31edo: Difference between revisions
Wikispaces>csvargas31 **Imported revision 485217024 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 491179802 - 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: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-02-22 03:56:52 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>491179802</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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|| 15\31 || 580.65 || [[Tritonic]]/[[Tritoni]] || | || 15\31 || 580.65 || [[Tritonic]]/[[Tritoni]] || | ||
=Intervals= | =Intervals= | ||
==Intervals by error== | |||
|| Interval || Error || | |||
|| 5/4 || 0.783 || | |||
|| 11/9 || 0.979 || | |||
|| 8/7 || 1.084 || | |||
|| 7/5 || -1.867 || | |||
|| 15/14 || -3.314 || | |||
|| 7/6 || 4.097 || | |||
|| 12/11 || 4.202 || | |||
|| 16/15 || 4.398 || | |||
|| 15/11 || 4.985 || | |||
|| 4/3 || 5.181 || | |||
|| 6/5 || -5.964 || | |||
|| 14/11 || 8.298 || | |||
|| 9/7 || -9.278 || | |||
|| 11/8 || -9.382 || | |||
|| 11/10 || -10.166 || | |||
|| 13/10 || 10.302 || | |||
|| 9/8 || -10.362 || | |||
|| 16/13 || -11.085 || | |||
|| 10/9 || 11.145 || | |||
|| 14/13 || -12.169 || | |||
|| 15/13 || -15.483 || | |||
|| 13/12 || 16.266 || | |||
|| 13/11 || 20.468 || | |||
|| 18/13 || -21.447 || | |||
===1\31 octave - approx. 38.71¢ - Diesis=== | ===1\31 octave - approx. 38.71¢ - Diesis=== | ||
A single step of 31-edo is about 38.71¢. Intervals around this size are called //dieses// (singular '//diesis//'). In 31 it is equivalent to the difference between one octave and three stacked major thirds (C to E, to G#, to B#, but B# ≠ C), or four minor thirds (C to Eb to Gb to Bbb to Dbb ≠ C). In the [[11-limit]], the diesis stands in for just ratios 56:55 (31.19); 55:54 (31.77¢); 49:48 (39.70¢); 45:44 (38.91¢); 36:35 (48.77¢); 33:32 (53.27¢) and others. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. Demonstrated in [[SpiralProgressions]]. | A single step of 31-edo is about 38.71¢. Intervals around this size are called //dieses// (singular '//diesis//'). In 31 it is equivalent to the difference between one octave and three stacked major thirds (C to E, to G#, to B#, but B# ≠ C), or four minor thirds (C to Eb to Gb to Bbb to Dbb ≠ C). In the [[11-limit]], the diesis stands in for just ratios 56:55 (31.19); 55:54 (31.77¢); 49:48 (39.70¢); 45:44 (38.91¢); 36:35 (48.77¢); 33:32 (53.27¢) and others. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. Demonstrated in [[SpiralProgressions]]. | ||
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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:84:&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:84 --><!-- ws:start:WikiTextTocRule:85: --><a href="#Linear temperaments">Linear temperaments</a><!-- ws:end:WikiTextTocRule:85 --><!-- ws:start:WikiTextTocRule:86: --> | <a href="#Intervals">Intervals</a><!-- 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: --><!-- ws:end:WikiTextTocRule:113 --><!-- ws:start:WikiTextTocRule:114: --><!-- ws:end:WikiTextTocRule:114 --><!-- ws:start:WikiTextTocRule:115: --><!-- ws:end:WikiTextTocRule:115 --><!-- ws:start:WikiTextTocRule:116: --><!-- ws:end:WikiTextTocRule:116 --><!-- ws:start:WikiTextTocRule:117: --><!-- ws:end:WikiTextTocRule:117 --><!-- ws:start:WikiTextTocRule:118: --> | <a href="#Harmonic Scale">Harmonic Scale</a><!-- ws:end:WikiTextTocRule:118 --><!-- ws:start:WikiTextTocRule:119: --> | <a href="#Commas">Commas</a><!-- ws:end:WikiTextTocRule:119 --><!-- ws:start:WikiTextTocRule:120: --> | <a href="#Modes">Modes</a><!-- ws:end:WikiTextTocRule:120 --><!-- ws:start:WikiTextTocRule:121: --><!-- ws:end:WikiTextTocRule:121 --><!-- ws:start:WikiTextTocRule:122: --> | <a href="#Music in 31-edo">Music in 31-edo</a><!-- ws:end:WikiTextTocRule:122 --><!-- ws:start:WikiTextTocRule:123: --><!-- ws:end:WikiTextTocRule:123 --><!-- ws:start:WikiTextTocRule:124: --><!-- ws:end:WikiTextTocRule:124 --><!-- ws:start:WikiTextTocRule:125: --> | <a href="#Practical Theory / Books">Practical Theory / Books</a><!-- ws:end:WikiTextTocRule:125 --><!-- ws:start:WikiTextTocRule:126: --> | <a href="#Other Articles">Other Articles</a><!-- ws:end:WikiTextTocRule:126 --><!-- ws:start:WikiTextTocRule:127: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:127 --><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 also deals with the 11-limit fairly well, and 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 casablanca 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 also deals with the 11-limit fairly well, and 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 casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.<br /> | ||
<br /> | <br /> | ||
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<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:2 -->Intervals</h1> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:2 -->Intervals</h1> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id=" | <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Intervals-Intervals by error"></a><!-- ws:end:WikiTextHeadingRule:4 -->Intervals by error</h2> | ||
<table class="wiki_table"> | |||
<tr> | |||
<td>Interval<br /> | |||
</td> | |||
<td>Error<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>5/4<br /> | |||
</td> | |||
<td>0.783<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>11/9<br /> | |||
</td> | |||
<td>0.979<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>8/7<br /> | |||
</td> | |||
<td>1.084<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>7/5<br /> | |||
</td> | |||
<td>-1.867<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>15/14<br /> | |||
</td> | |||
<td>-3.314<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>7/6<br /> | |||
</td> | |||
<td>4.097<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>12/11<br /> | |||
</td> | |||
<td>4.202<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>16/15<br /> | |||
</td> | |||
<td>4.398<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>15/11<br /> | |||
</td> | |||
<td>4.985<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>4/3<br /> | |||
</td> | |||
<td>5.181<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>6/5<br /> | |||
</td> | |||
<td>-5.964<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>14/11<br /> | |||
</td> | |||
<td>8.298<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>9/7<br /> | |||
</td> | |||
<td>-9.278<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>11/8<br /> | |||
</td> | |||
<td>-9.382<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>11/10<br /> | |||
</td> | |||
<td>-10.166<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>13/10<br /> | |||
</td> | |||
<td>10.302<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>9/8<br /> | |||
</td> | |||
<td>-10.362<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>16/13<br /> | |||
</td> | |||
<td>-11.085<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>10/9<br /> | |||
</td> | |||
<td>11.145<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>14/13<br /> | |||
</td> | |||
<td>-12.169<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>15/13<br /> | |||
</td> | |||
<td>-15.483<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>13/12<br /> | |||
</td> | |||
<td>16.266<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>13/11<br /> | |||
</td> | |||
<td>20.468<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>18/13<br /> | |||
</td> | |||
<td>-21.447<br /> | |||
</td> | |||
</tr> | |||
</table> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="Intervals-Intervals by error-1\31 octave - approx. 38.71¢ - Diesis"></a><!-- ws:end:WikiTextHeadingRule:6 -->1\31 octave - approx. 38.71¢ - Diesis</h3> | |||
A single step of 31-edo is about 38.71¢. Intervals around this size are called <em>dieses</em> (singular '<em>diesis</em>'). In 31 it is equivalent to the difference between one octave and three stacked major thirds (C to E, to G#, to B#, but B# ≠ C), or four minor thirds (C to Eb to Gb to Bbb to Dbb ≠ C). In the <a class="wiki_link" href="/11-limit">11-limit</a>, the diesis stands in for just ratios 56:55 (31.19); 55:54 (31.77¢); 49:48 (39.70¢); 45:44 (38.91¢); 36:35 (48.77¢); 33:32 (53.27¢) and others. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. Demonstrated in <a class="wiki_link" href="/SpiralProgressions">SpiralProgressions</a>.<br /> | A single step of 31-edo is about 38.71¢. Intervals around this size are called <em>dieses</em> (singular '<em>diesis</em>'). In 31 it is equivalent to the difference between one octave and three stacked major thirds (C to E, to G#, to B#, but B# ≠ C), or four minor thirds (C to Eb to Gb to Bbb to Dbb ≠ C). In the <a class="wiki_link" href="/11-limit">11-limit</a>, the diesis stands in for just ratios 56:55 (31.19); 55:54 (31.77¢); 49:48 (39.70¢); 45:44 (38.91¢); 36:35 (48.77¢); 33:32 (53.27¢) and others. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. Demonstrated in <a class="wiki_link" href="/SpiralProgressions">SpiralProgressions</a>.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="Intervals-Intervals by error-2\31 octave - approx. 77.42¢ - Minor Semitone or Chromatic Semitone or Small Minor Second"></a><!-- ws:end:WikiTextHeadingRule:8 -->2\31 octave - approx. 77.42¢ - Minor Semitone or Chromatic Semitone or Small Minor Second</h3> | ||
The difference between a major and minor third. The more 'expressive' of the 'half steps,' and the larger of 31's two &quot;microtones&quot;. In meantone, it is the <em>chromatic semitone</em>, the interval that distinguishes major and minor intervals of the same generic interval class (eg. thirds). 2\31 stands in for just ratios 28:27 (62.96¢); 25:24 (70.67¢); 22:21 (80.54¢); 21:20 (84.45¢) and others. Generates <a class="wiki_link" href="/Starling%20temperaments#Valentine%20temperament">valentine temperament</a> - aka <a class="wiki_link" href="/Armodue%20theory#Semi-equalized%20Armodue">semi-equalized Armodue</a>.<br /> | The difference between a major and minor third. The more 'expressive' of the 'half steps,' and the larger of 31's two &quot;microtones&quot;. In meantone, it is the <em>chromatic semitone</em>, the interval that distinguishes major and minor intervals of the same generic interval class (eg. thirds). 2\31 stands in for just ratios 28:27 (62.96¢); 25:24 (70.67¢); 22:21 (80.54¢); 21:20 (84.45¢) and others. Generates <a class="wiki_link" href="/Starling%20temperaments#Valentine%20temperament">valentine temperament</a> - aka <a class="wiki_link" href="/Armodue%20theory#Semi-equalized%20Armodue">semi-equalized Armodue</a>.<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h4&gt; --><h4 id="toc5"><a name="Intervals-Intervals by error-2\31 octave - approx. 77.42¢ - Minor Semitone or Chromatic Semitone or Small Minor Second-MOS Scales generated by 2\31:"></a><!-- ws:end:WikiTextHeadingRule:10 -->MOS Scales generated by 2\31:</h4> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="Intervals-Intervals by error-3\31 octave - approx. 116.13 - Major Semitone or Diatonic Semitone or Large Major Second"></a><!-- ws:end:WikiTextHeadingRule:12 -->3\31 octave - approx. 116.13 - Major Semitone or Diatonic Semitone or Large Major Second</h3> | ||
The larger and clunkier of the 31edo semitones. In meantone, it is the <em>diatonic semitone</em> which appears in the diatonic scale between, for instance, the major third and perfect fourth, and the major seventh and octave. 3\31 stands in for just ratios 16:15 (111.73¢); 15:14 (119.44¢) and others. It is notable that two of these make an 8/7; this implies that the 3\31 is a <em>secor</em> and generates <a class="wiki_link" href="/Gamelismic%20clan">miracle temperament</a>.<br /> | The larger and clunkier of the 31edo semitones. In meantone, it is the <em>diatonic semitone</em> which appears in the diatonic scale between, for instance, the major third and perfect fourth, and the major seventh and octave. 3\31 stands in for just ratios 16:15 (111.73¢); 15:14 (119.44¢) and others. It is notable that two of these make an 8/7; this implies that the 3\31 is a <em>secor</em> and generates <a class="wiki_link" href="/Gamelismic%20clan">miracle temperament</a>.<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h4&gt; --><h4 id="toc7"><a name="Intervals-Intervals by error-3\31 octave - approx. 116.13 - Major Semitone or Diatonic Semitone or Large Major Second-MOS Scales generated by 3\31:"></a><!-- ws:end:WikiTextHeadingRule:14 -->MOS Scales generated by 3\31:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="Intervals-Intervals by error-4\31 octave - approx. 154.84¢ - Neutral Tone or Neutral Second"></a><!-- ws:end:WikiTextHeadingRule:16 -->4\31 octave - approx. 154.84¢ - Neutral Tone or Neutral Second</h3> | ||
Exactly one half of the minor third and twice the minor semitone. 4\31 stands in for 12:11 (150.64¢); 35:32 (155.14¢); 11:10 (165.00¢) and others. Although neutral seconds are typically associated with the 11-limit, 4\31 approximates the <a class="wiki_link" href="/7-limit">7-limit</a> interval 35/32 quite well, as the 5th harmonic of the 7th harmonic or vice versa, both of which are closely approximated in 31edo. And although 31 is not extremely accurate in the 11-limit, it is notable that since 11 and 3 are both flat, the interval that distinguishes them (12/11) is only about 4.5¢ off. Generates <a class="wiki_link" href="/Starling%20temperaments">nusecond temperament</a>.<br /> | Exactly one half of the minor third and twice the minor semitone. 4\31 stands in for 12:11 (150.64¢); 35:32 (155.14¢); 11:10 (165.00¢) and others. Although neutral seconds are typically associated with the 11-limit, 4\31 approximates the <a class="wiki_link" href="/7-limit">7-limit</a> interval 35/32 quite well, as the 5th harmonic of the 7th harmonic or vice versa, both of which are closely approximated in 31edo. And although 31 is not extremely accurate in the 11-limit, it is notable that since 11 and 3 are both flat, the interval that distinguishes them (12/11) is only about 4.5¢ off. Generates <a class="wiki_link" href="/Starling%20temperaments">nusecond temperament</a>.<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h4&gt; --><h4 id="toc9"><a name="Intervals-Intervals by error-4\31 octave - approx. 154.84¢ - Neutral Tone or Neutral Second-MOS Scales generated by 4\31:"></a><!-- ws:end:WikiTextHeadingRule:18 -->MOS Scales generated by 4\31:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:20:&lt;h3&gt; --><h3 id="toc10"><a name="Intervals-Intervals by error-5\31 octave - approx. 193.55¢ - Whole Tone or Major Second"></a><!-- ws:end:WikiTextHeadingRule:20 -->5\31 octave - approx. 193.55¢ - Whole Tone or Major Second</h3> | ||
A rather smallish whole tone. Sometimes called melodically dull. As it falls between (and functions as) just whole tones 9:8 and 10:9, 5\31 is considered a &quot;meantone&quot;. Two meantones make a near-just major third. Perhaps it is worth noting that its relative narrowness (to JI 9/8) makes it easier to distinguish from the 8/7 approximation. And although it is over 10¢ flat of 9/8, 5\31 can function as a somewhat &quot;active&quot; (as opposed to perfectly stable) harmonic ninth, and it can be effective in combination with the also-narrow 11th harmonic. Indeed, the 11/9 approximation is excellent. Try, for instance 31's version of a 4:6:9:11 chord (steps 0-18-36-45). Generates <a class="wiki_link" href="/Gamelismic%20clan">hemithirds temperament</a> and <a class="wiki_link" href="/Wuerschmidt%20family">hermiwuerschmidt temperament</a>.<br /> | A rather smallish whole tone. Sometimes called melodically dull. As it falls between (and functions as) just whole tones 9:8 and 10:9, 5\31 is considered a &quot;meantone&quot;. Two meantones make a near-just major third. Perhaps it is worth noting that its relative narrowness (to JI 9/8) makes it easier to distinguish from the 8/7 approximation. And although it is over 10¢ flat of 9/8, 5\31 can function as a somewhat &quot;active&quot; (as opposed to perfectly stable) harmonic ninth, and it can be effective in combination with the also-narrow 11th harmonic. Indeed, the 11/9 approximation is excellent. Try, for instance 31's version of a 4:6:9:11 chord (steps 0-18-36-45). Generates <a class="wiki_link" href="/Gamelismic%20clan">hemithirds temperament</a> and <a class="wiki_link" href="/Wuerschmidt%20family">hermiwuerschmidt temperament</a>.<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:22:&lt;h4&gt; --><h4 id="toc11"><a name="Intervals-Intervals by error-5\31 octave - approx. 193.55¢ - Whole Tone or Major Second-MOS Scales generated by 5\31:"></a><!-- ws:end:WikiTextHeadingRule:22 -->MOS Scales generated by 5\31:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc12"><a name="Intervals-Intervals by error-6\31 octave - approx. 232.26¢ - Supermajor Second"></a><!-- ws:end:WikiTextHeadingRule:24 -->6\31 octave - approx. 232.26¢ - Supermajor Second</h3> | ||
Exactly one half of a narrow fourth, twice a major semitone, or thrice a minor semitone. In 7-limit tonal music, 6\31 closely represents 8:7 (231.17¢). In meantone, it is a diminished third, eg. C to Ebb. Generates <a class="wiki_link" href="/Meantone%20family">mothra temperament</a>.<br /> | Exactly one half of a narrow fourth, twice a major semitone, or thrice a minor semitone. In 7-limit tonal music, 6\31 closely represents 8:7 (231.17¢). In meantone, it is a diminished third, eg. C to Ebb. Generates <a class="wiki_link" href="/Meantone%20family">mothra temperament</a>.<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:26:&lt;h4&gt; --><h4 id="toc13"><a name="Intervals-Intervals by error-6\31 octave - approx. 232.26¢ - Supermajor Second-MOS Scales generated by 6\31:"></a><!-- ws:end:WikiTextHeadingRule:26 -->MOS Scales generated by 6\31:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:28:&lt;h3&gt; --><h3 id="toc14"><a name="Intervals-Intervals by error-7\31 octave - approx. 270.97¢ - Subminor Third"></a><!-- ws:end:WikiTextHeadingRule:28 -->7\31 octave - approx. 270.97¢ - Subminor Third</h3> | ||
Exactly one half of a superfourth (11:8 approximation). In 7-limit tonal music, 7\31 stands in for 7:6 (266.87¢). In meantone temperament, it is an augmented 2nd, eg. C to D#. Generates <a class="wiki_link" href="/Semicomma%20family">orwell temperament</a>.<br /> | Exactly one half of a superfourth (11:8 approximation). In 7-limit tonal music, 7\31 stands in for 7:6 (266.87¢). In meantone temperament, it is an augmented 2nd, eg. C to D#. Generates <a class="wiki_link" href="/Semicomma%20family">orwell temperament</a>.<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:30:&lt;h4&gt; --><h4 id="toc15"><a name="Intervals-Intervals by error-7\31 octave - approx. 270.97¢ - Subminor Third-MOS Scales generated by 7\31:"></a><!-- ws:end:WikiTextHeadingRule:30 -->MOS Scales generated by 7\31:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:32:&lt;h3&gt; --><h3 id="toc16"><a name="Intervals-Intervals by error-8\31 octave - approx. 309.68¢ - Minor Third"></a><!-- ws:end:WikiTextHeadingRule:32 -->8\31 octave - approx. 309.68¢ - Minor Third</h3> | ||
A minor third, closer to the just 6:5 (315.64¢) than 12-edo, but still on the flat side. Exactly twice a neutral second, four times a minor semitone, and half of a large tritone. Generates <a class="wiki_link" href="/Starling%20temperaments">myna temperament</a>.<br /> | A minor third, closer to the just 6:5 (315.64¢) than 12-edo, but still on the flat side. Exactly twice a neutral second, four times a minor semitone, and half of a large tritone. Generates <a class="wiki_link" href="/Starling%20temperaments">myna temperament</a>.<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:34:&lt;h4&gt; --><h4 id="toc17"><a name="Intervals-Intervals by error-8\31 octave - approx. 309.68¢ - Minor Third-MOS Scales generated by 8\31:"></a><!-- ws:end:WikiTextHeadingRule:34 -->MOS Scales generated by 8\31:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:36:&lt;h3&gt; --><h3 id="toc18"><a name="Intervals-Intervals by error-9\31 octave - approx. 348.39¢ - Neutral Third"></a><!-- ws:end:WikiTextHeadingRule:36 -->9\31 octave - approx. 348.39¢ - Neutral Third</h3> | ||
A neutral 3rd, about 1¢ away from 11:9 (347.41¢). 9\31 is half a perfect fifth (making it a suitable generator for <a class="wiki_link" href="/mohajira">mohajira temperament</a>), and also thrice a major semitone. It is closer in quality to a minor third than a major third, but indeed, it is distinct. It is 11¢ shy of 16/13 (359.47¢), suggesting a <a class="wiki_link" href="/13-limit">13-limit</a> interpretation for 31edo. However, its close proximity to 11/9 makes it hard to hear it as 16/13, which in JI has a different quality (and, as a neutral third, is more &quot;major-like&quot; than &quot;minor-like&quot;). Also, its inversion, 22\31 (851.61¢) is wide of the 13th harmonic by about 11¢, while all the lower harmonics are either near-just or narrow. This means the errors can accumulate, for instance, with 13/9 (636.62¢) represented by 17\31 (658.06¢), a good 21.4¢ sharp.<br /> | A neutral 3rd, about 1¢ away from 11:9 (347.41¢). 9\31 is half a perfect fifth (making it a suitable generator for <a class="wiki_link" href="/mohajira">mohajira temperament</a>), and also thrice a major semitone. It is closer in quality to a minor third than a major third, but indeed, it is distinct. It is 11¢ shy of 16/13 (359.47¢), suggesting a <a class="wiki_link" href="/13-limit">13-limit</a> interpretation for 31edo. However, its close proximity to 11/9 makes it hard to hear it as 16/13, which in JI has a different quality (and, as a neutral third, is more &quot;major-like&quot; than &quot;minor-like&quot;). Also, its inversion, 22\31 (851.61¢) is wide of the 13th harmonic by about 11¢, while all the lower harmonics are either near-just or narrow. This means the errors can accumulate, for instance, with 13/9 (636.62¢) represented by 17\31 (658.06¢), a good 21.4¢ sharp.<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:38:&lt;h4&gt; --><h4 id="toc19"><a name="Intervals-Intervals by error-9\31 octave - approx. 348.39¢ - Neutral Third-MOS Scales generated by 9\31:"></a><!-- ws:end:WikiTextHeadingRule:38 -->MOS Scales generated by 9\31:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:40:&lt;h3&gt; --><h3 id="toc20"><a name="Intervals-Intervals by error-10\31 octave - approx. 387.10¢ - Major Third"></a><!-- ws:end:WikiTextHeadingRule:40 -->10\31 octave - approx. 387.10¢ - Major Third</h3> | ||
A near-just major 3rd (compare with 5:4 = 386.31¢). Has led to the characterization of 31-edo as &quot;smooth&quot;. Generates <a class="wiki_link" href="/Wuerschmidt%20family">wurshmidt/worshmidt temperaments</a>.<br /> | A near-just major 3rd (compare with 5:4 = 386.31¢). Has led to the characterization of 31-edo as &quot;smooth&quot;. Generates <a class="wiki_link" href="/Wuerschmidt%20family">wurshmidt/worshmidt temperaments</a>.<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:42:&lt;h4&gt; --><h4 id="toc21"><a name="Intervals-Intervals by error-10\31 octave - approx. 387.10¢ - Major Third-MOS Scales generated by 10\31:"></a><!-- ws:end:WikiTextHeadingRule:42 -->MOS Scales generated by 10\31:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:44:&lt;h3&gt; --><h3 id="toc22"><a name="Intervals-Intervals by error-11\31 octave - approx. 425.806¢ - Supermajor Third"></a><!-- ws:end:WikiTextHeadingRule:44 -->11\31 octave - approx. 425.806¢ - Supermajor Third</h3> | ||
11\31 functions as 14:11 (417.51¢), 23:18 (424.36¢), 32:25 (427.37¢), 9:7 (435.08¢) and others. In meantone temperament, it is a diminished fourth, eg. C to Fb. It is notable as closely approximating an interval of the <a class="wiki_link" href="/23-limit">23-limit</a>, suggesting the possibility of treating 16\31 (619.35¢) as a flat version of 23/16 (628.27¢). It is perhaps also notable for being close to 6\17, the bright major third of the ever-popular <a class="wiki_link" href="/17edo">17edo</a>. Generates <a class="wiki_link" href="/Meantone%20family">squares temperament</a>.<br /> | 11\31 functions as 14:11 (417.51¢), 23:18 (424.36¢), 32:25 (427.37¢), 9:7 (435.08¢) and others. In meantone temperament, it is a diminished fourth, eg. C to Fb. It is notable as closely approximating an interval of the <a class="wiki_link" href="/23-limit">23-limit</a>, suggesting the possibility of treating 16\31 (619.35¢) as a flat version of 23/16 (628.27¢). It is perhaps also notable for being close to 6\17, the bright major third of the ever-popular <a class="wiki_link" href="/17edo">17edo</a>. Generates <a class="wiki_link" href="/Meantone%20family">squares temperament</a>.<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:46:&lt;h4&gt; --><h4 id="toc23"><a name="Intervals-Intervals by error-11\31 octave - approx. 425.806¢ - Supermajor Third-MOS Scales generated by 11\31:"></a><!-- ws:end:WikiTextHeadingRule:46 -->MOS Scales generated by 11\31:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:48:&lt;h3&gt; --><h3 id="toc24"><a name="Intervals-Intervals by error-12\31 octave - approx. 464.52¢ - Narrow Fourth or Subfourth"></a><!-- ws:end:WikiTextHeadingRule:48 -->12\31 octave - approx. 464.52¢ - Narrow Fourth or Subfourth</h3> | ||
Exactly twice a supermajor second, thrice a neutral second, or four times a minor second. In the 7-limit, 12\31 functions as 21:16 (470.78¢). It is also quite close to the <a class="wiki_link" href="/17-limit">17-limit</a> interval 17/13 (464.43¢), although 31edo does not offer up reasonable approximations of the 17th or 13th harmonics to help make this identity clear. This interval and its inversion 19\31 (735.48¢, a superfifth) are notable for being the only intervals in the 31edo octave larger than the 3\31 diatonic semitone (and smaller than its inversion, 28\31) that are not 11-limit consonances. Generates <a class="wiki_link" href="/semisept">semisept</a> temperament.<br /> | Exactly twice a supermajor second, thrice a neutral second, or four times a minor second. In the 7-limit, 12\31 functions as 21:16 (470.78¢). It is also quite close to the <a class="wiki_link" href="/17-limit">17-limit</a> interval 17/13 (464.43¢), although 31edo does not offer up reasonable approximations of the 17th or 13th harmonics to help make this identity clear. This interval and its inversion 19\31 (735.48¢, a superfifth) are notable for being the only intervals in the 31edo octave larger than the 3\31 diatonic semitone (and smaller than its inversion, 28\31) that are not 11-limit consonances. Generates <a class="wiki_link" href="/semisept">semisept</a> temperament.<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:50:&lt;h4&gt; --><h4 id="toc25"><a name="Intervals-Intervals by error-12\31 octave - approx. 464.52¢ - Narrow Fourth or Subfourth-MOS Scales generated by 12\31:"></a><!-- ws:end:WikiTextHeadingRule:50 -->MOS Scales generated by 12\31:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:52:&lt;h3&gt; --><h3 id="toc26"><a name="Intervals-Intervals by error-13\31 octave - approx. 503.23¢ - Perfect Fourth"></a><!-- ws:end:WikiTextHeadingRule:52 -->13\31 octave - approx. 503.23¢ - Perfect Fourth</h3> | ||
A slightly wide perfect fourth (compare to 4:3 = 498.04¢). As such, it functions marvelously as a generator for <a class="wiki_link" href="/meantone">meantone</a> temperament.<br /> | A slightly wide perfect fourth (compare to 4:3 = 498.04¢). As such, it functions marvelously as a generator for <a class="wiki_link" href="/meantone">meantone</a> temperament.<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:54:&lt;h4&gt; --><h4 id="toc27"><a name="Intervals-Intervals by error-13\31 octave - approx. 503.23¢ - Perfect Fourth-MOS Scales generated by 13\31:"></a><!-- ws:end:WikiTextHeadingRule:54 -->MOS Scales generated by 13\31:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:56:&lt;h3&gt; --><h3 id="toc28"><a name="Intervals-Intervals by error-14\31 octave - approx. 541.94¢ - Superfourth"></a><!-- ws:end:WikiTextHeadingRule:56 -->14\31 octave - approx. 541.94¢ - Superfourth</h3> | ||
Exactly twice a subminor third. Functions as both the 11:8 (551.32¢) and 15:11 (536.95¢) undecimal superfourths (121/120 is tempered out). Thus it makes possible a symmetrical tempered version of an 8:11:15 triad. As 11/8, 14\31 is about 9¢ flat; however, it fits nicely with the also-flat 9/8, allowing a near-just 11/9. Nonetheless, most 11-limit chords in 31edo have a somewhat unstable quality which distinguishes them from their just counterparts. Generates <a class="wiki_link" href="/Starling%20temperaments">casablanca temperament</a>.<br /> | Exactly twice a subminor third. Functions as both the 11:8 (551.32¢) and 15:11 (536.95¢) undecimal superfourths (121/120 is tempered out). Thus it makes possible a symmetrical tempered version of an 8:11:15 triad. As 11/8, 14\31 is about 9¢ flat; however, it fits nicely with the also-flat 9/8, allowing a near-just 11/9. Nonetheless, most 11-limit chords in 31edo have a somewhat unstable quality which distinguishes them from their just counterparts. Generates <a class="wiki_link" href="/Starling%20temperaments">casablanca temperament</a>.<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:58:&lt;h4&gt; --><h4 id="toc29"><a name="Intervals-Intervals by error-14\31 octave - approx. 541.94¢ - Superfourth-MOS Scales generated by 14\31:"></a><!-- ws:end:WikiTextHeadingRule:58 -->MOS Scales generated by 14\31:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:60:&lt;h3&gt; --><h3 id="toc30"><a name="Intervals-Intervals by error-15\31 octave - approx. 580.65¢ - Small Tritone or Augmented Fourth or Subdiminished Fifth"></a><!-- ws:end:WikiTextHeadingRule:60 -->15\31 octave - approx. 580.65¢ - Small Tritone or Augmented Fourth or Subdiminished Fifth</h3> | ||
In 7-limit tonal music, functions quite well as 7:5 (582.51¢). Exactly thrice a whole tone. Generates <a class="wiki_link" href="/tritonic">tritonic</a> temperament.<br /> | In 7-limit tonal music, functions quite well as 7:5 (582.51¢). Exactly thrice a whole tone. Generates <a class="wiki_link" href="/tritonic">tritonic</a> temperament.<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:62:&lt;h4&gt; --><h4 id="toc31"><a name="Intervals-Intervals by error-15\31 octave - approx. 580.65¢ - Small Tritone or Augmented Fourth or Subdiminished Fifth-MOS Scales generated by 15\31:"></a><!-- ws:end:WikiTextHeadingRule:62 -->MOS Scales generated by 15\31:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:64:&lt;h3&gt; --><h3 id="toc32"><a name="Intervals-Intervals by error-16\31 octave"></a><!-- ws:end:WikiTextHeadingRule:64 -->16\31 octave</h3> | ||
The large tritone. Etc.<br /> | The large tritone. Etc.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:66:&lt;h1&gt; --><h1 id="toc33"><a name="Harmonic Scale"></a><!-- ws:end:WikiTextHeadingRule:66 -->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. 31's version of 13/8 is quite wide and only vaguely suggests the <a class="wiki_link" href="/13-limit">13-limit</a>.<br /> | 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. 31's version of 13/8 is quite wide and only vaguely suggests the <a class="wiki_link" href="/13-limit">13-limit</a>.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:68:&lt;h1&gt; --><h1 id="toc34"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:68 -->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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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:70:&lt;h1&gt; --><h1 id="toc35"><a name="Modes"></a><!-- ws:end:WikiTextHeadingRule:70 -->Modes</h1> | ||
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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>. Interesting (to somebody) <a class="wiki_link" href="/9-note%2031edo%20scales">9-note 31edo scales</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>. Interesting (to somebody) <a class="wiki_link" href="/9-note%2031edo%20scales">9-note 31edo scales</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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<td colspan="2"><!-- ws:start:WikiTextHeadingRule: | <td colspan="2"><!-- ws:start:WikiTextHeadingRule:72:&lt;h4&gt; --><h4 id="toc36"><a name="Modes---Some 31 tone equal modes:"></a><!-- ws:end:WikiTextHeadingRule:72 -->Some 31 tone equal modes:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:74:&lt;h1&gt; --><h1 id="toc37"><a name="Music in 31-edo"></a><!-- ws:end:WikiTextHeadingRule:74 -->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 /> | ||
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<a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/wp-content/uploads/2012/03/Claudi_Meneghin_Chaconne_G_001.mp3" rel="nofollow">Chaconna en G=, La Padana, ou la septimala (‘The Padanian, or the septimal’)</a> by <a class="wiki_link" href="/Claudi%20Meneghin">Claudi Meneghin</a><br /> | <a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/wp-content/uploads/2012/03/Claudi_Meneghin_Chaconne_G_001.mp3" rel="nofollow">Chaconna en G=, La Padana, ou la septimala (‘The Padanian, or the septimal’)</a> by <a class="wiki_link" href="/Claudi%20Meneghin">Claudi Meneghin</a><br /> | ||
<a class="wiki_link" href="/earwig">earwig</a> by <a class="wiki_link" href="/Andrew%20Heathwaite">Andrew Heathwaite</a><br /> | <a class="wiki_link" href="/earwig">earwig</a> by <a class="wiki_link" href="/Andrew%20Heathwaite">Andrew Heathwaite</a><br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:76:&lt;h2&gt; --><h2 id="toc38"><!-- ws:end:WikiTextHeadingRule:76 --> </h2> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:78:&lt;h2&gt; --><h2 id="toc39"><a name="Music in 31-edo-Thirty-one tone pedagogy"></a><!-- ws:end:WikiTextHeadingRule:78 -->Thirty-one tone pedagogy</h2> | ||
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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 /> | ||
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See also: <a class="wiki_link" href="/31edo%20solfege">31edo solfege</a>, <a class="wiki_link" href="/Tricesimoprimal%20Tetrachordal%20Tesseract">Tricesimoprimal Tetrachordal Tesseract</a>, <a class="wiki_link" href="/Pentachords%20of%2031edo">Pentachords of 31edo</a>.<br /> | See also: <a class="wiki_link" href="/31edo%20solfege">31edo solfege</a>, <a class="wiki_link" href="/Tricesimoprimal%20Tetrachordal%20Tesseract">Tricesimoprimal Tetrachordal Tesseract</a>, <a class="wiki_link" href="/Pentachords%20of%2031edo">Pentachords of 31edo</a>.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:80:&lt;h1&gt; --><h1 id="toc40"><a name="Practical Theory / Books"></a><!-- ws:end:WikiTextHeadingRule:80 -->Practical Theory / Books</h1> | ||
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<!-- ws:start:WikiTextRemoteImageRule: | <!-- ws:start:WikiTextRemoteImageRule:8096:&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:8096 --><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:82:&lt;h1&gt; --><h1 id="toc41"><a name="Other Articles"></a><!-- ws:end:WikiTextHeadingRule:82 -->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> | ||