Just intonation subgroup: Difference between revisions
Wikispaces>genewardsmith **Imported revision 143688563 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 143693319 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | 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>2010-05-21 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-21 05:13:38 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>143693319</tt>.<br> | ||
: The revision comment was: <tt></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> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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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. | ||
===7-limit subgroups=== | |||
[2, 3, 7] | [2, 3, 7] | ||
Ets: 5, 31, 36, 135, 571 | Ets: 5, 31, 36, 135, 571 | ||
Archytas [8/7, 32/27, 4/3, 3/2, 12/7, 16/9, 2/1] | |||
Septimal [8/7, 9/7, 4/3, 32/21, 12/7, 16/9, 2/1] | |||
[2, 5, 7] | [2, 5, 7] | ||
| Line 24: | Line 28: | ||
[2, 5/3, 7] | [2, 5/3, 7] | ||
Ets: 12, 15, 42, 57, 270, 327 | 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] | [2, 5/3, 7/3] | ||
Ets: 27, 68, 72, 99, 171, 517 | Ets: 27, 68, 72, 99, 171, 517 | ||
===11-limit subgroups=== | |||
[2, 3, 11] | [2, 3, 11] | ||
Ets: 7, 15, 17, 24, 159, 494, 518, 653 | Ets: 7, 15, 17, 24, 159, 494, 518, 653 | ||
Zalzal [9/8, 27/22, 4/3, 3/2, 18/11, 16/9, 2/1] | |||
[2, 5, 11] | [2, 5, 11] | ||
| Line 38: | Line 47: | ||
[2, 7, 11] | [2, 7, 11] | ||
Ets: 6, 9, 11, 20, 26, 135, 161, 296 | Ets: 6, 9, 11, 20, 26, 135, 161, 296 | ||
</pre></div> | |||
[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 [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 [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 [8/7, 16/13, 4/3, 32/21, 64/39, 16/9, 2/1] | |||
Ibn [14/13, 7/6, 4/3, 3/2, 21/13, 7/4, 2]</pre></div> | |||
<h4>Original HTML content:</h4> | <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"><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 /> | <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"><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 /> | ||
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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.<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.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt; | <!-- 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 /> | [2, 3, 7]<br /> | ||
Ets: 5, 31, 36, 135, 571<br /> | Ets: 5, 31, 36, 135, 571<br /> | ||
<br /> | |||
Archytas [8/7, 32/27, 4/3, 3/2, 12/7, 16/9, 2/1]<br /> | |||
Septimal [8/7, 9/7, 4/3, 32/21, 12/7, 16/9, 2/1]<br /> | |||
<br /> | <br /> | ||
[2, 5, 7]<br /> | [2, 5, 7]<br /> | ||
| Line 58: | Line 95: | ||
[2, 5/3, 7]<br /> | [2, 5/3, 7]<br /> | ||
Ets: 12, 15, 42, 57, 270, 327<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 /> | <br /> | ||
[2, 5/3, 7/3]<br /> | [2, 5/3, 7/3]<br /> | ||
Ets: 27, 68, 72, 99, 171, 517<br /> | Ets: 27, 68, 72, 99, 171, 517<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt; | <!-- 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 /> | <br /> | ||
[2, 3, 11]<br /> | [2, 3, 11]<br /> | ||
Ets: 7, 15, 17, 24, 159, 494, 518, 653<br /> | Ets: 7, 15, 17, 24, 159, 494, 518, 653<br /> | ||
<br /> | |||
Zalzal [9/8, 27/22, 4/3, 3/2, 18/11, 16/9, 2/1]<br /> | |||
<br /> | <br /> | ||
[2, 5, 11]<br /> | [2, 5, 11]<br /> | ||
| Line 71: | Line 113: | ||
<br /> | <br /> | ||
[2, 7, 11]<br /> | [2, 7, 11]<br /> | ||
Ets: 6, 9, 11, 20, 26, 135, 161, 296</body></html></pre></div> | 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 [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 [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 [8/7, 16/13, 4/3, 32/21, 64/39, 16/9, 2/1]<br /> | |||
Ibn [14/13, 7/6, 4/3, 3/2, 21/13, 7/4, 2]</body></html></pre></div> | |||