31edo: Difference between revisions
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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:guest|guest]] and made on <tt>2011-08-15 02:15:21 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>245978733</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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For more encyclopedic info, see [[http://en.wikipedia.org/wiki/31_equal_temperament|Wikipedia's article]]. | For more encyclopedic info, see [[http://en.wikipedia.org/wiki/31_equal_temperament|Wikipedia's article]]. | ||
=Quick index to intervals and linear temperaments= | |||
||~ Generator ||~ Cents ||~ Temperaments || | |||
|| 1\31 || 38.71 || [[Slender]] || | |||
|| 2\31 || 77.42 || [[Valentine]]/[[Lupercalia]] || | |||
|| 3\31 || 116.13 || [[Miracle]] || | |||
|| 4\31 || 154.84 || [[Nusecond]] || | |||
|| 5\31 || 193.55 || [[Hemiwuerschmidt]] || | |||
|| 6\31 || 232.26 || [[Cynder]]/[[Mothra]] || | |||
|| 7\31 || 270.97 || [[Orson]]/[[Orwell]] || | |||
|| 8\31 || 309.68 || [[Myna]] || | |||
|| 9\31 || 348.39 || [[Vicentino]]/[[Mohajira]]/[[Migration]] || | |||
|| 10\31 || 387.10 || [[Wuerschmidt]]/[[Worschmidt]] || | |||
|| 11\31 || 425.81 || [[Squares]]/[[Sentinel]] || | |||
|| 12\31 || 464.52 || [[Semisept]] || | |||
|| 13\31 || 503.23 || [[Meantone]] || | |||
|| 14\31 || 541.94 || [[Cassablanca]]/[[Marrakesh]] || | |||
|| 15\31 || 580.65 || [[Tritonic]] || | |||
=Intervals= | =Intervals= | ||
===1\31 octave - approx. 38.71¢ - Diesis=== | ===1\31 octave - approx. 38.71¢ - Diesis=== | ||
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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:78:&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:78 --><!-- ws:start:WikiTextTocRule:79: --><a href="#Quick index to intervals and linear temperaments">Quick index to intervals and linear temperaments</a><!-- ws:end:WikiTextTocRule:79 --><!-- ws:start:WikiTextTocRule:80: --> | <a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:80 --><!-- ws:start:WikiTextTocRule:81: --><!-- ws:end:WikiTextTocRule:81 --><!-- ws:start:WikiTextTocRule:82: --><!-- 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: --> | <a href="#Commas">Commas</a><!-- ws:end:WikiTextTocRule:111 --><!-- ws:start:WikiTextTocRule:112: --> | <a href="#Modes">Modes</a><!-- ws:end:WikiTextTocRule:112 --><!-- ws:start:WikiTextTocRule:113: --><!-- ws:end:WikiTextTocRule:113 --><!-- ws:start:WikiTextTocRule:114: --> | <a href="#Music in 31-edo">Music in 31-edo</a><!-- ws:end:WikiTextTocRule:114 --><!-- ws:start:WikiTextTocRule:115: --><!-- ws:end:WikiTextTocRule:115 --><!-- ws:start:WikiTextTocRule:116: --> | <a href="#Practical Theory / Books">Practical Theory / Books</a><!-- ws:end:WikiTextTocRule:116 --><!-- ws:start:WikiTextTocRule:117: --> | <a href="#Other Articles">Other Articles</a><!-- ws:end:WikiTextTocRule:117 --><!-- ws:start:WikiTextTocRule:118: --> | ||
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<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 EDO lists">zeta integral edo</a>. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course). It is consistent through the 11-limit, 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 EDO lists">zeta integral edo</a>. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course). It is consistent through the 11-limit, 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 /> | ||
For more encyclopedic info, see <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/31_equal_temperament" rel="nofollow">Wikipedia's article</a>.<br /> | For more encyclopedic info, see <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/31_equal_temperament" rel="nofollow">Wikipedia's article</a>.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Quick index to intervals and linear temperaments"></a><!-- ws:end:WikiTextHeadingRule:0 -->Quick index to intervals and linear temperaments</h1> | ||
<!-- ws:start:WikiTextHeadingRule: | |||
<table class="wiki_table"> | |||
<tr> | |||
<th>Generator<br /> | |||
</th> | |||
<th>Cents<br /> | |||
</th> | |||
<th>Temperaments<br /> | |||
</th> | |||
</tr> | |||
<tr> | |||
<td>1\31<br /> | |||
</td> | |||
<td>38.71<br /> | |||
</td> | |||
<td><a class="wiki_link" href="/Slender">Slender</a><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>2\31<br /> | |||
</td> | |||
<td>77.42<br /> | |||
</td> | |||
<td><a class="wiki_link" href="/Valentine">Valentine</a>/<a class="wiki_link" href="/Lupercalia">Lupercalia</a><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>3\31<br /> | |||
</td> | |||
<td>116.13<br /> | |||
</td> | |||
<td><a class="wiki_link" href="/Miracle">Miracle</a><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>4\31<br /> | |||
</td> | |||
<td>154.84<br /> | |||
</td> | |||
<td><a class="wiki_link" href="/Nusecond">Nusecond</a><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>5\31<br /> | |||
</td> | |||
<td>193.55<br /> | |||
</td> | |||
<td><a class="wiki_link" href="/Hemiwuerschmidt">Hemiwuerschmidt</a><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>6\31<br /> | |||
</td> | |||
<td>232.26<br /> | |||
</td> | |||
<td><a class="wiki_link" href="/Cynder">Cynder</a>/<a class="wiki_link" href="/Mothra">Mothra</a><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>7\31<br /> | |||
</td> | |||
<td>270.97<br /> | |||
</td> | |||
<td><a class="wiki_link" href="/Orson">Orson</a>/<a class="wiki_link" href="/Orwell">Orwell</a><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>8\31<br /> | |||
</td> | |||
<td>309.68<br /> | |||
</td> | |||
<td><a class="wiki_link" href="/Myna">Myna</a><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>9\31<br /> | |||
</td> | |||
<td>348.39<br /> | |||
</td> | |||
<td><a class="wiki_link" href="/Vicentino">Vicentino</a>/<a class="wiki_link" href="/Mohajira">Mohajira</a>/<a class="wiki_link" href="/Migration">Migration</a><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>10\31<br /> | |||
</td> | |||
<td>387.10<br /> | |||
</td> | |||
<td><a class="wiki_link" href="/Wuerschmidt">Wuerschmidt</a>/<a class="wiki_link" href="/Worschmidt">Worschmidt</a><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>11\31<br /> | |||
</td> | |||
<td>425.81<br /> | |||
</td> | |||
<td><a class="wiki_link" href="/Squares">Squares</a>/<a class="wiki_link" href="/Sentinel">Sentinel</a><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>12\31<br /> | |||
</td> | |||
<td>464.52<br /> | |||
</td> | |||
<td><a class="wiki_link" href="/Semisept">Semisept</a><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>13\31<br /> | |||
</td> | |||
<td>503.23<br /> | |||
</td> | |||
<td><a class="wiki_link" href="/Meantone">Meantone</a><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>14\31<br /> | |||
</td> | |||
<td>541.94<br /> | |||
</td> | |||
<td><a class="wiki_link" href="/Cassablanca">Cassablanca</a>/<a class="wiki_link" href="/Marrakesh">Marrakesh</a><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>15\31<br /> | |||
</td> | |||
<td>580.65<br /> | |||
</td> | |||
<td><a class="wiki_link" href="/Tritonic">Tritonic</a><br /> | |||
</td> | |||
</tr> | |||
</table> | |||
<!-- 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="toc2"><a name="Intervals--1\31 octave - approx. 38.71¢ - Diesis"></a><!-- ws:end:WikiTextHeadingRule:4 -->1\31 octave - approx. 38.71¢ - Diesis</h3> | |||
A single step of 31-edo is about 38.71¢. Intervals around this size are called <a class="wiki_link" href="/diesis">dieses</a> (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 11-limit tonal music, the single step 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. 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 <a class="wiki_link" href="/diesis">dieses</a> (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 11-limit tonal music, the single step 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. Demonstrated in <a class="wiki_link" href="/SpiralProgressions">SpiralProgressions</a>.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="Intervals--2\31 octave - approx. 77.42¢ - Minor Semitone or Chromatic Semitone or Small Minor Second"></a><!-- ws:end:WikiTextHeadingRule:6 -->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'. In 11-limit tonal music, 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 temperament">valentine temperament</a> - aka <a class="wiki_link" href="/Armodue%20theory#Semi-equalized Armodue">semi-equalized Armodue</a>.<br /> | The difference between a major and minor third. The more 'expressive' of the 'half steps'. In 11-limit tonal music, 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 temperament">valentine temperament</a> - aka <a class="wiki_link" href="/Armodue%20theory#Semi-equalized Armodue">semi-equalized Armodue</a>.<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h4&gt; --><h4 id="toc4"><a name="Intervals--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:8 -->MOS Scales generated by 2\31:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="Intervals--3\31 octave - approx. 116.13 - Major Semitone or Diatonic Semitone or Large Major Second"></a><!-- ws:end:WikiTextHeadingRule:10 -->3\31 octave - approx. 116.13 - Major Semitone or Diatonic Semitone or Large Major Second</h3> | ||
The difference between a perfect fourth and a major third. The larger and clunkier of the 'half steps'. In 11-limit tonal music, 3\31 stands in for just ratios 16:15 (111.73¢); 15:14 (199.44¢) and others. Generates <a class="wiki_link" href="/Gamelismic%20clan">miracle temperament</a>.<br /> | The difference between a perfect fourth and a major third. The larger and clunkier of the 'half steps'. In 11-limit tonal music, 3\31 stands in for just ratios 16:15 (111.73¢); 15:14 (199.44¢) and others. Generates <a class="wiki_link" href="/Gamelismic%20clan">miracle temperament</a>.<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h4&gt; --><h4 id="toc6"><a name="Intervals--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:12 -->MOS Scales generated by 3\31:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="Intervals--4\31 octave - approx. 154.84¢ - Neutral Tone or Neutral Second"></a><!-- ws:end:WikiTextHeadingRule:14 -->4\31 octave - approx. 154.84¢ - Neutral Tone or Neutral Second</h3> | ||
Exactly one half of the minor third (and twice the minor semitone). In 11-limit tonal music, 4\31 stands in for 12:11 (150.64¢); 35:32 (155.14¢); 11:10 (165.00¢) and others. Generates <a class="wiki_link" href="/Starling%20temperaments">nusecond temperament</a>.<br /> | Exactly one half of the minor third (and twice the minor semitone). In 11-limit tonal music, 4\31 stands in for 12:11 (150.64¢); 35:32 (155.14¢); 11:10 (165.00¢) and others. Generates <a class="wiki_link" href="/Starling%20temperaments">nusecond temperament</a>.<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h4&gt; --><h4 id="toc8"><a name="Intervals--4\31 octave - approx. 154.84¢ - Neutral Tone or Neutral Second-MOS Scales generated by 4\31:"></a><!-- ws:end:WikiTextHeadingRule:16 -->MOS Scales generated by 4\31:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h3&gt; --><h3 id="toc9"><a name="Intervals--5\31 octave - approx. 193.55¢ - Whole Tone or Major Second"></a><!-- ws:end:WikiTextHeadingRule:18 -->5\31 octave - approx. 193.55¢ - Whole Tone or Major Second</h3> | ||
A rather smallish whole tone. Often 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. 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. Often 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. Generates <a class="wiki_link" href="/Gamelismic%20clan">hemithirds temperament</a> and <a class="wiki_link" href="/Wuerschmidt%20family">hermiwuerschmidt temperament</a>.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:22:&lt;h3&gt; --><h3 id="toc11"><a name="Intervals--6\31 octave - approx. 232.26¢ - Supermajor Second"></a><!-- ws:end:WikiTextHeadingRule:22 -->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 stands in for 8:7 (231.17¢). 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 stands in for 8:7 (231.17¢). Generates <a class="wiki_link" href="/Meantone%20family">mothra temperament</a>.<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:24:&lt;h4&gt; --><h4 id="toc12"><a name="Intervals--6\31 octave - approx. 232.26¢ - Supermajor Second-MOS Scales generated by 6\31:"></a><!-- ws:end:WikiTextHeadingRule:24 -->MOS Scales generated by 6\31:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc13"><a name="Intervals--7\31 octave - approx. 270.97¢ - Subminor Third"></a><!-- ws:end:WikiTextHeadingRule:26 -->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¢). A generator for Orwell temperament (but not as good as 12\53 or 19\84). 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¢). A generator for Orwell temperament (but not as good as 12\53 or 19\84). Generates <a class="wiki_link" href="/Semicomma%20family">orwell temperament</a>.<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:28:&lt;h4&gt; --><h4 id="toc14"><a name="Intervals--7\31 octave - approx. 270.97¢ - Subminor Third-MOS Scales generated by 7\31:"></a><!-- ws:end:WikiTextHeadingRule:28 -->MOS Scales generated by 7\31:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:30:&lt;h3&gt; --><h3 id="toc15"><a name="Intervals--8\31 octave - approx. 309.68¢ - Minor Third"></a><!-- ws:end:WikiTextHeadingRule:30 -->8\31 octave - approx. 309.68¢ - Minor Third</h3> | ||
A minor third, closer to the just 6:5 (315.64¢) than 12-edo. 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. 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:32:&lt;h4&gt; --><h4 id="toc16"><a name="Intervals--8\31 octave - approx. 309.68¢ - Minor Third-MOS Scales generated by 8\31:"></a><!-- ws:end:WikiTextHeadingRule:32 -->MOS Scales generated by 8\31:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:34:&lt;h3&gt; --><h3 id="toc17"><a name="Intervals--9\31 octave - approx. 348.39¢ - Neutral Third"></a><!-- ws:end:WikiTextHeadingRule:34 -->9\31 octave - approx. 348.39¢ - Neutral Third</h3> | ||
A neutral 3rd, practically equivalent to 11:9 (347.41¢). Exactly half a perfect fifth (making it a suitable generator for neutral third scales such as <a class="wiki_link" href="/3L%204s">3L 4s</a>). Is also thrice a major semitone. Generates <a class="wiki_link" href="/Meantone%20family">mohajira temperament</a>.<br /> | A neutral 3rd, practically equivalent to 11:9 (347.41¢). Exactly half a perfect fifth (making it a suitable generator for neutral third scales such as <a class="wiki_link" href="/3L%204s">3L 4s</a>). Is also thrice a major semitone. Generates <a class="wiki_link" href="/Meantone%20family">mohajira temperament</a>.<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:36:&lt;h4&gt; --><h4 id="toc18"><a name="Intervals--9\31 octave - approx. 348.39¢ - Neutral Third-MOS Scales generated by 9\31:"></a><!-- ws:end:WikiTextHeadingRule:36 -->MOS Scales generated by 9\31:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:38:&lt;h3&gt; --><h3 id="toc19"><a name="Intervals--10\31 octave - approx. 387.10¢ - Major Third"></a><!-- ws:end:WikiTextHeadingRule:38 -->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:40:&lt;h4&gt; --><h4 id="toc20"><a name="Intervals--10\31 octave - approx. 387.10¢ - Major Third-MOS Scales generated by 10\31:"></a><!-- ws:end:WikiTextHeadingRule:40 -->MOS Scales generated by 10\31:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:42:&lt;h3&gt; --><h3 id="toc21"><a name="Intervals--11\31 octave - approx. 425.806¢ - Supermajor Third"></a><!-- ws:end:WikiTextHeadingRule:42 -->11\31 octave - approx. 425.806¢ - Supermajor Third</h3> | ||
In 11-limit tonal music, 11\31 functions as 14:11 (417.51¢), 32:25 (427.37¢), 9:7 (435.08¢) and others. Generates <a class="wiki_link" href="/Meantone%20family">squares temperament</a>.<br /> | In 11-limit tonal music, 11\31 functions as 14:11 (417.51¢), 32:25 (427.37¢), 9:7 (435.08¢) and others. Generates <a class="wiki_link" href="/Meantone%20family">squares temperament</a>.<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:44:&lt;h4&gt; --><h4 id="toc22"><a name="Intervals--11\31 octave - approx. 425.806¢ - Supermajor Third-MOS Scales generated by 11\31:"></a><!-- ws:end:WikiTextHeadingRule:44 -->MOS Scales generated by 11\31:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:46:&lt;h3&gt; --><h3 id="toc23"><a name="Intervals--12\31 octave - approx. 464.52¢ - Narrow Fourth or Subfourth"></a><!-- ws:end:WikiTextHeadingRule:46 -->12\31 octave - approx. 464.52¢ - Narrow Fourth or Subfourth</h3> | ||
Exactly twice a supermajor second, thrice a neutral second, or four times a major second. In 7-limit tonal music, 12\31 functions as 21:16 (470.78¢). Generates semisept temperament.<br /> | Exactly twice a supermajor second, thrice a neutral second, or four times a major second. In 7-limit tonal music, 12\31 functions as 21:16 (470.78¢). Generates semisept temperament.<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:48:&lt;h4&gt; --><h4 id="toc24"><a name="Intervals--12\31 octave - approx. 464.52¢ - Narrow Fourth or Subfourth-MOS Scales generated by 12\31:"></a><!-- ws:end:WikiTextHeadingRule:48 -->MOS Scales generated by 12\31:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:50:&lt;h3&gt; --><h3 id="toc25"><a name="Intervals--13\31 octave - approx. 503.23¢ - Perfect Fourth"></a><!-- ws:end:WikiTextHeadingRule:50 -->13\31 octave - approx. 503.23¢ - Perfect Fourth</h3> | ||
A sharp perfect fourth (compare to 4:3 = 498.04¢). As such, it functions marvelously as a generator for meantone temperament.<br /> | A sharp perfect fourth (compare to 4:3 = 498.04¢). As such, it functions marvelously as a generator for meantone temperament.<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:52:&lt;h4&gt; --><h4 id="toc26"><a name="Intervals--13\31 octave - approx. 503.23¢ - Perfect Fourth-MOS Scales generated by 13\31:"></a><!-- ws:end:WikiTextHeadingRule:52 -->MOS Scales generated by 13\31:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:54:&lt;h3&gt; --><h3 id="toc27"><a name="Intervals--14\31 octave - approx. 541.94¢ - Superfourth"></a><!-- ws:end:WikiTextHeadingRule:54 -->14\31 octave - approx. 541.94¢ - Superfourth</h3> | ||
10¢ off from a just 11:8 (551.32¢); barely functional as such. Exactly twice a subminor third. Generates <a class="wiki_link" href="/Starling%20temperaments">casablanca temperament</a>.<br /> | 10¢ off from a just 11:8 (551.32¢); barely functional as such. Exactly twice a subminor third. Generates <a class="wiki_link" href="/Starling%20temperaments">casablanca temperament</a>.<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:56:&lt;h4&gt; --><h4 id="toc28"><a name="Intervals--14\31 octave - approx. 541.94¢ - Superfourth-MOS Scales generated by 14\31:"></a><!-- ws:end:WikiTextHeadingRule:56 -->MOS Scales generated by 14\31:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:58:&lt;h3&gt; --><h3 id="toc29"><a name="Intervals--15\31 octave - approx. 580.65¢ - Small Tritone or Augmented Fourth or Subdiminished Fifth"></a><!-- ws:end:WikiTextHeadingRule:58 -->15\31 octave - approx. 580.65¢ - Small Tritone or Augmented Fourth or Subdiminished Fifth</h3> | ||
In 7-limit tonal music, functions as 7:5 (582.51¢). Exactly thrice a whole tone. Generates tritonic temperament.<br /> | In 7-limit tonal music, functions as 7:5 (582.51¢). Exactly thrice a whole tone. Generates tritonic temperament.<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:60:&lt;h4&gt; --><h4 id="toc30"><a name="Intervals--15\31 octave - approx. 580.65¢ - Small Tritone or Augmented Fourth or Subdiminished Fifth-MOS Scales generated by 15\31:"></a><!-- ws:end:WikiTextHeadingRule:60 -->MOS Scales generated by 15\31:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:62:&lt;h3&gt; --><h3 id="toc31"><a name="Intervals--16\31 octave"></a><!-- ws:end:WikiTextHeadingRule:62 -->16\31 octave</h3> | ||
The large tritone.<br /> | The large tritone.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:64:&lt;h1&gt; --><h1 id="toc32"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:64 -->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:66:&lt;h1&gt; --><h1 id="toc33"><a name="Modes"></a><!-- ws:end:WikiTextHeadingRule:66 -->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>. 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:68:&lt;h4&gt; --><h4 id="toc34"><a name="Modes---Some 31 tone equal modes:"></a><!-- ws:end:WikiTextHeadingRule:68 -->Some 31 tone equal modes:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:70:&lt;h1&gt; --><h1 id="toc35"><a name="Music in 31-edo"></a><!-- ws:end:WikiTextHeadingRule:70 -->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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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:72:&lt;h2&gt; --><h2 id="toc36"><a name="Music in 31-edo-Thirty-one tone pedagogy"></a><!-- ws:end:WikiTextHeadingRule:72 -->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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:74:&lt;h1&gt; --><h1 id="toc37"><a name="Practical Theory / Books"></a><!-- ws:end:WikiTextHeadingRule:74 -->Practical Theory / Books</h1> | ||
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<!-- ws:start:WikiTextRemoteImageRule: | <!-- ws:start:WikiTextRemoteImageRule:7823:&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:7823 --><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:76:&lt;h1&gt; --><h1 id="toc38"><a name="Other Articles"></a><!-- ws:end:WikiTextHeadingRule:76 -->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> | ||