 11/7, 12/7, 2/1]&lt;br /&gt;
 
&lt;br /&gt;
=== Higher-limit subgroups ===
2.5.7.11&lt;br /&gt;
* [[2.11.13.17.19 subgroup]]
Ets: 6, 15, 31, 35, 37, 109, 618, 960&lt;br /&gt;
* [[2.17/13.19/13 subgroup]]
&lt;br /&gt;
 
===13-limit subgroups&lt;br /&gt;
== See also ==
&lt;br /&gt;
* [[Subgroup basis matrices]] – a formal discussion on matrix representations of subgroup bases
2.3.13&lt;br /&gt;
 
Ets: 7, 10, 17, 60, 70, 130, 147, 277, 424&lt;br /&gt;
== Notes ==
&lt;br /&gt;
 
Mustaqim mode, Ibn Sina [9/8, 39/32, 4/3, 3/2, 13/8, 16/9, 2/1]&lt;br /&gt;
[[Category:Subgroup| ]] <!-- main article -->
&lt;br /&gt;
[[Category:Just intonation]]
2.3.7.13&lt;br /&gt;
Ets: 10, 26, 27, 36, 77, 94, 104, 130, 234&lt;br /&gt;
&lt;br /&gt;
Buzurg [14/13, 16/13, 4/3, 56/39, 3/2]&lt;br /&gt;
Safi al-Din tuning [8/7, 16/13, 4/3, 32/21, 64/39, 16/9, 2/1]&lt;br /&gt;
Ibn Sina tuning [14/13, 7/6, 4/3, 3/2, 21/13, 7/4, 2]&lt;/body&gt;&lt;/html&gt;</pre></div>
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