31edo: Difference between revisions
Wikispaces>guest **Imported revision 315396508 - Original comment: ** |
Wikispaces>guest **Imported revision 315405798 - 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:guest|guest]] and made on <tt>2012-03-28 | : This revision was by author [[User:guest|guest]] and made on <tt>2012-03-28 09:21:01 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>315405798</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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The large tritone. | The large tritone. | ||
=Harmonic Scale= | |||
31edo approximates Mode 8 of the [[OverToneSeries|harmonic series]] O.K., but many intervals between the harmonics aren't distinguished, most importantly 9/8 (major tone) and 10/9 (minor tone), as 31EDO is a meantone temperament. The interval between the 8th and 11th harmonics is approximated O.K., but the intervals between the 11th harmonic and closer harmonics such as the 12th and 9th harmonics are approximated even better. 13/8 isn't distinguished from the 11-limit 18/11, as 144/143 is tempered out, so 13-limit intervals can't be distinguished from 11-limit ones. | |||
|| Overtones in "Mode 8": || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16 || | |||
|| ...as JI Ratio from 1/1: || 1/1 || 9/8 || 5/4 || 11/8 || 3/2 || 13/8 || 7/4 || 15/8 || 2/1 || | |||
|| ...in cents: || 0 || 203.9 || 386.3 || 551.3 || 702.0 || 840.5 || 968.8 || 1088.3 || 1200.0 || | |||
|| Nearest degree of 41edo: || 0 || 5 || 10 || 14 || 18 || 22 || 25 || 28 || 31 || | |||
|| ...in cents: || 0 || 193.5 || 387.1 || 541.9 || 696.8 || 851.6 || 967.7 || 1083.9 || 1200.0 || | |||
=Commas= | =Commas= | ||
31 EDO tempers out the following commas. (Note: This assumes the val < 31 49 72 87 107 115 |, comma values roundet to 5 significant digits.) | 31 EDO tempers out the following commas. (Note: This assumes the val < 31 49 72 87 107 115 |, comma values roundet to 5 significant digits.) | ||
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* <span class="wiki_link_ext">[[http://tonalsoft.com/enc/number/31edo.aspx|Tonalsoft Encyclopedia article]]</span> [[http://www.webcitation.org/5xeGYj7QU|Permalink]]</pre></div> | * <span class="wiki_link_ext">[[http://tonalsoft.com/enc/number/31edo.aspx|Tonalsoft Encyclopedia article]]</span> [[http://www.webcitation.org/5xeGYj7QU|Permalink]]</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>31edo</title></head><body><!-- ws:start:WikiTextTocRule: | <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>31edo</title></head><body><!-- ws:start:WikiTextTocRule:80:&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:80 --><!-- ws:start:WikiTextTocRule:81: --><a href="#Quick index to intervals and linear temperaments">Quick index to intervals and linear temperaments</a><!-- ws:end:WikiTextTocRule:81 --><!-- ws:start:WikiTextTocRule:82: --> | <a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:82 --><!-- ws:start:WikiTextTocRule:83: --><!-- ws:end:WikiTextTocRule:83 --><!-- ws:start:WikiTextTocRule:84: --><!-- ws:end:WikiTextTocRule:84 --><!-- ws:start:WikiTextTocRule:85: --><!-- ws:end:WikiTextTocRule:85 --><!-- ws:start:WikiTextTocRule:86: --><!-- ws:end:WikiTextTocRule:86 --><!-- ws:start:WikiTextTocRule:87: --><!-- ws:end:WikiTextTocRule:87 --><!-- ws:start:WikiTextTocRule:88: --><!-- ws:end:WikiTextTocRule:88 --><!-- ws:start:WikiTextTocRule:89: --><!-- ws:end:WikiTextTocRule:89 --><!-- ws:start:WikiTextTocRule:90: --><!-- ws:end:WikiTextTocRule:90 --><!-- ws:start:WikiTextTocRule:91: --><!-- ws:end:WikiTextTocRule:91 --><!-- ws:start:WikiTextTocRule:92: --><!-- ws:end:WikiTextTocRule:92 --><!-- ws:start:WikiTextTocRule:93: --><!-- ws:end:WikiTextTocRule:93 --><!-- ws:start:WikiTextTocRule:94: --><!-- ws:end:WikiTextTocRule:94 --><!-- ws:start:WikiTextTocRule:95: --><!-- ws:end:WikiTextTocRule:95 --><!-- ws:start:WikiTextTocRule:96: --><!-- ws:end:WikiTextTocRule:96 --><!-- ws:start:WikiTextTocRule:97: --><!-- ws:end:WikiTextTocRule:97 --><!-- ws:start:WikiTextTocRule:98: --><!-- ws:end:WikiTextTocRule:98 --><!-- ws:start:WikiTextTocRule:99: --><!-- ws:end:WikiTextTocRule:99 --><!-- ws:start:WikiTextTocRule:100: --><!-- ws:end:WikiTextTocRule:100 --><!-- ws:start:WikiTextTocRule:101: --><!-- ws:end:WikiTextTocRule:101 --><!-- ws:start:WikiTextTocRule:102: --><!-- ws:end:WikiTextTocRule:102 --><!-- ws:start:WikiTextTocRule:103: --><!-- ws:end:WikiTextTocRule:103 --><!-- ws:start:WikiTextTocRule:104: --><!-- ws:end:WikiTextTocRule:104 --><!-- ws:start:WikiTextTocRule:105: --><!-- ws:end:WikiTextTocRule:105 --><!-- ws:start:WikiTextTocRule:106: --><!-- ws:end:WikiTextTocRule:106 --><!-- ws:start:WikiTextTocRule:107: --><!-- ws:end:WikiTextTocRule:107 --><!-- ws:start:WikiTextTocRule:108: --><!-- ws:end:WikiTextTocRule:108 --><!-- ws:start:WikiTextTocRule:109: --><!-- ws:end:WikiTextTocRule:109 --><!-- ws:start:WikiTextTocRule:110: --><!-- ws:end:WikiTextTocRule:110 --><!-- ws:start:WikiTextTocRule:111: --><!-- ws:end:WikiTextTocRule:111 --><!-- ws:start:WikiTextTocRule:112: --><!-- ws:end:WikiTextTocRule:112 --><!-- ws:start:WikiTextTocRule:113: --> | <a href="#Harmonic Scale">Harmonic Scale</a><!-- ws:end:WikiTextTocRule:113 --><!-- ws:start:WikiTextTocRule:114: --> | <a href="#Commas">Commas</a><!-- ws:end:WikiTextTocRule:114 --><!-- ws:start:WikiTextTocRule:115: --> | <a href="#Modes">Modes</a><!-- ws:end:WikiTextTocRule:115 --><!-- ws:start:WikiTextTocRule:116: --><!-- ws:end:WikiTextTocRule:116 --><!-- ws:start:WikiTextTocRule:117: --> | <a href="#Music in 31-edo">Music in 31-edo</a><!-- ws:end:WikiTextTocRule:117 --><!-- ws:start:WikiTextTocRule:118: --><!-- ws:end:WikiTextTocRule:118 --><!-- ws:start:WikiTextTocRule:119: --> | <a href="#Practical Theory / Books">Practical Theory / Books</a><!-- ws:end:WikiTextTocRule:119 --><!-- ws:start:WikiTextTocRule:120: --> | <a href="#Other Articles">Other Articles</a><!-- ws:end:WikiTextTocRule:120 --><!-- ws:start:WikiTextTocRule:121: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:121 --><hr /> | ||
<em>Thirty-one tone equal temperament</em>, also called <em>31-tET</em>, <em>31-EDO</em>, <em>31-et</em>, or <em>tricesimoprimal meantone temperament</em>, is the scale derived by dividing the octave into 31 <a class="wiki_link" href="/equal">equally</a> large steps. The term 'Tricesimoprimal' was first used by <a class="wiki_link" href="/Adriaan%20Fokker">Adriaan Fokker</a>. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 <a class="wiki_link" href="/cents">cents</a>. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists">zeta integral edo</a>. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course). It's not bad for an 11-limit approximation, as it is consistent through it, but is the <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a> for the rank five temperament tempering out the 13-limit comma 66/65. It also provides the optimal patent val for mohajira, squares and casablance in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.<br /> | <em>Thirty-one tone equal temperament</em>, also called <em>31-tET</em>, <em>31-EDO</em>, <em>31-et</em>, or <em>tricesimoprimal meantone temperament</em>, is the scale derived by dividing the octave into 31 <a class="wiki_link" href="/equal">equally</a> large steps. The term 'Tricesimoprimal' was first used by <a class="wiki_link" href="/Adriaan%20Fokker">Adriaan Fokker</a>. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 <a class="wiki_link" href="/cents">cents</a>. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists">zeta integral edo</a>. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course). It's not bad for an 11-limit approximation, as it is consistent through it, but is the <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a> for the rank five temperament tempering out the 13-limit comma 66/65. It also provides the optimal patent val for mohajira, squares and casablance in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.<br /> | ||
<br /> | <br /> | ||
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The large tritone.<br /> | The large tritone.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:64:&lt;h1&gt; --><h1 id="toc32"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:64:&lt;h1&gt; --><h1 id="toc32"><a name="Harmonic Scale"></a><!-- ws:end:WikiTextHeadingRule:64 -->Harmonic Scale</h1> | ||
31edo approximates Mode 8 of the <a class="wiki_link" href="/OverToneSeries">harmonic series</a> O.K., but many intervals between the harmonics aren't distinguished, most importantly 9/8 (major tone) and 10/9 (minor tone), as 31EDO is a meantone temperament. The interval between the 8th and 11th harmonics is approximated O.K., but the intervals between the 11th harmonic and closer harmonics such as the 12th and 9th harmonics are approximated even better. 13/8 isn't distinguished from the 11-limit 18/11, as 144/143 is tempered out, so 13-limit intervals can't be distinguished from 11-limit ones.<br /> | |||
<br /> | |||
<table class="wiki_table"> | |||
<tr> | |||
<td>Overtones in &quot;Mode 8&quot;:<br /> | |||
</td> | |||
<td>8<br /> | |||
</td> | |||
<td>9<br /> | |||
</td> | |||
<td>10<br /> | |||
</td> | |||
<td>11<br /> | |||
</td> | |||
<td>12<br /> | |||
</td> | |||
<td>13<br /> | |||
</td> | |||
<td>14<br /> | |||
</td> | |||
<td>15<br /> | |||
</td> | |||
<td>16<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>...as JI Ratio from 1/1:<br /> | |||
</td> | |||
<td>1/1<br /> | |||
</td> | |||
<td>9/8<br /> | |||
</td> | |||
<td>5/4<br /> | |||
</td> | |||
<td>11/8<br /> | |||
</td> | |||
<td>3/2<br /> | |||
</td> | |||
<td>13/8<br /> | |||
</td> | |||
<td>7/4<br /> | |||
</td> | |||
<td>15/8<br /> | |||
</td> | |||
<td>2/1<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>...in cents:<br /> | |||
</td> | |||
<td>0<br /> | |||
</td> | |||
<td>203.9<br /> | |||
</td> | |||
<td>386.3<br /> | |||
</td> | |||
<td>551.3<br /> | |||
</td> | |||
<td>702.0<br /> | |||
</td> | |||
<td>840.5<br /> | |||
</td> | |||
<td>968.8<br /> | |||
</td> | |||
<td>1088.3<br /> | |||
</td> | |||
<td>1200.0<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>Nearest degree of 41edo:<br /> | |||
</td> | |||
<td>0<br /> | |||
</td> | |||
<td>5<br /> | |||
</td> | |||
<td>10<br /> | |||
</td> | |||
<td>14<br /> | |||
</td> | |||
<td>18<br /> | |||
</td> | |||
<td>22<br /> | |||
</td> | |||
<td>25<br /> | |||
</td> | |||
<td>28<br /> | |||
</td> | |||
<td>31<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>...in cents:<br /> | |||
</td> | |||
<td>0<br /> | |||
</td> | |||
<td>193.5<br /> | |||
</td> | |||
<td>387.1<br /> | |||
</td> | |||
<td>541.9<br /> | |||
</td> | |||
<td>696.8<br /> | |||
</td> | |||
<td>851.6<br /> | |||
</td> | |||
<td>967.7<br /> | |||
</td> | |||
<td>1083.9<br /> | |||
</td> | |||
<td>1200.0<br /> | |||
</td> | |||
</tr> | |||
</table> | |||
<!-- ws:start:WikiTextHeadingRule:66:&lt;h1&gt; --><h1 id="toc33"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:66 -->Commas</h1> | |||
31 EDO tempers out the following commas. (Note: This assumes the val &lt; 31 49 72 87 107 115 |, comma values roundet to 5 significant digits.)<br /> | 31 EDO tempers out the following commas. (Note: This assumes the val &lt; 31 49 72 87 107 115 |, comma values roundet to 5 significant digits.)<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:68:&lt;h1&gt; --><h1 id="toc34"><a name="Modes"></a><!-- ws:end:WikiTextHeadingRule:68 -->Modes</h1> | ||
<br /> | <br /> | ||
A large open list of modes (subsets) from 31edo that people have named: <a class="wiki_link" href="/31edo%20modes">31edo modes</a>. <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rothenberg_propriety" rel="nofollow">Strictly proper</a> <a class="wiki_link" href="/Strictly%20proper%207-note%2031edo%20scales">7-note 31edo scales</a> in the sense of <a class="wiki_link" href="/David%20Rothenberg">David Rothenberg</a>. See also <a class="wiki_link" href="/31edo%20MOS%20scales">31edo MOS scales</a>. Some of the popular ones:<br /> | A large open list of modes (subsets) from 31edo that people have named: <a class="wiki_link" href="/31edo%20modes">31edo modes</a>. <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rothenberg_propriety" rel="nofollow">Strictly proper</a> <a class="wiki_link" href="/Strictly%20proper%207-note%2031edo%20scales">7-note 31edo scales</a> in the sense of <a class="wiki_link" href="/David%20Rothenberg">David Rothenberg</a>. See also <a class="wiki_link" href="/31edo%20MOS%20scales">31edo MOS scales</a>. Some of the popular ones:<br /> | ||
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<table class="wiki_table"> | <table class="wiki_table"> | ||
<tr> | <tr> | ||
<td colspan="2"><!-- ws:start:WikiTextHeadingRule: | <td colspan="2"><!-- ws:start:WikiTextHeadingRule:70:&lt;h4&gt; --><h4 id="toc35"><a name="Modes---Some 31 tone equal modes:"></a><!-- ws:end:WikiTextHeadingRule:70 -->Some 31 tone equal modes:</h4> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:72:&lt;h1&gt; --><h1 id="toc36"><a name="Music in 31-edo"></a><!-- ws:end:WikiTextHeadingRule:72 -->Music in 31-edo</h1> | ||
<a class="wiki_link" href="/31-edo%20compositions">An alphabetical list of Tricesimoprimal Compositions</a>.<br /> | <a class="wiki_link" href="/31-edo%20compositions">An alphabetical list of Tricesimoprimal Compositions</a>.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:74:&lt;h2&gt; --><h2 id="toc37"><a name="Music in 31-edo-Thirty-one tone pedagogy"></a><!-- ws:end:WikiTextHeadingRule:74 -->Thirty-one tone pedagogy</h2> | ||
The <a class="wiki_link" href="/MicroPedagogyCollective">MicroPedagogyCollective</a> is currently at work producing demonstrative material which will encourage and enable more people to learn this system. There have been two <a class="wiki_link" href="/ThirtyOneToneSinginCamp">ThirtyOneToneSinginCamp</a>s as well.<br /> | The <a class="wiki_link" href="/MicroPedagogyCollective">MicroPedagogyCollective</a> is currently at work producing demonstrative material which will encourage and enable more people to learn this system. There have been two <a class="wiki_link" href="/ThirtyOneToneSinginCamp">ThirtyOneToneSinginCamp</a>s as well.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:76:&lt;h1&gt; --><h1 id="toc38"><a name="Practical Theory / Books"></a><!-- ws:end:WikiTextHeadingRule:76 -->Practical Theory / Books</h1> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextRemoteImageRule: | <!-- ws:start:WikiTextRemoteImageRule:7938:&lt;a href=&quot;http://www.ronsword.com/books.html&quot; target=&quot;_blank&quot; rel=&quot;nofollow&quot;&gt;&lt;img src=&quot;http://ronsword.com/images/TSG_sm.jpg&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 116px; width: 87px;&quot; /&gt;&lt;/a&gt; --><a href="http://www.ronsword.com/books.html" target="_blank" rel="nofollow"><img src="http://ronsword.com/images/TSG_sm.jpg" alt="external image TSG_sm.jpg" title="external image TSG_sm.jpg" style="height: 116px; width: 87px;" /></a><!-- ws:end:WikiTextRemoteImageRule:7938 --><a class="wiki_link_ext" href="http://www.ronsword.com/books.html" rel="nofollow" target="_blank">Sword, Ronald. &quot;Tricesimoprimal Scales for Guitar.&quot; IAAA Press, UK-USA. First Ed: March 2009.</a> - A comprehensive approach to 31-EDO and all the families associated for Guitar. Features over 300 scale charts / scale examples.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:78:&lt;h1&gt; --><h1 id="toc39"><a name="Other Articles"></a><!-- ws:end:WikiTextHeadingRule:78 -->Other Articles</h1> | ||
<ul><li><span class="wiki_link_ext"><a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/beerart.html" rel="nofollow">de Beer, Anton, ''The Development of 31-tone Music</a></span> <a class="wiki_link_ext" href="http://www.webcitation.org/5xeFzBM9b" rel="nofollow">Permalink</a></li><li><span class="wiki_link_ext"><a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/fokkerorg.html" rel="nofollow">Fokker, Adriaan Daniël, ''Equal Temperament and the Thirty-one-keyed organ</a></span> <a class="wiki_link_ext" href="http://www.webcitation.org/5xeG6Tmli" rel="nofollow">Permalink</a></li><li>Fokker, A.D., &quot;New Music with 31 Notes&quot; translated by Leigh Gerdine</li><li><span class="wiki_link_ext"><a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/rap31.html" rel="nofollow">Rapoport, Paul, ''About 31-tone Equal Temperament</a></span> <a class="wiki_link_ext" href="http://www.webcitation.org/5xeGH4uBH" rel="nofollow">Permalink</a></li><li><span class="wiki_link_ext"><a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/terp31.html" rel="nofollow">Terpstra, Siemen, ''Toward a Theory of Meantone (and 31-et) Harmony''</a></span> <a class="wiki_link_ext" href="http://www.webcitation.org/5xeGMeCMd" rel="nofollow">Permalink</a></li><li><span class="wiki_link_ext"><a class="wiki_link_ext" href="http://tonalsoft.com/enc/number/31edo.aspx" rel="nofollow">Tonalsoft Encyclopedia article</a></span> <a class="wiki_link_ext" href="http://www.webcitation.org/5xeGYj7QU" rel="nofollow">Permalink</a></li></ul></body></html></pre></div> | <ul><li><span class="wiki_link_ext"><a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/beerart.html" rel="nofollow">de Beer, Anton, ''The Development of 31-tone Music</a></span> <a class="wiki_link_ext" href="http://www.webcitation.org/5xeFzBM9b" rel="nofollow">Permalink</a></li><li><span class="wiki_link_ext"><a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/fokkerorg.html" rel="nofollow">Fokker, Adriaan Daniël, ''Equal Temperament and the Thirty-one-keyed organ</a></span> <a class="wiki_link_ext" href="http://www.webcitation.org/5xeG6Tmli" rel="nofollow">Permalink</a></li><li>Fokker, A.D., &quot;New Music with 31 Notes&quot; translated by Leigh Gerdine</li><li><span class="wiki_link_ext"><a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/rap31.html" rel="nofollow">Rapoport, Paul, ''About 31-tone Equal Temperament</a></span> <a class="wiki_link_ext" href="http://www.webcitation.org/5xeGH4uBH" rel="nofollow">Permalink</a></li><li><span class="wiki_link_ext"><a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/terp31.html" rel="nofollow">Terpstra, Siemen, ''Toward a Theory of Meantone (and 31-et) Harmony''</a></span> <a class="wiki_link_ext" href="http://www.webcitation.org/5xeGMeCMd" rel="nofollow">Permalink</a></li><li><span class="wiki_link_ext"><a class="wiki_link_ext" href="http://tonalsoft.com/enc/number/31edo.aspx" rel="nofollow">Tonalsoft Encyclopedia article</a></span> <a class="wiki_link_ext" href="http://www.webcitation.org/5xeGYj7QU" rel="nofollow">Permalink</a></li></ul></body></html></pre></div> | ||