Swetismic chords: Difference between revisions

Revision as of 22:45, 12 December 2011

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This revision was by author genewardsmith and made on 2011-12-12 22:45:27 UTC.

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A //swetismic chord// is a swetismic (540/539) 11-limit [[dyadic chord|essentially tempered chord]]. 

There are six swetismic triads, consisting of three pairs of chords in inverse relationship: 1-7/5-12/7 with steps 7/5-11/9-7/6 and 1-7/6-10/7 with steps 7/6-11/9-7/5; 1-7/6-9/7 with steps 7/6-11/10-14/9 and 1-14/9-12/7 with steps 14/9-11/10-7/6; and 1-9/7-7/5 with steps 9/7-12/11-10/7 and 1-10/7-14/9 with steps 10/7-12/11-9/7.

There are fifteen swetismic tetrads, consisting of three palindromic (self-inverse) chords and six pairs of chords in inverse relationship. Then palindromic tetrads are 1-7/6-9/7-3/2 with steps 7/6-11/10-7/6-4/3; 1-6/5-7/5-12/7 with steps 6/5-7/6-11/9-7/6 and 1-9/7-7/5-9/5 with steps 9/7-12/11-9/7-10/9. The six pairs are 1-9/7-3/2-11/6 with steps 9/7-7/6-11/9-12/11 and 1-7/6-3/2-18/11 with steps 7/6-9/7-12/11-11/9; 1-7/6-9/7-10/7 with steps 7/6-11/10-10/9-7/5 and 1-7/5-14/9-12/7 with steps 7/5-10/9-11/10-7/6; 1-7/6-9/7-18/11 with steps 7/6-11/10-14/11-11/9 and 1-14/11-7/5-18/11 with steps 14/11-11/10-7/6-11/9; 1-7/6-10/7-11/6 with steps 7/6-11/9-9/7-12/11 and 1-9/7-11/7-11/6 with steps 9/7-11/9-7/6-12/11; 1-7/6-9/7-11/6 with steps 7/6-11/10-10/7-12/11 and 1-10/7-11/7-11/6 with steps 10/7-11/10-7/6-12/11; and 1-10/7-14/9-12/7 with steps 10/7-12/11-11/10-7/6 and 1-7/6-9/7-7/5 with steps 7/6-11/10-12/11-10/7.

Finally, there are six swetismic pentads coming in three pairs: 1-7/6-9/7-3/2-11/6 with steps 7/6-11/10-7/6-11/9-12/11 and 1-7/6-9/7-3/2-18/11 with steps 7/6-11/10-7/6-12/11-11/9; 1-7/6-10/7-5/3-11/6 with steps 7/6-11/9-7/6-11/10-12/11 and 1-7/6-10/7-5/3-20/11 with steps 7/6-11/9-7/6-12/11-11/10; and 1-7/6-9/7-10/7-11/6 with steps 7/6-11/10-10/9-9/7-12/11 and 1-9/7-10/7-11/7-11/6 with steps 9/7-10/9-11/10-7/6-12/11.

If we are willing to consider the 15-limit, there are also 15-limit swetismic tetrads of 1-9/7-3/2-7/4, 1-7/6-3/2-12/7, 1-11/9-10/7-5/3 and 1-7/6-15/11-5/3 with steps 9/7-7/6-7/6-8/7, 7/6-9/7-8/7-7/6, 11/9-7/6-7/6-6/5 and 7/6-7/6-11/9-6/5.

Equal temperaments with swetismic chords include 19, 22, 31, 41, 53, 58, 72, 80, 94, 103, 111, 121, 130, 152, 183, 224, 354, 537, 578, 761d, 1115de, 1339de, 1491de, 1715de and 1845de.

Original HTML content:

<html><head><title>swetismic chords</title></head><body>A <em>swetismic chord</em> is a swetismic (540/539) 11-limit <a class="wiki_link" href="/dyadic%20chord">essentially tempered chord</a>. <br />
<br />
There are six swetismic triads, consisting of three pairs of chords in inverse relationship: 1-7/5-12/7 with steps 7/5-11/9-7/6 and 1-7/6-10/7 with steps 7/6-11/9-7/5; 1-7/6-9/7 with steps 7/6-11/10-14/9 and 1-14/9-12/7 with steps 14/9-11/10-7/6; and 1-9/7-7/5 with steps 9/7-12/11-10/7 and 1-10/7-14/9 with steps 10/7-12/11-9/7.<br />
<br />
There are fifteen swetismic tetrads, consisting of three palindromic (self-inverse) chords and six pairs of chords in inverse relationship. Then palindromic tetrads are 1-7/6-9/7-3/2 with steps 7/6-11/10-7/6-4/3; 1-6/5-7/5-12/7 with steps 6/5-7/6-11/9-7/6 and 1-9/7-7/5-9/5 with steps 9/7-12/11-9/7-10/9. The six pairs are 1-9/7-3/2-11/6 with steps 9/7-7/6-11/9-12/11 and 1-7/6-3/2-18/11 with steps 7/6-9/7-12/11-11/9; 1-7/6-9/7-10/7 with steps 7/6-11/10-10/9-7/5 and 1-7/5-14/9-12/7 with steps 7/5-10/9-11/10-7/6; 1-7/6-9/7-18/11 with steps 7/6-11/10-14/11-11/9 and 1-14/11-7/5-18/11 with steps 14/11-11/10-7/6-11/9; 1-7/6-10/7-11/6 with steps 7/6-11/9-9/7-12/11 and 1-9/7-11/7-11/6 with steps 9/7-11/9-7/6-12/11; 1-7/6-9/7-11/6 with steps 7/6-11/10-10/7-12/11 and 1-10/7-11/7-11/6 with steps 10/7-11/10-7/6-12/11; and 1-10/7-14/9-12/7 with steps 10/7-12/11-11/10-7/6 and 1-7/6-9/7-7/5 with steps 7/6-11/10-12/11-10/7.<br />
<br />
Finally, there are six swetismic pentads coming in three pairs: 1-7/6-9/7-3/2-11/6 with steps 7/6-11/10-7/6-11/9-12/11 and 1-7/6-9/7-3/2-18/11 with steps 7/6-11/10-7/6-12/11-11/9; 1-7/6-10/7-5/3-11/6 with steps 7/6-11/9-7/6-11/10-12/11 and 1-7/6-10/7-5/3-20/11 with steps 7/6-11/9-7/6-12/11-11/10; and 1-7/6-9/7-10/7-11/6 with steps 7/6-11/10-10/9-9/7-12/11 and 1-9/7-10/7-11/7-11/6 with steps 9/7-10/9-11/10-7/6-12/11.<br />
<br />
If we are willing to consider the 15-limit, there are also 15-limit swetismic tetrads of 1-9/7-3/2-7/4, 1-7/6-3/2-12/7, 1-11/9-10/7-5/3 and 1-7/6-15/11-5/3 with steps 9/7-7/6-7/6-8/7, 7/6-9/7-8/7-7/6, 11/9-7/6-7/6-6/5 and 7/6-7/6-11/9-6/5.<br />
<br />
Equal temperaments with swetismic chords include 19, 22, 31, 41, 53, 58, 72, 80, 94, 103, 111, 121, 130, 152, 183, 224, 354, 537, 578, 761d, 1115de, 1339de, 1491de, 1715de and 1845de.</body></html>

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 <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>2011-08-05 01:28:26 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-12-12 22:45:27 UTC</tt>.<br>
-: The original revision id was <tt>244442099</tt>.<br>
+: The original revision id was <tt>285322490</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">The 11-limit swetismic triad is either the 540/539-tempered version of a 7/6-11/9-7/5 chord or its inversion, an 11/9-7/6-7/5 chord. It is an 11-limit [[dyadic chord|essentially tempered triad]], and can also be characterized as the tempering of 1-7/6-10/7 or 1-11/9-10/7. It can be extended to the 11-limit swetismic tetrad, the 7/6-11/9-7/6-6/5 chord, the swetismic tempering of 1-7/6-10/7-5/3. There are also 15-limit swetismic tetrads of 1-9/7-3/2-7/4, 1-7/6-3/2-12/7, 1-11/9-10/7-5/3 and 1-7/6-15/11-5/3 with steps 9/7-7/6-7/6-8/7, 7/6-9/7-8/7-7/6, 11/9-7/6-7/6-6/5 and 7/6-7/6-11/9-6/5.
+<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">A //swetismic chord// is a swetismic (540/539) 11-limit [[dyadic chord|essentially tempered chord]].
-Equal temperaments with swetismic tetrads include 19, 22, 31, 41, 53, 58, 72, 80, 94, 103, 111, 121, 130, 152, 183, 224, 354, 537 and 578.</pre></div>
+There are six swetismic triads, consisting of three pairs of chords in inverse relationship: 1-7/5-12/7 with steps 7/5-11/9-7/6 and 1-7/6-10/7 with steps 7/6-11/9-7/5; 1-7/6-9/7 with steps 7/6-11/10-14/9 and 1-14/9-12/7 with steps 14/9-11/10-7/6; and 1-9/7-7/5 with steps 9/7-12/11-10/7 and 1-10/7-14/9 with steps 10/7-12/11-9/7.
+There are fifteen swetismic tetrads, consisting of three palindromic (self-inverse) chords and six pairs of chords in inverse relationship. Then palindromic tetrads are 1-7/6-9/7-3/2 with steps 7/6-11/10-7/6-4/3; 1-6/5-7/5-12/7 with steps 6/5-7/6-11/9-7/6 and 1-9/7-7/5-9/5 with steps 9/7-12/11-9/7-10/9. The six pairs are 1-9/7-3/2-11/6 with steps 9/7-7/6-11/9-12/11 and 1-7/6-3/2-18/11 with steps 7/6-9/7-12/11-11/9; 1-7/6-9/7-10/7 with steps 7/6-11/10-10/9-7/5 and 1-7/5-14/9-12/7 with steps 7/5-10/9-11/10-7/6; 1-7/6-9/7-18/11 with steps 7/6-11/10-14/11-11/9 and 1-14/11-7/5-18/11 with steps 14/11-11/10-7/6-11/9; 1-7/6-10/7-11/6 with steps 7/6-11/9-9/7-12/11 and 1-9/7-11/7-11/6 with steps 9/7-11/9-7/6-12/11; 1-7/6-9/7-11/6 with steps 7/6-11/10-10/7-12/11 and 1-10/7-11/7-11/6 with steps 10/7-11/10-7/6-12/11; and 1-10/7-14/9-12/7 with steps 10/7-12/11-11/10-7/6 and 1-7/6-9/7-7/5 with steps 7/6-11/10-12/11-10/7.
+Finally, there are six swetismic pentads coming in three pairs: 1-7/6-9/7-3/2-11/6 with steps 7/6-11/10-7/6-11/9-12/11 and 1-7/6-9/7-3/2-18/11 with steps 7/6-11/10-7/6-12/11-11/9; 1-7/6-10/7-5/3-11/6 with steps 7/6-11/9-7/6-11/10-12/11 and 1-7/6-10/7-5/3-20/11 with steps 7/6-11/9-7/6-12/11-11/10; and 1-7/6-9/7-10/7-11/6 with steps 7/6-11/10-10/9-9/7-12/11 and 1-9/7-10/7-11/7-11/6 with steps 9/7-10/9-11/10-7/6-12/11.
+If we are willing to consider the 15-limit, there are also 15-limit swetismic tetrads of 1-9/7-3/2-7/4, 1-7/6-3/2-12/7, 1-11/9-10/7-5/3 and 1-7/6-15/11-5/3 with steps 9/7-7/6-7/6-8/7, 7/6-9/7-8/7-7/6, 11/9-7/6-7/6-6/5 and 7/6-7/6-11/9-6/5.
+Equal temperaments with swetismic chords include 19, 22, 31, 41, 53, 58, 72, 80, 94, 103, 111, 121, 130, 152, 183, 224, 354, 537, 578, 761d, 1115de, 1339de, 1491de, 1715de and 1845de.</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;swetismic chords&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 11-limit swetismic triad is either the 540/539-tempered version of a 7/6-11/9-7/5 chord or its inversion, an 11/9-7/6-7/5 chord. It is an 11-limit &lt;a class="wiki_link" href="/dyadic%20chord"&gt;essentially tempered triad&lt;/a&gt;, and can also be characterized as the tempering of 1-7/6-10/7 or 1-11/9-10/7. It can be extended to the 11-limit swetismic tetrad, the 7/6-11/9-7/6-6/5 chord, the swetismic tempering of 1-7/6-10/7-5/3. There are also 15-limit swetismic tetrads of 1-9/7-3/2-7/4, 1-7/6-3/2-12/7, 1-11/9-10/7-5/3 and 1-7/6-15/11-5/3 with steps 9/7-7/6-7/6-8/7, 7/6-9/7-8/7-7/6, 11/9-7/6-7/6-6/5 and 7/6-7/6-11/9-6/5.&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;swetismic chords&lt;/title&gt;&lt;/head&gt;&lt;body&gt;A &lt;em&gt;swetismic chord&lt;/em&gt; is a swetismic (540/539) 11-limit &lt;a class="wiki_link" href="/dyadic%20chord"&gt;essentially tempered chord&lt;/a&gt;. &lt;br /&gt;
+&lt;br /&gt;
+There are six swetismic triads, consisting of three pairs of chords in inverse relationship: 1-7/5-12/7 with steps 7/5-11/9-7/6 and 1-7/6-10/7 with steps 7/6-11/9-7/5; 1-7/6-9/7 with steps 7/6-11/10-14/9 and 1-14/9-12/7 with steps 14/9-11/10-7/6; and 1-9/7-7/5 with steps 9/7-12/11-10/7 and 1-10/7-14/9 with steps 10/7-12/11-9/7.&lt;br /&gt;
+&lt;br /&gt;
+There are fifteen swetismic tetrads, consisting of three palindromic (self-inverse) chords and six pairs of chords in inverse relationship. Then palindromic tetrads are 1-7/6-9/7-3/2 with steps 7/6-11/10-7/6-4/3; 1-6/5-7/5-12/7 with steps 6/5-7/6-11/9-7/6 and 1-9/7-7/5-9/5 with steps 9/7-12/11-9/7-10/9. The six pairs are 1-9/7-3/2-11/6 with steps 9/7-7/6-11/9-12/11 and 1-7/6-3/2-18/11 with steps 7/6-9/7-12/11-11/9; 1-7/6-9/7-10/7 with steps 7/6-11/10-10/9-7/5 and 1-7/5-14/9-12/7 with steps 7/5-10/9-11/10-7/6; 1-7/6-9/7-18/11 with steps 7/6-11/10-14/11-11/9 and 1-14/11-7/5-18/11 with steps 14/11-11/10-7/6-11/9; 1-7/6-10/7-11/6 with steps 7/6-11/9-9/7-12/11 and 1-9/7-11/7-11/6 with steps 9/7-11/9-7/6-12/11; 1-7/6-9/7-11/6 with steps 7/6-11/10-10/7-12/11 and 1-10/7-11/7-11/6 with steps 10/7-11/10-7/6-12/11; and 1-10/7-14/9-12/7 with steps 10/7-12/11-11/10-7/6 and 1-7/6-9/7-7/5 with steps 7/6-11/10-12/11-10/7.&lt;br /&gt;
+&lt;br /&gt;
+Finally, there are six swetismic pentads coming in three pairs: 1-7/6-9/7-3/2-11/6 with steps 7/6-11/10-7/6-11/9-12/11 and 1-7/6-9/7-3/2-18/11 with steps 7/6-11/10-7/6-12/11-11/9; 1-7/6-10/7-5/3-11/6 with steps 7/6-11/9-7/6-11/10-12/11 and 1-7/6-10/7-5/3-20/11 with steps 7/6-11/9-7/6-12/11-11/10; and 1-7/6-9/7-10/7-11/6 with steps 7/6-11/10-10/9-9/7-12/11 and 1-9/7-10/7-11/7-11/6 with steps 9/7-10/9-11/10-7/6-12/11.&lt;br /&gt;
+&lt;br /&gt;
+If we are willing to consider the 15-limit, there are also 15-limit swetismic tetrads of 1-9/7-3/2-7/4, 1-7/6-3/2-12/7, 1-11/9-10/7-5/3 and 1-7/6-15/11-5/3 with steps 9/7-7/6-7/6-8/7, 7/6-9/7-8/7-7/6, 11/9-7/6-7/6-6/5 and 7/6-7/6-11/9-6/5.&lt;br /&gt;
 &lt;br /&gt;
-Equal temperaments with swetismic tetrads include 19, 22, 31, 41, 53, 58, 72, 80, 94, 103, 111, 121, 130, 152, 183, 224, 354, 537 and 578.&lt;/body&gt;&lt;/html&gt;</pre></div>
+Equal temperaments with swetismic chords include 19, 22, 31, 41, 53, 58, 72, 80, 94, 103, 111, 121, 130, 152, 183, 224, 354, 537, 578, 761d, 1115de, 1339de, 1491de, 1715de and 1845de.&lt;/body&gt;&lt;/html&gt;</pre></div>

Swetismic chords: Difference between revisions

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