31edo: Difference between revisions
Wikispaces>genewardsmith **Imported revision 160462393 - Original comment: ** |
Wikispaces>Andrew_Heathwaite **Imported revision 160891375 - 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:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2010-09-07 13:45:21 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>160891375</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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===3\31 octave - approx. 116.13 - Major Semitone or Diatonic Semitone or Large Major Second=== | ===3\31 octave - approx. 116.13 - Major Semitone or Diatonic Semitone or Large Major Second=== | ||
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 [[Gamelismic clan|miracle temperament]]. | 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 [[Gamelismic clan|miracle temperament]]. | ||
====MOS Scales generated by 3\31:==== | |||
10-tone (quasi-equal 10): [[9L 1s]]: 3 3 3 3 3 3 3 3 3 4. | |||
11-tone: [[10L 1s]]: 3 3 3 3 3 3 3 3 3 3 1. | |||
21-tone, [[11L 10s]]: 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 (Blackjack). | |||
===4\31 octave - approx. 154.84¢ - Neutral Tone or Neutral Second=== | ===4\31 octave - approx. 154.84¢ - Neutral Tone or Neutral Second=== | ||
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 [[Starling temperaments|nusecond temperament]]. | 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 [[Starling temperaments|nusecond temperament]]. | ||
====MOS Scales generated by 4\31:==== | |||
7-tone: [[1L 6s]]: 4 4 4 4 4 4 7. | |||
8-tone (quasi-equal 8): [[7L 1s]]: 4 4 4 4 4 4 4 3. | |||
15-tone: [[8L 7s]]: 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3. | |||
===5\31 octave - approx. 193.55¢ - Whole Tone or Major Second=== | ===5\31 octave - approx. 193.55¢ - Whole Tone or Major Second=== | ||
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 "meantone". Two meantones make a near-just major third. Generates [[Gamelismic clan|hemithirds temperament]] and [[Wuerschmidt family|hermiwuerschmidt temperament]]. | 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 "meantone". Two meantones make a near-just major third. Generates [[Gamelismic clan|hemithirds temperament]] and [[Wuerschmidt family|hermiwuerschmidt temperament]]. | ||
====MOS Scales generated by 5\31:==== | |||
6-tone (quasi-equal 6): [[1L 5s]]: 5 5 5 5 5 6. | |||
7-tone: [[6L 1s]]: 5 5 5 5 5 5 1. | |||
13-tone: [[6L 7s]]: 1 4 1 4 1 4 1 4 1 4 1 4 1. | |||
===6\31 octave - approx. 232.26¢ - Supermajor Second=== | ===6\31 octave - approx. 232.26¢ - Supermajor Second=== | ||
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 [[Meantone family|mothra temperament]]. | 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 [[Meantone family|mothra temperament]]. | ||
====MOS Scales generated by 6\31:==== | |||
5-tone (quasi-equal 5): [[1L 4s]]: 6 6 6 6 7. | |||
6-tone: [[5L 1s]]: 6 6 6 6 6 1. | |||
11-tone: [[5L 6s]]: 1 5 1 5 1 5 1 5 1 5 1. | |||
===7\31 octave - approx. 270.97¢ - Subminor Third=== | ===7\31 octave - approx. 270.97¢ - Subminor Third=== | ||
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=Modes= | =Modes= | ||
A large open list of modes (subsets) from 31edo that people have named: [[31edo modes]]. [[http://en.wikipedia.org/wiki/Rothenberg_propriety|Strictly proper]] [[Strictly proper 7-note 31edo scales|7-note 31edo scales]] in the sense of [[David Rothenberg]]. | A large open list of modes (subsets) from 31edo that people have named: [[31edo modes]]. [[http://en.wikipedia.org/wiki/Rothenberg_propriety|Strictly proper]] [[Strictly proper 7-note 31edo scales|7-note 31edo scales]] in the sense of [[David Rothenberg]]. See also [[31edo MOS scales]]. Some of the popular ones: | ||
* 31-tone major: 5 5 3 5 5 5 3 | * 31-tone major: 5 5 3 5 5 5 3 | ||
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* <span class="wiki_link_ext">[[http://tonalsoft.com/enc/number/31edo.aspx|Tonalsoft Encyclopedia article]]</span></pre></div> | * <span class="wiki_link_ext">[[http://tonalsoft.com/enc/number/31edo.aspx|Tonalsoft Encyclopedia article]]</span></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:54:&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:54 --><!-- ws:start:WikiTextTocRule:55: --><a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextTocRule:56: --><!-- ws:end:WikiTextTocRule:56 --><!-- ws:start:WikiTextTocRule:57: --><!-- ws:end:WikiTextTocRule:57 --><!-- ws:start:WikiTextTocRule:58: --><!-- ws:end:WikiTextTocRule:58 --><!-- ws:start:WikiTextTocRule:59: --><!-- ws:end:WikiTextTocRule:59 --><!-- ws:start:WikiTextTocRule:60: --><!-- ws:end:WikiTextTocRule:60 --><!-- ws:start:WikiTextTocRule:61: --><!-- ws:end:WikiTextTocRule:61 --><!-- ws:start:WikiTextTocRule:62: --><!-- ws:end:WikiTextTocRule:62 --><!-- ws:start:WikiTextTocRule:63: --><!-- ws:end:WikiTextTocRule:63 --><!-- ws:start:WikiTextTocRule:64: --><!-- ws:end:WikiTextTocRule:64 --><!-- ws:start:WikiTextTocRule:65: --><!-- ws:end:WikiTextTocRule:65 --><!-- ws:start:WikiTextTocRule:66: --><!-- ws:end:WikiTextTocRule:66 --><!-- ws:start:WikiTextTocRule:67: --><!-- ws:end:WikiTextTocRule:67 --><!-- ws:start:WikiTextTocRule:68: --><!-- ws:end:WikiTextTocRule:68 --><!-- ws:start:WikiTextTocRule:69: --><!-- ws:end:WikiTextTocRule:69 --><!-- ws:start:WikiTextTocRule:70: --><!-- ws:end:WikiTextTocRule:70 --><!-- ws:start:WikiTextTocRule:71: --><!-- ws:end:WikiTextTocRule:71 --><!-- ws:start:WikiTextTocRule:72: --><!-- ws:end:WikiTextTocRule:72 --><!-- ws:start:WikiTextTocRule:73: --><!-- ws:end:WikiTextTocRule:73 --><!-- ws:start:WikiTextTocRule:74: --><!-- ws:end:WikiTextTocRule:74 --><!-- ws:start:WikiTextTocRule:75: --><!-- ws:end:WikiTextTocRule:75 --><!-- ws:start:WikiTextTocRule:76: --> | <a href="#Modes">Modes</a><!-- ws:end:WikiTextTocRule:76 --><!-- ws:start:WikiTextTocRule:77: --><!-- ws:end:WikiTextTocRule:77 --><!-- ws:start:WikiTextTocRule:78: --> | <a href="#Music in 31-edo">Music in 31-edo</a><!-- ws:end:WikiTextTocRule:78 --><!-- ws:start:WikiTextTocRule:79: --><!-- ws:end:WikiTextTocRule:79 --><!-- ws:start:WikiTextTocRule:80: --> | <a href="#Practical Theory / Books">Practical Theory / Books</a><!-- ws:end:WikiTextTocRule:80 --><!-- ws:start:WikiTextTocRule:81: --> | <a href="#Other Articles">Other Articles</a><!-- ws:end:WikiTextTocRule:81 --><!-- ws:start:WikiTextTocRule:82: --> | ||
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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 Zeta function integral tuning, <!-- ws:start:WikiTextUrlRule: | <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 Zeta function integral tuning, <!-- ws:start:WikiTextUrlRule:914:http://www.research.att.com/~njas/sequences/A117538 --><a class="wiki_link_ext" href="http://www.research.att.com/~njas/sequences/A117538" rel="nofollow">http://www.research.att.com/~njas/sequences/A117538</a><!-- ws:end:WikiTextUrlRule:914 -->. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course). For JI that uses primes 3 and 7, but no 5, try <a class="wiki_link" href="/36edo">36edo</a>.<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 /> | ||
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<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="Intervals--3\31 octave - approx. 116.13 - Major Semitone or Diatonic Semitone or Large Major Second"></a><!-- ws:end:WikiTextHeadingRule:6 -->3\31 octave - approx. 116.13 - Major Semitone or Diatonic Semitone or Large Major Second</h3> | <!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="Intervals--3\31 octave - approx. 116.13 - Major Semitone or Diatonic Semitone or Large Major Second"></a><!-- ws:end:WikiTextHeadingRule:6 -->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:8:&lt;h4&gt; --><h4 id="toc4"><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:8 -->MOS Scales generated by 3\31:</h4> | |||
10-tone (quasi-equal 10): <a class="wiki_link" href="/9L%201s">9L 1s</a>: 3 3 3 3 3 3 3 3 3 4.<br /> | |||
11-tone: <a class="wiki_link" href="/10L%201s">10L 1s</a>: 3 3 3 3 3 3 3 3 3 3 1.<br /> | |||
21-tone, <a class="wiki_link" href="/11L%2010s">11L 10s</a>: 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 (Blackjack).<br /> | |||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="Intervals--4\31 octave - approx. 154.84¢ - Neutral Tone or Neutral Second"></a><!-- ws:end:WikiTextHeadingRule:10 -->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:12:&lt;h4&gt; --><h4 id="toc6"><a name="Intervals--4\31 octave - approx. 154.84¢ - Neutral Tone or Neutral Second-MOS Scales generated by 4\31:"></a><!-- ws:end:WikiTextHeadingRule:12 -->MOS Scales generated by 4\31:</h4> | |||
7-tone: <a class="wiki_link" href="/1L%206s">1L 6s</a>: 4 4 4 4 4 4 7.<br /> | |||
8-tone (quasi-equal 8): <a class="wiki_link" href="/7L%201s">7L 1s</a>: 4 4 4 4 4 4 4 3.<br /> | |||
15-tone: <a class="wiki_link" href="/8L%207s">8L 7s</a>: 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3.<br /> | |||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="Intervals--5\31 octave - approx. 193.55¢ - Whole Tone or Major Second"></a><!-- ws:end:WikiTextHeadingRule:14 -->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 /> | ||
<!-- ws:start:WikiTextHeadingRule:16:&lt;h4&gt; --><h4 id="toc8"><a name="Intervals--5\31 octave - approx. 193.55¢ - Whole Tone or Major Second-MOS Scales generated by 5\31:"></a><!-- ws:end:WikiTextHeadingRule:16 -->MOS Scales generated by 5\31:</h4> | |||
6-tone (quasi-equal 6): <a class="wiki_link" href="/1L%205s">1L 5s</a>: 5 5 5 5 5 6.<br /> | |||
7-tone: <a class="wiki_link" href="/6L%201s">6L 1s</a>: 5 5 5 5 5 5 1.<br /> | |||
13-tone: <a class="wiki_link" href="/6L%207s">6L 7s</a>: 1 4 1 4 1 4 1 4 1 4 1 4 1.<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h3&gt; --><h3 id="toc9"><a name="Intervals--6\31 octave - approx. 232.26¢ - Supermajor Second"></a><!-- ws:end:WikiTextHeadingRule:18 -->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:20:&lt;h4&gt; --><h4 id="toc10"><a name="Intervals--6\31 octave - approx. 232.26¢ - Supermajor Second-MOS Scales generated by 6\31:"></a><!-- ws:end:WikiTextHeadingRule:20 -->MOS Scales generated by 6\31:</h4> | |||
5-tone (quasi-equal 5): <a class="wiki_link" href="/1L%204s">1L 4s</a>: 6 6 6 6 7.<br /> | |||
6-tone: <a class="wiki_link" href="/5L%201s">5L 1s</a>: 6 6 6 6 6 1.<br /> | |||
11-tone: <a class="wiki_link" href="/5L%206s">5L 6s</a>: 1 5 1 5 1 5 1 5 1 5 1.<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:22:&lt;h3&gt; --><h3 id="toc11"><a name="Intervals--7\31 octave - approx. 270.97¢ - Subminor Third"></a><!-- ws:end:WikiTextHeadingRule:22 -->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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc12"><a name="Intervals--8\31 octave - approx. 309.68¢ - Minor Third"></a><!-- ws:end:WikiTextHeadingRule:24 -->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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc13"><a name="Intervals--9\31 octave - approx. 348.39¢ - Neutral Third"></a><!-- ws:end:WikiTextHeadingRule:26 -->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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:28:&lt;h3&gt; --><h3 id="toc14"><a name="Intervals--10\31 octave - approx. 387.10¢ - Major Third"></a><!-- ws:end:WikiTextHeadingRule:28 -->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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:30:&lt;h3&gt; --><h3 id="toc15"><a name="Intervals--11\31 octave - approx. 425.806¢ - Supermajor Third"></a><!-- ws:end:WikiTextHeadingRule:30 -->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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:32:&lt;h3&gt; --><h3 id="toc16"><a name="Intervals--12\31 octave - approx. 464.52¢ - Narrow Fourth or Subfourth"></a><!-- ws:end:WikiTextHeadingRule:32 -->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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:34:&lt;h3&gt; --><h3 id="toc17"><a name="Intervals--13\31 octave - approx. 503.23¢ - Perfect Fourth"></a><!-- ws:end:WikiTextHeadingRule:34 -->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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:36:&lt;h3&gt; --><h3 id="toc18"><a name="Intervals--14\31 octave - approx. 541.94¢ - Superfourth"></a><!-- ws:end:WikiTextHeadingRule:36 -->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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:38:&lt;h3&gt; --><h3 id="toc19"><a name="Intervals--15\31 octave - approx. 580.65¢ - Small Tritone or Augmented Fourth or Subdiminished Fifth"></a><!-- ws:end:WikiTextHeadingRule:38 -->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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:40:&lt;h3&gt; --><h3 id="toc20"><a name="Intervals--16\31 octave"></a><!-- ws:end:WikiTextHeadingRule:40 -->16\31 octave</h3> | ||
The large tritone.<br /> | The large tritone.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:42:&lt;h1&gt; --><h1 id="toc21"><a name="Modes"></a><!-- ws:end:WikiTextHeadingRule:42 -->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>. | 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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<ul><li>31-tone major: 5 5 3 5 5 5 3</li><li>Meantone[12] (Eb-G#): 2 3 3 2 3 2 3 2 3 3 2 3</li><li>Harmonic scale 8: 5 5 4 4 4 4 3 3</li><li>the <a class="wiki_link" href="/Euler-Fokker%20genera">Euler-Fokker genera</a> (technically <a class="wiki_link" href="/JI">JI</a> but representable in 31)</li></ul><br /> | <ul><li>31-tone major: 5 5 3 5 5 5 3</li><li>Meantone[12] (Eb-G#): 2 3 3 2 3 2 3 2 3 3 2 3</li><li>Harmonic scale 8: 5 5 4 4 4 4 3 3</li><li>the <a class="wiki_link" href="/Euler-Fokker%20genera">Euler-Fokker genera</a> (technically <a class="wiki_link" href="/JI">JI</a> but representable in 31)</li></ul><br /> | ||
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<table class="wiki_table"> | <table class="wiki_table"> | ||
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<td colspan="2"><!-- ws:start:WikiTextHeadingRule: | <td colspan="2"><!-- ws:start:WikiTextHeadingRule:44:&lt;h4&gt; --><h4 id="toc22"><a name="Modes---Some 31 tone equal modes:"></a><!-- ws:end:WikiTextHeadingRule:44 -->Some 31 tone equal modes:</h4> | ||
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</tr> | </tr> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:46:&lt;h1&gt; --><h1 id="toc23"><a name="Music in 31-edo"></a><!-- ws:end:WikiTextHeadingRule:46 -->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:48:&lt;h2&gt; --><h2 id="toc24"><a name="Music in 31-edo-Thirty-one tone pedagogy"></a><!-- ws:end:WikiTextHeadingRule:48 -->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:50:&lt;h1&gt; --><h1 id="toc25"><a name="Practical Theory / Books"></a><!-- ws:end:WikiTextHeadingRule:50 -->Practical Theory / Books</h1> | ||
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<!-- ws:start:WikiTextRemoteImageRule: | <!-- ws:start:WikiTextRemoteImageRule:625:&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:625 --><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:52:&lt;h1&gt; --><h1 id="toc26"><a name="Other Articles"></a><!-- ws:end:WikiTextHeadingRule:52 -->Other Articles</h1> | ||
<ul><li><span class="wiki_link_ext"><a class="wiki_link_ext" href="http://www.xs4all.nl/%7Ehuygensf/english/index.html" rel="nofollow">de Beer, Anton, ''The Development of 31-tone Music</a></span></li><li><span class="wiki_link_ext"><a class="wiki_link_ext" href="http://www.xs4all.nl/%7Ehuygensf/doc/fokkerorg.html" rel="nofollow">Fokker, Adriaan Daniël, ''Equal Temperament and the Thirty-one-keyed organ</a></span></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.xs4all.nl/%7Ehuygensf/doc/rap31.html" rel="nofollow">Rapoport, Paul, ''About 31-tone Equal Temperament</a></span></li><li><span class="wiki_link_ext"><a class="wiki_link_ext" href="http://www.xs4all.nl/%7Ehuygensf/doc/terp31.html" rel="nofollow">Terpstra, Siemen, ''Toward a Theory of Meantone (and 31-et) Harmony''</a></span></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></li></ul></body></html></pre></div> | <ul><li><span class="wiki_link_ext"><a class="wiki_link_ext" href="http://www.xs4all.nl/%7Ehuygensf/english/index.html" rel="nofollow">de Beer, Anton, ''The Development of 31-tone Music</a></span></li><li><span class="wiki_link_ext"><a class="wiki_link_ext" href="http://www.xs4all.nl/%7Ehuygensf/doc/fokkerorg.html" rel="nofollow">Fokker, Adriaan Daniël, ''Equal Temperament and the Thirty-one-keyed organ</a></span></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.xs4all.nl/%7Ehuygensf/doc/rap31.html" rel="nofollow">Rapoport, Paul, ''About 31-tone Equal Temperament</a></span></li><li><span class="wiki_link_ext"><a class="wiki_link_ext" href="http://www.xs4all.nl/%7Ehuygensf/doc/terp31.html" rel="nofollow">Terpstra, Siemen, ''Toward a Theory of Meantone (and 31-et) Harmony''</a></span></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></li></ul></body></html></pre></div> | ||