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31st-octave temperaments: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
__FORCETOC__
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>2011-12-01 18:23:43 UTC</tt>.<br>
: The original revision id was <tt>281304484</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">[[toc|flat]]
 
 
=31-3-comma=
=31-3-comma=
By the //31-3-comma// is meant 617673396283947/562949953421312 = &lt;-49 31|, the amount (160.605 cents) by which 31 just (3/2) fifths exceed 18 octaves. This may not seem like much of a comma, but since 31et is such a strong 7-limit system, 11- and 13-limit temperaments based on the cycle of 31 fifths actually make sense.
By the ''31-3-comma'' is meant 617673396283947/562949953421312 = &lt;-49 31|, the amount (160.605 cents) by which 31 just (3/2) fifths exceed 18 octaves. This may not seem like much of a comma, but since 31et is such a strong 7-limit system, 11- and 13-limit temperaments based on the cycle of 31 fifths actually make sense.


=Prajapati=
=Prajapati=
Line 20: Line 11:


Map: [[&lt;31 49 72 87 107|, &lt;0 0 0 0 1|]
Map: [[&lt;31 49 72 87 107|, &lt;0 0 0 0 1|]
EDOs: 31, 93, 124b, 155b, 186b
EDOs: 31, 93, 124b, 155b, 186b
Badness: 0.0430
Badness: 0.0430


Line 29: Line 22:


Map: [[&lt;31 49 72 87 107 115|, &lt;0 0 0 0 1 0|]
Map: [[&lt;31 49 72 87 107 115|, &lt;0 0 0 0 1 0|]
EDOs: 31, 93f, 124bf
EDOs: 31, 93f, 124bf
Badness: 0.0379
Badness: 0.0379


Line 38: Line 33:


Map: [[&lt;31 49 72 87 107 115|, &lt;0 0 0 0 1 -1|]
Map: [[&lt;31 49 72 87 107 115|, &lt;0 0 0 0 1 -1|]
EDOs: 31, 62e, 93, 124b, 341b
EDOs: 31, 62e, 93, 124b, 341b
Badness: 0.0486
Badness: 0.0486


Line 49: Line 46:


Map: [&lt;31 49 72 87 107 0|, &lt;0 0 0 0 0 1|]
Map: [&lt;31 49 72 87 107 0|, &lt;0 0 0 0 0 1|]
EDOs: 31, 62, 93e, 155bef
EDOs: 31, 62, 93e, 155bef
Badness: 0.0255
Badness: 0.0255


=Birds=
=Birds=
The birds temperament tempers out the //31-5 comma//, |72 0 -31&gt;, which is the amount (24.275 cents) by which 31 just (5/4) major thirds fall short of ten octaves. The name comes from Isaiah 31:5 "As birds flying, so wil the Lord of hostes defend Ierusalem, defending also hee will deliuer it, and passing ouer, he will preserue it." It also tempers out the 31-7 comma, but sadly, combining the two commas leads to torsion.
The birds temperament tempers out the ''31-5 comma'', |72 0 -31&gt;, which is the amount (24.275 cents) by which 31 just (5/4) major thirds fall short of ten octaves. The name comes from Isaiah 31:5 "As birds flying, so wil the Lord of hostes defend Ierusalem, defending also hee will deliuer it, and passing ouer, he will preserue it." It also tempers out the 31-7 comma, but sadly, combining the two commas leads to torsion.


Commas: 3136/3125, 823543/819200
Commas: 3136/3125, 823543/819200
Line 60: Line 59:


Map: [&lt;31 49 72 87|, &lt;0 1 0 0|]
Map: [&lt;31 49 72 87|, &lt;0 1 0 0|]
Wedgie: &lt;&lt;31 0 0 -72 -87 0||
Wedgie: &lt;&lt;31 0 0 -72 -87 0||
Edos: 31, 124, 155, 186, 217, 248, 465
Edos: 31, 124, 155, 186, 217, 248, 465
Badness: 0.1000
Badness: 0.1000


Line 70: Line 72:


Map: [&lt;31 49 72 87 107|, &lt;0 1 0 0 2|]
Map: [&lt;31 49 72 87 107|, &lt;0 1 0 0 2|]
EDOs: 31, 186e, 217, 248, 961cd
EDOs: 31, 186e, 217, 248, 961cd
Badness: 0.0399
Badness: 0.0399


Line 79: Line 83:


Map: [&lt;31 49 72 87 107 115|, &lt;0 1 0 0 2 -2|]
Map: [&lt;31 49 72 87 107 115|, &lt;0 1 0 0 2 -2|]
EDOs: 31, 186e, 217, 248, 465
EDOs: 31, 186e, 217, 248, 465
Badness: 0.0357
Badness: 0.0357
</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;31 comma temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:16:&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:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt;&lt;a href="#x31-3-comma"&gt;31-3-comma&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt; | &lt;a href="#Prajapati"&gt;Prajapati&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt; | &lt;a href="#Gallium"&gt;Gallium&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt; | &lt;a href="#Birds"&gt;Birds&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt;
&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x31-3-comma"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;31-3-comma&lt;/h1&gt;
By the &lt;em&gt;31-3-comma&lt;/em&gt; is meant 617673396283947/562949953421312 = &amp;lt;-49 31|, the amount (160.605 cents) by which 31 just (3/2) fifths exceed 18 octaves. This may not seem like much of a comma, but since 31et is such a strong 7-limit system, 11- and 13-limit temperaments based on the cycle of 31 fifths actually make sense.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Prajapati"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Prajapati&lt;/h1&gt;
The Hindu  god Pradjapati is said to have created the universe by speaking aloud the odd numbers from 1 to 31. People with an interest in 31 may want to try this method themselves.&lt;br /&gt;
&lt;br /&gt;
Commas: 81/80, 126/125, 1029/1024&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~176/175 = 6.519&lt;br /&gt;
&lt;br /&gt;
Map: [[&amp;lt;31 49 72 87 107|, &amp;lt;0 0 0 0 1|]&lt;br /&gt;
EDOs: 31, 93, 124b, 155b, 186b&lt;br /&gt;
Badness: 0.0430&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Prajapati-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;13-limit&lt;/h2&gt;
Commas: 81/80, 126/125, 105/104, 512/507&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~66/65 = 9.171&lt;br /&gt;
&lt;br /&gt;
Map: [[&amp;lt;31 49 72 87 107 115|, &amp;lt;0 0 0 0 1 0|]&lt;br /&gt;
EDOs: 31, 93f, 124bf&lt;br /&gt;
Badness: 0.0379&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Prajapati-Kumhar"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Kumhar&lt;/h2&gt;
Commas: 81/80, 126/125, 1029/1024, 144/143&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~196/195 = 10.120&lt;br /&gt;
&lt;br /&gt;
Map: [[&amp;lt;31 49 72 87 107 115|, &amp;lt;0 0 0 0 1 -1|]&lt;br /&gt;
EDOs: 31, 62e, 93, 124b, 341b&lt;br /&gt;
Badness: 0.0486&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Gallium"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Gallium&lt;/h1&gt;
The name of gallium temperament comes from Gallium, the 31st element.&lt;br /&gt;
&lt;br /&gt;
Commas: 81/80, 126/125, 99/98, 121/120&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~16807/16640 = 15.541&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;31 49 72 87 107 0|, &amp;lt;0 0 0 0 0 1|]&lt;br /&gt;
EDOs: 31, 62, 93e, 155bef&lt;br /&gt;
Badness: 0.0255&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Birds"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Birds&lt;/h1&gt;
The birds temperament tempers out the &lt;em&gt;31-5 comma&lt;/em&gt;, |72 0 -31&amp;gt;, which is the amount (24.275 cents) by which 31 just (5/4) major thirds fall short of ten octaves. The name comes from Isaiah 31:5 &amp;quot;As birds flying, so wil the Lord of hostes defend Ierusalem, defending also hee will deliuer it, and passing ouer, he will preserue it.&amp;quot; It also tempers out the 31-7 comma, but sadly, combining the two commas leads to torsion.&lt;br /&gt;
&lt;br /&gt;
Commas: 3136/3125, 823543/819200&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~1029/1024 = 5.155&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;31 49 72 87|, &amp;lt;0 1 0 0|]&lt;br /&gt;
Wedgie: &amp;lt;&amp;lt;31 0 0 -72 -87 0||&lt;br /&gt;
Edos: 31, 124, 155, 186, 217, 248, 465&lt;br /&gt;
Badness: 0.1000&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Birds-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;11-limit&lt;/h2&gt;
Commas: 441/440, 3136/3125, 41503/41472&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~385/384 = 4.938&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;31 49 72 87 107|, &amp;lt;0 1 0 0 2|]&lt;br /&gt;
EDOs: 31, 186e, 217, 248, 961cd&lt;br /&gt;
Badness: 0.0399&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Birds-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;13-limit&lt;/h2&gt;
Commas: 441/440, 3136/3125, 1001/1000, 13720/13689&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~385/384 = 5.170&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;31 49 72 87 107 115|, &amp;lt;0 1 0 0 2 -2|]&lt;br /&gt;
EDOs: 31, 186e, 217, 248, 465&lt;br /&gt;
Badness: 0.0357&lt;/body&gt;&lt;/html&gt;</pre></div>
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