Just intonation subgroup: Difference between revisions

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IMPORTED REVISION FROM WIKISPACES

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[[toc|flat]]
----
=Definition=
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. The index can be computed by taking the determinant of the matrix whose rows are the [[monzos]] of the generators.

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.

=7-limit subgroups=

2.3.7
Ets: 5, 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
Ets: 6, 25, 31, 171, 239, 379, 410, 789

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

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

2.5.7/3
Ets: 9, 31, 40, 50, 81, 90, 171, 261

2.5/3.7/3
Ets: 27, 68, 72, 99, 171, 517

2.27/25.7/3
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]

2.9/5.9/7
Ets: 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558

The [[Chromatic pairs|Terrain temperament]] subgroup.

=11-limit subgroups=

2.3.11
Ets: 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
Ets: 6, 7, 9, 13, 15, 22, 37, 87, 320

2.7.11
Ets: 6, 9, 11, 20, 26, 135, 161, 296

2.3.5.11
Ets: 7, 15, 22, 31, 65, 72, 87, 270, 342, 407, 494

2.3.7.11
Ets: 9, 17, 26, 31, 41, 46, 63, 72, 135

The [[Chromatic pairs|Radon temperament]] subgroup, generated by the Ptolemy Intense Chromatic [22/21, 8/7, 4/3, 3/2, 11/7, 12/7, 2/1]

2.5.7.11
Ets: 6, 15, 31, 35, 37, 109, 618, 960

2.5/3.7/3.11/3
Ets: 33, 41, 49, 57, 106, 204, 253

The [[Chromatic pairs|Indium temperament]] subgroup.

=13-limit subgroups=

2.3.13
Ets: 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
Ets: 15, 19, 34, 53, 87, 130, 140, 246, 270

The [[Chromatic pairs|Cata]], [[The Archipelago|Trinidad]] and [[The Archipelago|Parizekmic]] temperaments subgroup.

2.3.7.13
Ets: 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
Ets: 7, 10, 17, 27, 37, 84, 121, 400

The [[Chromatic pairs|Huntington temperament]] subgroup.

2.5.7.11.13
Ets: 6, 7, 13, 19, 25, 31, 37

The [[Chromatic pairs|Roulette temperament]] subgroup

2.3.13/5
Ets: 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362

The [[The Archipelago|Barbados temperament]] subgroup.

2.3.11/5.13/5
5, 9, 14, 19, 24, 29

The [[Chromatic pairs|Bridgetown temperament]] subgroup.

2.3.11/7.13/7
Ets: 5, 7, 12, 17, 29, 46, 75, 196, 271

The [[Chromatic pairs|Pepperoni temperament]] subgroup.

2.7/5.11/5.13/5
Ets: 5, 8, 21, 29, 37, 66, 169, 235

The [[Chromatic pairs|Tridec temperament]] subgroup.

Original HTML content:

<html><head><title>Just intonation subgroups</title></head><body><!-- ws:start:WikiTextTocRule:8:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --><a href="#Definition">Definition</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --> | <a href="#x7-limit subgroups">7-limit subgroups</a><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --> | <a href="#x11-limit subgroups">11-limit subgroups</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#x13-limit subgroups">13-limit subgroups</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: -->
<!-- ws:end:WikiTextTocRule:13 --><hr />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h1>
By a just intonation subgroup is meant a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Free_abelian_group" rel="nofollow">group</a> generated by a finite set of positive rational numbers via arbitrary multiplications and divisions. Any such group will be contained in a <a class="wiki_link" href="/Harmonic%20Limit">p-limit</a> group for some minimal choice of prime p, which is the prime limit of the subgroup. <br />
<br />
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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Index_of_a_subgroup" rel="nofollow">index</a> 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 <a class="wiki_link" href="/3-limit">3-limit</a> (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 <a class="wiki_link" href="/monzos">monzos</a> of the generators.<br />
<br />
A canonical naming system for just intonation subgroups is to give a <a class="wiki_link" href="/Normal%20lists">normal interval list</a> for the generators of the group, which will also show the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rank_of_an_abelian_group" rel="nofollow">rank</a> 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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="x7-limit subgroups"></a><!-- ws:end:WikiTextHeadingRule:2 -->7-limit subgroups</h1>
<br />
2.3.7<br />
Ets: 5, 31, 36, 135, 571<br />
<br />
Archytas Diatonic  [8/7, 32/27, 4/3, 3/2, 12/7, 16/9, 2/1]<br />
Safi al-Din Septimal [8/7, 9/7, 4/3, 32/21, 12/7, 16/9, 2/1]<br />
<br />
2.5.7<br />
Ets: 6, 25, 31, 171, 239, 379, 410, 789<br />
<br />
2.5.7/5<br />
Ets: 10, 29, 31, 41, 70, 171, 241, 412<br />
<br />
2.5/3.7<br />
Ets: 12, 15, 42, 57, 270, 327<br />
<br />
2.5.7/3<br />
Ets: 9, 31, 40, 50, 81, 90, 171, 261<br />
<br />
2.5/3.7/3<br />
Ets: 27, 68, 72, 99, 171, 517<br />
<br />
2.27/25.7/3<br />
Ets: 9<br />
<br />
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]<br />
<br />
2.9/5.9/7<br />
Ets: 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558<br />
<br />
The <a class="wiki_link" href="/Chromatic%20pairs">Terrain temperament</a> subgroup.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="x11-limit subgroups"></a><!-- ws:end:WikiTextHeadingRule:4 -->11-limit subgroups</h1>
<br />
2.3.11<br />
Ets: 7, 15, 17, 24, 159, 494, 518, 653<br />
<br />
Zalzal, al-Farabi's version [9/8, 27/22, 4/3, 3/2, 18/11, 16/9, 2/1]<br />
<br />
2.5.11<br />
Ets: 6, 7, 9, 13, 15, 22, 37, 87, 320<br />
<br />
2.7.11<br />
Ets: 6, 9, 11, 20, 26, 135, 161, 296<br />
<br />
2.3.5.11<br />
Ets: 7, 15, 22, 31, 65, 72, 87, 270, 342, 407, 494<br />
<br />
2.3.7.11<br />
Ets: 9, 17, 26, 31, 41, 46, 63, 72, 135<br />
<br />
The <a class="wiki_link" href="/Chromatic%20pairs">Radon temperament</a> subgroup, generated by the Ptolemy Intense Chromatic [22/21, 8/7, 4/3, 3/2, 11/7, 12/7, 2/1]<br />
<br />
2.5.7.11<br />
Ets: 6, 15, 31, 35, 37, 109, 618, 960<br />
<br />
2.5/3.7/3.11/3<br />
Ets: 33, 41, 49, 57, 106, 204, 253<br />
<br />
The <a class="wiki_link" href="/Chromatic%20pairs">Indium temperament</a> subgroup.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="x13-limit subgroups"></a><!-- ws:end:WikiTextHeadingRule:6 -->13-limit subgroups</h1>
<br />
2.3.13<br />
Ets: 7, 10, 17, 60, 70, 130, 147, 277, 424<br />
<br />
Mustaqim mode, Ibn Sina [9/8, 39/32, 4/3, 3/2, 13/8, 16/9, 2/1]<br />
<br />
2.3.5.13<br />
Ets: 15, 19, 34, 53, 87, 130, 140, 246, 270<br />
<br />
The <a class="wiki_link" href="/Chromatic%20pairs">Cata</a>, <a class="wiki_link" href="/The%20Archipelago">Trinidad</a> and <a class="wiki_link" href="/The%20Archipelago">Parizekmic</a> temperaments subgroup.<br />
<br />
2.3.7.13<br />
Ets: 10, 26, 27, 36, 77, 94, 104, 130, 234<br />
<br />
Buzurg [14/13, 16/13, 4/3, 56/39, 3/2]<br />
Safi al-Din tuning [8/7, 16/13, 4/3, 32/21, 64/39, 16/9, 2/1]<br />
Ibn Sina tuning [14/13, 7/6, 4/3, 3/2, 21/13, 7/4, 2]<br />
<br />
2.5.7.13<br />
Ets: 7, 10, 17, 27, 37, 84, 121, 400<br />
<br />
The <a class="wiki_link" href="/Chromatic%20pairs">Huntington temperament</a> subgroup.<br />
<br />
2.5.7.11.13<br />
Ets: 6, 7, 13, 19, 25, 31, 37<br />
<br />
The <a class="wiki_link" href="/Chromatic%20pairs">Roulette temperament</a> subgroup<br />
<br />
2.3.13/5<br />
Ets: 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362<br />
<br />
The <a class="wiki_link" href="/The%20Archipelago">Barbados temperament</a> subgroup.<br />
<br />
2.3.11/5.13/5<br />
5, 9, 14, 19, 24, 29<br />
<br />
The <a class="wiki_link" href="/Chromatic%20pairs">Bridgetown temperament</a> subgroup.<br />
<br />
2.3.11/7.13/7<br />
Ets: 5, 7, 12, 17, 29, 46, 75, 196, 271<br />
<br />
The <a class="wiki_link" href="/Chromatic%20pairs">Pepperoni temperament</a> subgroup.<br />
<br />
2.7/5.11/5.13/5<br />
Ets: 5, 8, 21, 29, 37, 66, 169, 235<br />
<br />
The <a class="wiki_link" href="/Chromatic%20pairs">Tridec temperament</a> subgroup.</body></html>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-04-03 14:03:20 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-04-03 14:17:54 UTC</tt>.<br>
-: The original revision id was <tt>216626854</tt>.<br>
+: The original revision id was <tt>216629940</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 <h4>Original Wikitext 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;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.
+<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">[[toc|flat]]
+----
+=Definition=
+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. The index can be computed by taking the determinant of the matrix whose rows are the [[monzos]] of the generators.
@@ Line 12: / Line 15: @@
 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.
-===7-limit subgroups===
+=7-limit subgroups=
 .3.7
@@ Line 45: / Line 48: @@
 The [[Chromatic pairs|Terrain temperament]] subgroup.
-===11-limit subgroups===
+=11-limit subgroups=
 .3.11
@@ Line 74: / Line 77: @@
 The [[Chromatic pairs|Indium temperament]] subgroup.
-===13-limit subgroups
+=13-limit subgroups=
 .3.13
@@ Line 82: / Line 85: @@
 .3.5.13
-Ets: 15, 19, 34, 53, 87, 130, 140, 270
+Ets: 15, 19, 34, 53, 87, 130, 140, 246, 270
-The [[The Archipelago|Trinidad]] and [[The Archipelago|Parizekmic]] temperaments subgroup.
+The [[Chromatic pairs|Cata]], [[The Archipelago|Trinidad]] and [[The Archipelago|Parizekmic]] temperaments subgroup.
 .3.7.13
@@ Line 92: / Line 95: @@
 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]
+.5.7.13
+Ets: 7, 10, 17, 27, 37, 84, 121, 400
+The [[Chromatic pairs|Huntington temperament]] subgroup.
+.5.7.11.13
+Ets: 6, 7, 13, 19, 25, 31, 37
+The [[Chromatic pairs|Roulette temperament]] subgroup
 .3.13/5
@@ Line 101: / Line 114: @@
 , 9, 14, 19, 24, 29
-The [[Chromatic pairs|Bridgetown temperament]] subgroup.</pre></div>
+The [[Chromatic pairs|Bridgetown temperament]] subgroup.
+.3.11/7.13/7
+Ets: 5, 7, 12, 17, 29, 46, 75, 196, 271
+The [[Chromatic pairs|Pepperoni temperament]] subgroup.
+.7/5.11/5.13/5
+Ets: 5, 8, 21, 29, 37, 66, 169, 235
+The [[Chromatic pairs|Tridec temperament]] subgroup.</pre></div>
 <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;
+<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;&lt;!-- ws:start:WikiTextTocRule:8:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;!-- ws:start:WikiTextTocRule:10: --&gt; | &lt;a href="#x7-limit subgroups"&gt;7-limit subgroups&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt; | &lt;a href="#x11-limit subgroups"&gt;11-limit subgroups&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextTocRule:12: --&gt; | &lt;a href="#x13-limit subgroups"&gt;13-limit subgroups&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;
+&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;hr /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Definition&lt;/h1&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;
 &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. The index can be computed by taking the determinant of the matrix whose rows are the &lt;a class="wiki_link" href="/monzos"&gt;monzos&lt;/a&gt; of the generators.&lt;br /&gt;
@@ Line 109: / Line 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;
 &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;
+&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="x7-limit subgroups"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;7-limit subgroups&lt;/h1&gt;
 &lt;br /&gt;
 .3.7&lt;br /&gt;
@@ Line 142: / Line 168: @@
 The &lt;a class="wiki_link" href="/Chromatic%20pairs"&gt;Terrain temperament&lt;/a&gt; subgroup.&lt;br /&gt;
 &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;
+&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="x11-limit subgroups"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;11-limit subgroups&lt;/h1&gt;
 &lt;br /&gt;
 .3.11&lt;br /&gt;
@@ Line 171: / Line 197: @@
 The &lt;a class="wiki_link" href="/Chromatic%20pairs"&gt;Indium temperament&lt;/a&gt; subgroup.&lt;br /&gt;
 &lt;br /&gt;
-===13-limit subgroups&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="x13-limit subgroups"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;13-limit subgroups&lt;/h1&gt;
 &lt;br /&gt;
 .3.13&lt;br /&gt;
@@ Line 179: / Line 205: @@
 &lt;br /&gt;
 .3.5.13&lt;br /&gt;
-Ets: 15, 19, 34, 53, 87, 130, 140, 270&lt;br /&gt;
+Ets: 15, 19, 34, 53, 87, 130, 140, 246, 270&lt;br /&gt;
 &lt;br /&gt;
-The &lt;a class="wiki_link" href="/The%20Archipelago"&gt;Trinidad&lt;/a&gt; and &lt;a class="wiki_link" href="/The%20Archipelago"&gt;Parizekmic&lt;/a&gt; temperaments subgroup.&lt;br /&gt;
+The &lt;a class="wiki_link" href="/Chromatic%20pairs"&gt;Cata&lt;/a&gt;, &lt;a class="wiki_link" href="/The%20Archipelago"&gt;Trinidad&lt;/a&gt; and &lt;a class="wiki_link" href="/The%20Archipelago"&gt;Parizekmic&lt;/a&gt; temperaments subgroup.&lt;br /&gt;
 &lt;br /&gt;
 .3.7.13&lt;br /&gt;
@@ Line 189: / Line 215: @@
 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;br /&gt;
+&lt;br /&gt;
+.5.7.13&lt;br /&gt;
+Ets: 7, 10, 17, 27, 37, 84, 121, 400&lt;br /&gt;
+&lt;br /&gt;
+The &lt;a class="wiki_link" href="/Chromatic%20pairs"&gt;Huntington temperament&lt;/a&gt; subgroup.&lt;br /&gt;
+&lt;br /&gt;
+.5.7.11.13&lt;br /&gt;
+Ets: 6, 7, 13, 19, 25, 31, 37&lt;br /&gt;
+&lt;br /&gt;
+The &lt;a class="wiki_link" href="/Chromatic%20pairs"&gt;Roulette temperament&lt;/a&gt; subgroup&lt;br /&gt;
 &lt;br /&gt;
 .3.13/5&lt;br /&gt;
@@ Line 198: / Line 234: @@
 , 9, 14, 19, 24, 29&lt;br /&gt;
 &lt;br /&gt;
-The &lt;a class="wiki_link" href="/Chromatic%20pairs"&gt;Bridgetown temperament&lt;/a&gt; subgroup.&lt;/body&gt;&lt;/html&gt;</pre></div>
+The &lt;a class="wiki_link" href="/Chromatic%20pairs"&gt;Bridgetown temperament&lt;/a&gt; subgroup.&lt;br /&gt;
+&lt;br /&gt;
+.3.11/7.13/7&lt;br /&gt;
+Ets: 5, 7, 12, 17, 29, 46, 75, 196, 271&lt;br /&gt;
+&lt;br /&gt;
+The &lt;a class="wiki_link" href="/Chromatic%20pairs"&gt;Pepperoni temperament&lt;/a&gt; subgroup.&lt;br /&gt;
+&lt;br /&gt;
+.7/5.11/5.13/5&lt;br /&gt;
+Ets: 5, 8, 21, 29, 37, 66, 169, 235&lt;br /&gt;
+&lt;br /&gt;
+The &lt;a class="wiki_link" href="/Chromatic%20pairs"&gt;Tridec temperament&lt;/a&gt; subgroup.&lt;/body&gt;&lt;/html&gt;</pre></div>

Just intonation subgroup: Difference between revisions

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