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Just intonation subgroup: Difference between revisions

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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.
<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, 3, 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


===11-limit subgroups===
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.


[2, 3, 11]
== Generalization ==
Ets: 7, 15, 17, 24, 159, 494, 518, 653
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).


Zalzal, al-Farabi's version [9/8, 27/22, 4/3, 3/2, 18/11, 16/9, 2/1]
== List of selected subgroups ==
=== 7-limit subgroups ===
{{See also| 2.3.7 subgroup }}


[2, 5, 11]
; 2.3.7:
Ets: 6, 7, 9, 13, 15, 22, 37, 87, 320
* {{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, 7, 11]
; 2.5.7:
Ets: 6, 9, 11, 20, 26, 135, 161, 296
* {{EDOs|legend=1| 6, 25, 31, 35, 47, 171, 239, 379, 410, 789 }}


[2, 3, 5, 11]
; 2.3.7/5:
Ets: 7, 15, 22, 31, 65, 72, 87, 270, 342, 407, 494
* {{EDOs|legend=1| 10, 29, 31, 41, 70, 171, 241, 412 }}


[2, 3, 7, 11]
; 2.5/3.7:
Ets: 9, 17, 26, 31, 41, 46, 63, 72, 135
* {{EDOs|legend=1| 12, 15, 42, 57, 270, 327 }}


Ptolemy Intense Chromatic [22/21, 8/7, 4/3, 3/2, 11/7, 12/7, 2/1]
; 2.5.7/3:
* {{EDOs|legend=1| 9, 31, 40, 50, 81, 90, 171, 261 }}


[2, 5, 7, 11]
; 2.5/3.7/3:
Ets: 6, 15, 31, 35, 37, 109, 618, 960
* {{EDOs|legend=1| 27, 68, 72, 99, 171, 517 }}


===13-limit subgroups
; 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, 3, 13]
; 2.9/5.9/7:
Ets: 7, 10, 17, 60, 70, 130, 147, 277, 424
* {{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]]


Mustaqim mode, Ibn Sina [9/8, 39/32, 4/3, 3/2, 13/8, 16/9, 2/1]
; 3.5.7:
* Does not have octaves, commonly used for non-octave [[EDT]]s


[2, 3, 7, 13]
=== 11-limit subgroups ===
Ets: 10, 26, 27, 36, 77, 94, 104, 130, 234
{{See also| 2.3.7.11 subgroup }}
{{See also| Alpharabian tuning }}


Buzurg [14/13, 16/13, 4/3, 56/39, 3/2]
; 2.3.11:
Septimal tuning, Safi al-Din [8/7, 16/13, 4/3, 32/21, 64/39, 16/9, 2/1]
* {{EDOs|legend=1| 7, 15, 17, 24, 159, 494, 518, 653 }}
Septimal tuning, Ibn Sina [14/13, 7/6, 4/3, 3/2, 21/13, 7/4, 2]</pre></div>
* Zalzal, al-Farabi's version [9/8, 27/22, 4/3, 3/2, 18/11, 16/9, 2/1]
<h4>Original HTML content:</h4>
 
<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;
; 2.5.11:
&lt;br /&gt;
* {{EDOs|legend=1| 6, 7, 9, 13, 15, 22, 37, 87, 320 }}
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 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.&lt;br /&gt;
 
&lt;br /&gt;
; 2.7.11:
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.&lt;br /&gt;
* {{EDOs|legend=1| 6, 9, 11, 20, 26, 135, 161, 296 }}
&lt;br /&gt;
 
&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;
; 2.3.5.11:
&lt;br /&gt;
* {{EDOs|legend=1| 7, 15, 22, 31, 65, 72, 87, 270, 342, 407, 494 }}
[2, 3, 7]&lt;br /&gt;
 
Ets: 5, 31, 36, 135, 571&lt;br /&gt;
; 2.3.7.11:
&lt;br /&gt;
* {{EDOs|legend=1| 9, 17, 26, 31, 41, 46, 63, 72, 135 }}
Archytas Diatonic  [8/7, 32/27, 4/3, 3/2, 12/7, 16/9, 2/1]&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]
Safi al-Din Septimal [8/7, 9/7, 4/3, 32/21, 12/7, 16/9, 2/1]&lt;br /&gt;
* See: [[Gallery of 2.3.7.11 Subgroup Scales]]
&lt;br /&gt;
 
[2, 5, 7]&lt;br /&gt;
; 2.5.7.11:
Ets: 6, 25, 31, 171, 239, 379, 410, 789&lt;br /&gt;
* {{EDOs|legend=1| 6, 15, 31, 35, 37, 109, 618, 960 }}
&lt;br /&gt;
 
[2, 3, 7/5]&lt;br /&gt;
; 2.5/3.7/3.11/3:
Ets: 10, 29, 31, 41, 70, 171, 241, 412&lt;br /&gt;
* {{EDOs|legend=1| 33, 41, 49, 57, 106, 204, 253 }}
&lt;br /&gt;
* The [[Subgroup temperaments#Indium|Indium temperament]] subgroup.
[2, 5/3, 7]&lt;br /&gt;
 
Ets: 12, 15, 42, 57, 270, 327&lt;br /&gt;
=== 13-limit subgroups ===
&lt;br /&gt;
 
[2, 5, 7/3]&lt;br /&gt;
; 2.3.13:
Ets: 9, 31, 40, 50, 81, 90, 171, 261&lt;br /&gt;
* {{EDOs|legend=1| 7, 10, 17, 60, 70, 130, 147, 277, 424 }}
&lt;br /&gt;
* Mustaqim mode, Ibn Sina [9/8, 39/32, 4/3, 3/2, 13/8, 16/9, 2/1]
[2, 5/3, 7/3]&lt;br /&gt;
 
Ets: 27, 68, 72, 99, 171, 517&lt;br /&gt;
; 2.3.5.13:
&lt;br /&gt;
* {{EDOs|legend=1| 15, 19, 34, 53, 87, 130, 140, 246, 270 }}
&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;
* The [[Kleismic family#Cata|Cata]], [[The Archipelago#Trinidad|Trinidad]] and [[The Archipelago#Parizekmic|Parizekmic]] temperaments subgroup.
&lt;br /&gt;
 
[2, 3, 11]&lt;br /&gt;
; 2.3.7.13:
Ets: 7, 15, 17, 24, 159, 494, 518, 653&lt;br /&gt;
* {{EDOs|legend=1| 10, 26, 27, 36, 77, 94, 104, 130, 234 }}
&lt;br /&gt;
* Buzurg [14/13, 16/13, 4/3, 56/39, 3/2]
Zalzal, al-Farabi's version [9/8, 27/22, 4/3, 3/2, 18/11, 16/9, 2/1]&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, 5, 11]&lt;br /&gt;
 
Ets: 6, 7, 9, 13, 15, 22, 37, 87, 320&lt;br /&gt;
; 2.5.7.13:
&lt;br /&gt;
* {{EDOs|legend=1| 7, 10, 17, 27, 37, 84, 121, 400 }}
[2, 7, 11]&lt;br /&gt;
* The [[No-threes subgroup temperaments#Huntington|Huntington temperament]] subgroup.
Ets: 6, 9, 11, 20, 26, 135, 161, 296&lt;br /&gt;
 
&lt;br /&gt;
; 2.5.7.11.13:
[2, 3, 5, 11]&lt;br /&gt;
* {{EDOs|legend=1| 6, 7, 13, 19, 25, 31, 37 }}
Ets: 7, 15, 22, 31, 65, 72, 87, 270, 342, 407, 494&lt;br /&gt;
* The [[Hemimean clan#Roulette|Roulette temperament]] subgroup
&lt;br /&gt;
 
[2, 3, 7, 11]&lt;br /&gt;
; 2.3.13/5:
Ets: 9, 17, 26, 31, 41, 46, 63, 72, 135&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.
Ptolemy Intense Chromatic [22/21, 8/7, 4/3, 3/2, 11/7, 12/7, 2/1]&lt;br /&gt;
 
&lt;br /&gt;
; 2.3.11/5.13/5:
[2, 5, 7, 11]&lt;br /&gt;
* {{EDOs|legend=1| 5, 9, 14, 19, 24, 29 }}
Ets: 6, 15, 31, 35, 37, 109, 618, 960&lt;br /&gt;
* The [[Subgroup temperaments#Bridgetown|Bridgetown temperament]] subgroup.
&lt;br /&gt;
 
===13-limit subgroups&lt;br /&gt;
; 2.3.11/7.13/7:
&lt;br /&gt;
* {{EDOs|legend=1| 5, 7, 12, 17, 29, 46, 75, 196, 271 }}
[2, 3, 13]&lt;br /&gt;
* The [[Subgroup temperaments#Pepperoni|Pepperoni temperament]] subgroup.
Ets: 7, 10, 17, 60, 70, 130, 147, 277, 424&lt;br /&gt;
 
&lt;br /&gt;
; 2.7/5.11/5.13/5:
Mustaqim mode, Ibn Sina [9/8, 39/32, 4/3, 3/2, 13/8, 16/9, 2/1]&lt;br /&gt;
* {{EDOs|legend=1| 5, 8, 21, 29, 37, 66, 169, 235 }}
&lt;br /&gt;
* The [[Subgroup temperaments#Tridec|Tridec temperament]] subgroup.
[2, 3, 7, 13]&lt;br /&gt;
 
Ets: 10, 26, 27, 36, 77, 94, 104, 130, 234&lt;br /&gt;
=== Higher-limit subgroups ===
&lt;br /&gt;
* [[2.11.13.17.19 subgroup]]
Buzurg [14/13, 16/13, 4/3, 56/39, 3/2]&lt;br /&gt;
* [[2.17/13.19/13 subgroup]]
Septimal tuning, Safi al-Din [8/7, 16/13, 4/3, 32/21, 64/39, 16/9, 2/1]&lt;br /&gt;
 
Septimal tuning, Ibn Sina [14/13, 7/6, 4/3, 3/2, 21/13, 7/4, 2]&lt;/body&gt;&lt;/html&gt;</pre></div>
== See also ==
* [[Subgroup basis matrices]] – a formal discussion on matrix representations of subgroup bases
 
== Notes ==
 
[[Category:Subgroup| ]] <!-- main article -->
[[Category:Just intonation]]
Retrieved from "https://en.xen.wiki/w/Just_intonation_subgroup"