31edo: Difference between revisions
Wikispaces>guest **Imported revision 150596635 - Original comment: ** |
Wikispaces>Andrew_Heathwaite **Imported revision 160282345 - 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>2010-09-03 14:24:23 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>160282345</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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<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]] | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]] | ||
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//Thirty-one tone equal temperament//, also called //31-tET//, //31-EDO//, //31-et//, or //tricesimoprimal meantone temperament//, is the scale derived by dividing the octave into 31 [[equal|equally]] large steps. The term 'Tricesimoprimal' was first used by Adriaan Fokker. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. | //Thirty-one tone equal temperament//, also called //31-tET//, //31-EDO//, //31-et//, or //tricesimoprimal meantone temperament//, is the scale derived by dividing the octave into 31 [[equal|equally]] large steps. The term 'Tricesimoprimal' was first used by [[Adriaan Fokker]]. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. 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, http://www.research.att.com/~njas/sequences/A117538. 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 [[36edo]]. | ||
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]]. | ||
=Intervals= | =Intervals= | ||
=== | ===1\31 octave - approx. 38.71¢ - Diesis=== | ||
A single step of 31-edo is about 38. | A single step of 31-edo is about 38.71¢. Intervals around this size are called [[diesis|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 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 [[SpiralProgressions]]. | ||
===2 | ===2\31 octave - approx. 77.42¢ - Minor Semitone or Chromatic Semitone or Small Minor Second=== | ||
The difference between a major and minor third. The more 'expressive' of the 'half steps'. | 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. | ||
===3 | ===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'. | 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. | ||
=== | ===4\31 octave - approx. 154.84¢ - Neutral Tone or Neutral Second=== | ||
Exactly one half of the minor third. | 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. | ||
===5 | ===5\31 octave - approx. 193.55¢ - Whole Tone or Major Second=== | ||
A rather smallish whole tone. Often called melodically dull. | 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. | ||
===6 | ===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¢). | |||
===7 | ===7\31 octave - approx. 270.97¢ - Subminor Third=== | ||
Exactly one half of a superfourth (11:8 approximation). In 7-limit tonal music, 7\31 stands in for 7:6 (266.87¢). | |||
===8 | ===8\31 octave - approx. 309.68¢ - Minor Third=== | ||
A minor third, closer to the just 6:5 than 12-edo, | 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. | ||
===9 | ===9\31 octave - approx. 348.39¢ - Neutral Third=== | ||
A neutral 3rd, practically equivalent to 11:9. | 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 [[3L 4s]]). Is also thrice a major semitone. | ||
===10 | ===10\31 octave - approx. 387.10¢ - Major Third=== | ||
A near-just major 3rd. Has led to the characterization of 31-edo as "smooth". | A near-just major 3rd (compare with 5:4 = 386.31¢). Has led to the characterization of 31-edo as "smooth". | ||
===11 | ===11\31 octave - approx. 425.806¢ - Supermajor Third=== | ||
In 11-limit tonal music, 11\31 functions as 14:11 (417.51¢), 32:25 (427.37¢), 9:7 (235.08¢) and others. | |||
===12 | ===12\31 octave - approx. 464.52¢ - Narrow Fourth or Subfourth=== | ||
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¢). | |||
===13/31 octave=== | ===13/31 octave - approx. 503.23¢ - Perfect Fourth=== | ||
A sharp perfect fourth (compare to 4:3 = 498.04¢). As such, it functions marvelously as a generator for meantone temperament. | |||
===14/31 octave=== | ===14/31 octave - approx. 541.94¢ - Superfourth=== | ||
10¢ off from a just 11:8; barely functional as such. | 10¢ off from a just 11:8 (551.32¢); barely functional as such. | ||
===15/31 octave=== | ===15/31 octave=== | ||
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<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:50:&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:50 --><!-- ws:start:WikiTextTocRule:51: --><a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --><!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --><!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --><!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --><!-- 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: --> | <a href="#Modes">Modes</a><!-- ws:end:WikiTextTocRule:70 --><!-- ws:start:WikiTextTocRule:71: --><!-- ws:end:WikiTextTocRule:71 --><!-- ws:start:WikiTextTocRule:72: --> | <a href="#Music in 31-edo">Music in 31-edo</a><!-- ws:end:WikiTextTocRule:72 --><!-- ws:start:WikiTextTocRule:73: --><!-- ws:end:WikiTextTocRule:73 --><!-- ws:start:WikiTextTocRule:74: --> | <a href="#Practical Theory / Books">Practical Theory / Books</a><!-- ws:end:WikiTextTocRule:74 --><!-- ws:start:WikiTextTocRule:75: --> | <a href="#Other Articles">Other Articles</a><!-- ws:end:WikiTextTocRule:75 --><!-- ws:start:WikiTextTocRule:76: --> | <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:50:&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:50 --><!-- ws:start:WikiTextTocRule:51: --><a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --><!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --><!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --><!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --><!-- 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: --> | <a href="#Modes">Modes</a><!-- ws:end:WikiTextTocRule:70 --><!-- ws:start:WikiTextTocRule:71: --><!-- ws:end:WikiTextTocRule:71 --><!-- ws:start:WikiTextTocRule:72: --> | <a href="#Music in 31-edo">Music in 31-edo</a><!-- ws:end:WikiTextTocRule:72 --><!-- ws:start:WikiTextTocRule:73: --><!-- ws:end:WikiTextTocRule:73 --><!-- ws:start:WikiTextTocRule:74: --> | <a href="#Practical Theory / Books">Practical Theory / Books</a><!-- ws:end:WikiTextTocRule:74 --><!-- ws:start:WikiTextTocRule:75: --> | <a href="#Other Articles">Other Articles</a><!-- ws:end:WikiTextTocRule:75 --><!-- ws:start:WikiTextTocRule:76: --> | ||
<!-- ws:end:WikiTextTocRule:76 --><hr /> | <!-- ws:end:WikiTextTocRule:76 --><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 Adriaan Fokker. 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>. | <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:871: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:871 -->. 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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:0 -->Intervals</h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:0 -->Intervals</h1> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="Intervals- | <!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="Intervals--1\31 octave - approx. 38.71¢ - Diesis"></a><!-- ws:end:WikiTextHeadingRule:2 -->1\31 octave - approx. 38.71¢ - Diesis</h3> | ||
A single step of 31-edo is about 38. | 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:4:&lt;h3&gt; --><h3 id="toc2"><a name="Intervals--2 | <!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="Intervals--2\31 octave - approx. 77.42¢ - Minor Semitone or Chromatic Semitone or Small Minor Second"></a><!-- ws:end:WikiTextHeadingRule:4 -->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'.<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.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="Intervals--3 | <!-- 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'.<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.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="Intervals- | <!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="Intervals--4\31 octave - approx. 154.84¢ - Neutral Tone or Neutral Second"></a><!-- ws:end:WikiTextHeadingRule:8 -->4\31 octave - approx. 154.84¢ - Neutral Tone or Neutral Second</h3> | ||
Exactly one half of the minor third.<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.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="Intervals--5 | <!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="Intervals--5\31 octave - approx. 193.55¢ - Whole Tone or Major Second"></a><!-- ws:end:WikiTextHeadingRule:10 -->5\31 octave - approx. 193.55¢ - Whole Tone or Major Second</h3> | ||
A rather smallish whole tone. Often called melodically dull.<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.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="Intervals--6 | <!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="Intervals--6\31 octave - approx. 232.26¢ - Supermajor Second"></a><!-- ws:end:WikiTextHeadingRule:12 -->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¢).<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="Intervals--7 | <!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="Intervals--7\31 octave - approx. 270.97¢ - Subminor Third"></a><!-- ws:end:WikiTextHeadingRule:14 -->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¢).<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="Intervals--8 | <!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="Intervals--8\31 octave - approx. 309.68¢ - Minor Third"></a><!-- ws:end:WikiTextHeadingRule:16 -->8\31 octave - approx. 309.68¢ - Minor Third</h3> | ||
A minor third, closer to the just 6:5 than 12-edo, | 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.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:18:&lt;h3&gt; --><h3 id="toc9"><a name="Intervals--9 | <!-- ws:start:WikiTextHeadingRule:18:&lt;h3&gt; --><h3 id="toc9"><a name="Intervals--9\31 octave - approx. 348.39¢ - Neutral Third"></a><!-- ws:end:WikiTextHeadingRule:18 -->9\31 octave - approx. 348.39¢ - Neutral Third</h3> | ||
A neutral 3rd, practically equivalent to 11:9.<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.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:20:&lt;h3&gt; --><h3 id="toc10"><a name="Intervals--10 | <!-- ws:start:WikiTextHeadingRule:20:&lt;h3&gt; --><h3 id="toc10"><a name="Intervals--10\31 octave - approx. 387.10¢ - Major Third"></a><!-- ws:end:WikiTextHeadingRule:20 -->10\31 octave - approx. 387.10¢ - Major Third</h3> | ||
A near-just major 3rd. Has led to the characterization of 31-edo as &quot;smooth&quot;.<br /> | A near-just major 3rd (compare with 5:4 = 386.31¢). Has led to the characterization of 31-edo as &quot;smooth&quot;.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:22:&lt;h3&gt; --><h3 id="toc11"><a name="Intervals--11 | <!-- ws:start:WikiTextHeadingRule:22:&lt;h3&gt; --><h3 id="toc11"><a name="Intervals--11\31 octave - approx. 425.806¢ - Supermajor Third"></a><!-- ws:end:WikiTextHeadingRule:22 -->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 (235.08¢) and others.<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc12"><a name="Intervals--12 | <!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc12"><a name="Intervals--12\31 octave - approx. 464.52¢ - Narrow Fourth or Subfourth"></a><!-- ws:end:WikiTextHeadingRule:24 -->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¢).<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc13"><a name="Intervals--13/31 octave"></a><!-- ws:end:WikiTextHeadingRule:26 -->13/31 octave</h3> | <!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc13"><a name="Intervals--13/31 octave - approx. 503.23¢ - Perfect Fourth"></a><!-- ws:end:WikiTextHeadingRule:26 -->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 /> | |||
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<!-- ws:start:WikiTextHeadingRule:28:&lt;h3&gt; --><h3 id="toc14"><a name="Intervals--14/31 octave"></a><!-- ws:end:WikiTextHeadingRule:28 -->14/31 octave</h3> | <!-- ws:start:WikiTextHeadingRule:28:&lt;h3&gt; --><h3 id="toc14"><a name="Intervals--14/31 octave - approx. 541.94¢ - Superfourth"></a><!-- ws:end:WikiTextHeadingRule:28 -->14/31 octave - approx. 541.94¢ - Superfourth</h3> | ||
10¢ off from a just 11:8; barely functional as such.<br /> | 10¢ off from a just 11:8 (551.32¢); barely functional as such.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:30:&lt;h3&gt; --><h3 id="toc15"><a name="Intervals--15/31 octave"></a><!-- ws:end:WikiTextHeadingRule:30 -->15/31 octave</h3> | <!-- ws:start:WikiTextHeadingRule:30:&lt;h3&gt; --><h3 id="toc15"><a name="Intervals--15/31 octave"></a><!-- ws:end:WikiTextHeadingRule:30 -->15/31 octave</h3> | ||