Just intonation subgroup: Difference between revisions

Revision as of 13:40, 3 April 2011

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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.

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]

===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

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

===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]

2.3.7.13
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]

Original HTML content:

<html><head><title>Just intonation subgroups</title></head><body>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.<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:0:&lt;h3&gt; --><h3 id="toc0"><a name="x--7-limit subgroups"></a><!-- ws:end:WikiTextHeadingRule:0 -->7-limit subgroups</h3>
<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 />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x--11-limit subgroups"></a><!-- ws:end:WikiTextHeadingRule:2 -->11-limit subgroups</h3>
<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 />
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 />
===13-limit subgroups<br />
<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 />
2.3.7.13<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]</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:xenwolf|xenwolf]] and made on <tt>2011-03-31 03:45:58 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-04-03 13:40:53 UTC</tt>.<br>
-: The original revision id was <tt>215740780</tt>.<br>
+: The original revision id was <tt>216622730</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>
@@ Line 10: / Line 10: @@
 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.
-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.
+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]
+.3.7
 Ets: 5, 31, 36, 135, 571
@@ Line 20: / Line 20: @@
 Safi al-Din Septimal [8/7, 9/7, 4/3, 32/21, 12/7, 16/9, 2/1]
-[2, 5, 7]
+.5.7
 Ets: 6, 25, 31, 171, 239, 379, 410, 789
-[2, 3, 7/5]
+.5.7/5
 Ets: 10, 29, 31, 41, 70, 171, 241, 412
-[2, 5/3, 7]
+.5/3.7
 Ets: 12, 15, 42, 57, 270, 327
-[2, 5, 7/3]
+.5.7/3
 Ets: 9, 31, 40, 50, 81, 90, 171, 261
-[2, 5/3, 7/3]
+.5/3.7/3
 Ets: 27, 68, 72, 99, 171, 517
-[2, 27/25, 7/3]
+.27/25.7/3
 Ets: 9
@@ Line 42: / Line 42: @@
 ===11-limit subgroups===
-[2, 3, 11]
+.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]
+.5.11
 Ets: 6, 7, 9, 13, 15, 22, 37, 87, 320
-[2, 7, 11]
+.7.11
 Ets: 6, 9, 11, 20, 26, 135, 161, 296
-[2, 3, 5, 11]
+.3.5.11
 Ets: 7, 15, 22, 31, 65, 72, 87, 270, 342, 407, 494
-[2, 3, 7, 11]
+.3.7.11
 Ets: 9, 17, 26, 31, 41, 46, 63, 72, 135
 Ptolemy Intense Chromatic [22/21, 8/7, 4/3, 3/2, 11/7, 12/7, 2/1]
-[2, 5, 7, 11]
+.5.7.11
 Ets: 6, 15, 31, 35, 37, 109, 618, 960
 ===13-limit subgroups
-[2, 3, 13]
+.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, 7, 13]
+.3.7.13
 Ets: 10, 26, 27, 36, 77, 94, 104, 130, 234
@@ Line 82: / Line 82: @@
 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;
 &lt;br /&gt;
-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;
+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;br /&gt;
-[2, 3, 7]&lt;br /&gt;
+.3.7&lt;br /&gt;
 Ets: 5, 31, 36, 135, 571&lt;br /&gt;
 &lt;br /&gt;
@@ Line 92: / Line 92: @@
 Safi al-Din Septimal [8/7, 9/7, 4/3, 32/21, 12/7, 16/9, 2/1]&lt;br /&gt;
 &lt;br /&gt;
-[2, 5, 7]&lt;br /&gt;
+.5.7&lt;br /&gt;
 Ets: 6, 25, 31, 171, 239, 379, 410, 789&lt;br /&gt;
 &lt;br /&gt;
-[2, 3, 7/5]&lt;br /&gt;
+.5.7/5&lt;br /&gt;
 Ets: 10, 29, 31, 41, 70, 171, 241, 412&lt;br /&gt;
 &lt;br /&gt;
-[2, 5/3, 7]&lt;br /&gt;
+.5/3.7&lt;br /&gt;
 Ets: 12, 15, 42, 57, 270, 327&lt;br /&gt;
 &lt;br /&gt;
-[2, 5, 7/3]&lt;br /&gt;
+.5.7/3&lt;br /&gt;
 Ets: 9, 31, 40, 50, 81, 90, 171, 261&lt;br /&gt;
 &lt;br /&gt;
-[2, 5/3, 7/3]&lt;br /&gt;
+.5/3.7/3&lt;br /&gt;
 Ets: 27, 68, 72, 99, 171, 517&lt;br /&gt;
 &lt;br /&gt;
-[2, 27/25, 7/3]&lt;br /&gt;
+.27/25.7/3&lt;br /&gt;
 Ets: 9&lt;br /&gt;
 &lt;br /&gt;
@@ Line 114: / Line 114: @@
 &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;br /&gt;
-[2, 3, 11]&lt;br /&gt;
+.3.11&lt;br /&gt;
 Ets: 7, 15, 17, 24, 159, 494, 518, 653&lt;br /&gt;
 &lt;br /&gt;
 Zalzal, al-Farabi's version [9/8, 27/22, 4/3, 3/2, 18/11, 16/9, 2/1]&lt;br /&gt;
 &lt;br /&gt;
-[2, 5, 11]&lt;br /&gt;
+.5.11&lt;br /&gt;
 Ets: 6, 7, 9, 13, 15, 22, 37, 87, 320&lt;br /&gt;
 &lt;br /&gt;
-[2, 7, 11]&lt;br /&gt;
+.7.11&lt;br /&gt;
 Ets: 6, 9, 11, 20, 26, 135, 161, 296&lt;br /&gt;
 &lt;br /&gt;
-[2, 3, 5, 11]&lt;br /&gt;
+.3.5.11&lt;br /&gt;
 Ets: 7, 15, 22, 31, 65, 72, 87, 270, 342, 407, 494&lt;br /&gt;
 &lt;br /&gt;
-[2, 3, 7, 11]&lt;br /&gt;
+.3.7.11&lt;br /&gt;
 Ets: 9, 17, 26, 31, 41, 46, 63, 72, 135&lt;br /&gt;
 &lt;br /&gt;
 Ptolemy Intense Chromatic [22/21, 8/7, 4/3, 3/2, 11/7, 12/7, 2/1]&lt;br /&gt;
 &lt;br /&gt;
-[2, 5, 7, 11]&lt;br /&gt;
+.5.7.11&lt;br /&gt;
 Ets: 6, 15, 31, 35, 37, 109, 618, 960&lt;br /&gt;
 &lt;br /&gt;
 ===13-limit subgroups&lt;br /&gt;
 &lt;br /&gt;
-[2, 3, 13]&lt;br /&gt;
+.3.13&lt;br /&gt;
 Ets: 7, 10, 17, 60, 70, 130, 147, 277, 424&lt;br /&gt;
 &lt;br /&gt;
 Mustaqim mode, Ibn Sina [9/8, 39/32, 4/3, 3/2, 13/8, 16/9, 2/1]&lt;br /&gt;
 &lt;br /&gt;
-[2, 3, 7, 13]&lt;br /&gt;
+.3.7.13&lt;br /&gt;
 Ets: 10, 26, 27, 36, 77, 94, 104, 130, 234&lt;br /&gt;
 &lt;br /&gt;

Just intonation subgroup: Difference between revisions

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IMPORTED REVISION FROM WIKISPACES

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Original HTML content:

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