31edt: Difference between revisions
Wikispaces>JosephRuhf **Imported revision 596368846 - Original comment: ** |
Wikispaces>JosephRuhf **Imported revision 596368930 - 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:JosephRuhf|JosephRuhf]] and made on <tt>2016-10-21 15:49 | : This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-10-21 15:50:49 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>596368930</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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|| 17-tone || [[xenharmonic/7L 10s|7L 10s]] || 1 || 1 || 3 || || || 1 || 3 || || || 1 || 1 || 3 || || || 1 || 3 || || || 1 || 1 || 3 || || || 1 || 3 || || || 1 || 3 || || || | || 17-tone || [[xenharmonic/7L 10s|7L 10s]] || 1 || 1 || 3 || || || 1 || 3 || || || 1 || 1 || 3 || || || 1 || 3 || || || 1 || 1 || 3 || || || 1 || 3 || || || 1 || 3 || || || | ||
|| 24-tone || [[xenharmonic/7L 17s|7L 17s]] || 1 || 1 || 1 || 2 || || 1 || 1 || 2 || || 1 || 1 || 1 || 2 || || 1 || 1 || 2 || || 1 || 1 || 1 || 2 || || 1 || 1 || 2 || || 1 || 1 || 2 || || | || 24-tone || [[xenharmonic/7L 17s|7L 17s]] || 1 || 1 || 1 || 2 || || 1 || 1 || 2 || || 1 || 1 || 1 || 2 || || 1 || 1 || 2 || || 1 || 1 || 1 || 2 || || 1 || 1 || 2 || || 1 || 1 || 2 || || | ||
===<span style="background-color: #ffffff;">[[#Intervals-Selected just intervals by error-10\31 octave - approx. 387.10¢ - Major Third]]10\31 octave - approx. 387.10¢ - Major Third</span>=== | ===<span style="background-color: #ffffff;">[[#Intervals-Selected just intervals by error-10\31 octave - approx. 387.10¢ - Major Third]]10\31 octave - approx. 387.10¢ - Major Third</span>=== | ||
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|| 19-tone || [[xenharmonic/12L 7s|12L 7s]] || 1 || 2 || || 1 || 2 || || 2 || || 1 || 2 || || 2 || || 1 || 2 || || 1 || 2 || || 2 || || 1 || 2 || || 2 || || 1 || 2 || || 2 || || | || 19-tone || [[xenharmonic/12L 7s|12L 7s]] || 1 || 2 || || 1 || 2 || || 2 || || 1 || 2 || || 2 || || 1 || 2 || || 1 || 2 || || 2 || || 1 || 2 || || 2 || || 1 || 2 || || 2 || || | ||
<span style="background-color: #ffffff; font-size: 1.1em;">**14\31 octave - approx. 541.94¢ - Superfourth**</span> | |||
<span style="background-color: #ffffff;">Exactly twice a subminor third. Functions as both the 11:8 (551.32¢) and 15:11 (536.95¢) undecimal superfourths (121/120 is tempered out). Thus it makes possible a symmetrical tempered version of an 8:11:15 triad. As 11/8, 14\31 is about 9¢ flat; however, it fits nicely with the also-flat 9/8, allowing a near-just 11/9. Nonetheless, most 11-limit chords in 31edo have a somewhat unstable quality which distinguishes them from their just counterparts. Generates </span>[[xenharmonic/Starling temperaments|casablanca temperament]]<span style="background-color: #ffffff;">.</span> | <span style="background-color: #ffffff;">Exactly twice a subminor third. Functions as both the 11:8 (551.32¢) and 15:11 (536.95¢) undecimal superfourths (121/120 is tempered out). Thus it makes possible a symmetrical tempered version of an 8:11:15 triad. As 11/8, 14\31 is about 9¢ flat; however, it fits nicely with the also-flat 9/8, allowing a near-just 11/9. Nonetheless, most 11-limit chords in 31edo have a somewhat unstable quality which distinguishes them from their just counterparts. Generates </span>[[xenharmonic/Starling temperaments|casablanca temperament]]<span style="background-color: #ffffff;">.</span> | ||
====<span style="background-color: #ffffff;">[[#Intervals-Selected just intervals by error-14\31 octave - approx. 541.94¢ - Superfourth-MOS Scales generated by 14\31:]]MOS Scales generated by 14\31:</span>==== | ====<span style="background-color: #ffffff;">[[#Intervals-Selected just intervals by error-14\31 octave - approx. 541.94¢ - Superfourth-MOS Scales generated by 14\31:]]MOS Scales generated by 14\31:</span>==== | ||
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|| 11-tone (quasi-equal) || [[xenharmonic/9L 2s|9L 2s]] || 2 || || 3 || || || 3 || || || 3 || || || 3 || || || 2 || || 3 || || || 3 || || || 3 || || || 3 || || || 3 || || || | || 11-tone (quasi-equal) || [[xenharmonic/9L 2s|9L 2s]] || 2 || || 3 || || || 3 || || || 3 || || || 3 || || || 2 || || 3 || || || 3 || || || 3 || || || 3 || || || 3 || || || | ||
|| 20-tone || [[xenharmonic/11L 9s|11L 9s]] || 2 || || 2 || || 1 || 2 || || 1 || 2 || || 1 || 2 || || 1 || 2 || || 2 || || 1 || 2 || || 1 || 2 || || 1 || 2 || || 1 || 2 || || 1 || | || 20-tone || [[xenharmonic/11L 9s|11L 9s]] || 2 || || 2 || || 1 || 2 || || 1 || 2 || || 1 || 2 || || 1 || 2 || || 2 || || 1 || 2 || || 1 || 2 || || 1 || 2 || || 1 || 2 || || 1 || | ||
===<span style="background-color: #ffffff;">[[#Intervals-Selected just intervals by error-15\31 octave - approx. 580.65¢ - Small Tritone or Augmented Fourth or Subdiminished Fifth]]15\31 octave - approx. 580.65¢ - Small Tritone or Augmented Fourth or Subdiminished Fifth</span>=== | ===<span style="background-color: #ffffff;">[[#Intervals-Selected just intervals by error-15\31 octave - approx. 580.65¢ - Small Tritone or Augmented Fourth or Subdiminished Fifth]]15\31 octave - approx. 580.65¢ - Small Tritone or Augmented Fourth or Subdiminished Fifth</span>=== | ||
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<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="background-color: #ffffff;">Intervals</span></h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="background-color: #ffffff;">Intervals</span></h1> | ||
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<!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="Intervals--1\31 tritave- approx. 61.35¢ - Third tone"></a><!-- ws:end:WikiTextHeadingRule:2 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="Intervals--1\31 tritave- approx. 61.35¢ - Third tone"></a><!-- ws:end:WikiTextHeadingRule:2 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule:58:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Intervals-Selected just intervals by error-1\31 octave - approx. 38.71¢ - Diesis&quot; title=&quot;Anchor: Intervals-Selected just intervals by error-1\31 octave - approx. 38.71¢ - Diesis&quot;/&gt; --><a name="Intervals-Selected just intervals by error-1\31 octave - approx. 38.71¢ - Diesis"></a><!-- ws:end:WikiTextAnchorRule:58 -->1\31 tritave- approx. <span style="font-size: 1.1em;">61.35¢ - Third tone</span></span></h3> | ||
<span style="background-color: #ffffff;">A single step of 31-edt is about 6</span><span style="background-color: #ffffff;">1.35¢. Intervals around this size are called <em>third tones</em>. In 31 it is equivalent to the difference between one tritave 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). The third tone is a defining sound of 31edt; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size.</span><br /> | <span style="background-color: #ffffff;">A single step of 31-edt is about 6</span><span style="background-color: #ffffff;">1.35¢. Intervals around this size are called <em>third tones</em>. In 31 it is equivalent to the difference between one tritave 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). The third tone is a defining sound of 31edt; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size.</span><br /> | ||
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<!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="Intervals--2\31 tritave - approx. 122.71¢ - Two-third tone or Small Minor Second"></a><!-- ws:end:WikiTextHeadingRule:4 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="Intervals--2\31 tritave - approx. 122.71¢ - Two-third tone or Small Minor Second"></a><!-- ws:end:WikiTextHeadingRule:4 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule:59:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Intervals-Selected just intervals by error-2\31 octave - approx. 77.42¢ - Minor Semitone or Chromatic Semitone or Small Minor Second&quot; title=&quot;Anchor: Intervals-Selected just intervals by error-2\31 octave - approx. 77.42¢ - Minor Semitone or Chromatic Semitone or Small Minor Second&quot;/&gt; --><a name="Intervals-Selected just intervals by error-2\31 octave - approx. 77.42¢ - Minor Semitone or Chromatic Semitone or Small Minor Second"></a><!-- ws:end:WikiTextAnchorRule:59 -->2\31 tritave - approx. 122.71¢ - Two-third tone or Small Minor Second</span></h3> | ||
<span style="background-color: #ffffff;">The difference between a major and minor third and the thing to a 'half step'; in</span><br /> | <span style="background-color: #ffffff;">The difference between a major and minor third and the thing to a 'half step'; in</span><br /> | ||
<span style="background-color: #ffffff;">macromeantone, it is exactly analogus to the </span><em><span style="background-color: #ffffff;">chromatic semitone</span></em><span style="background-color: #ffffff;">, the interval that distinguishes major and minor intervals of the same generic interval class (eg. thirds). </span><br /> | <span style="background-color: #ffffff;">macromeantone, it is exactly analogus to the </span><em><span style="background-color: #ffffff;">chromatic semitone</span></em><span style="background-color: #ffffff;">, the interval that distinguishes major and minor intervals of the same generic interval class (eg. thirds). </span><br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h4&gt; --><h4 id="toc3"><a name="Intervals--2\31 tritave - approx. 122.71¢ - Two-third tone or Small Minor Second-MOS Scales generated by 2\31:"></a><!-- ws:end:WikiTextHeadingRule:6 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:6:&lt;h4&gt; --><h4 id="toc3"><a name="Intervals--2\31 tritave - approx. 122.71¢ - Two-third tone or Small Minor Second-MOS Scales generated by 2\31:"></a><!-- ws:end:WikiTextHeadingRule:6 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule:60:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Intervals-Selected just intervals by error-2\31 octave - approx. 77.42¢ - Minor Semitone or Chromatic Semitone or Small Minor Second-MOS Scales generated by 2\31:&quot; title=&quot;Anchor: Intervals-Selected just intervals by error-2\31 octave - approx. 77.42¢ - Minor Semitone or Chromatic Semitone or Small Minor Second-MOS Scales generated by 2\31:&quot;/&gt; --><a name="Intervals-Selected just intervals by error-2\31 octave - approx. 77.42¢ - Minor Semitone or Chromatic Semitone or Small Minor Second-MOS Scales generated by 2\31:"></a><!-- ws:end:WikiTextAnchorRule:60 -->MOS Scales generated by 2\31:</span></h4> | ||
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<!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="Intervals--3\31 tritave - approx. 184.06¢- Whole tone or Large Minor Second"></a><!-- ws:end:WikiTextHeadingRule:8 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="Intervals--3\31 tritave - approx. 184.06¢- Whole tone or Large Minor Second"></a><!-- ws:end:WikiTextHeadingRule:8 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule:61:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Intervals-Selected just intervals by error-3\31 octave - approx. 116.13¢- Major Semitone or Diatonic Semitone or Large Major Second&quot; title=&quot;Anchor: Intervals-Selected just intervals by error-3\31 octave - approx. 116.13¢- Major Semitone or Diatonic Semitone or Large Major Second&quot;/&gt; --><a name="Intervals-Selected just intervals by error-3\31 octave - approx. 116.13¢- Major Semitone or Diatonic Semitone or Large Major Second"></a><!-- ws:end:WikiTextAnchorRule:61 -->3\31 tritave - approx. 184.06¢- Whole tone or Large Minor Second</span></h3> | ||
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<span style="background-color: #ffffff;">A small whole tone only ~1.6 cents wide of 10:9 which is an interval sometimes called melodically dull; in macromeantone, it is exactly analogus to the </span><em><span style="background-color: #ffffff;">diatonic semitone</span></em><span style="background-color: #ffffff;"> which appears in the diatonic scale between, for instance, the major third and perfect fourth, and the major seventh and octave. </span><br /> | <span style="background-color: #ffffff;">A small whole tone only ~1.6 cents wide of 10:9 which is an interval sometimes called melodically dull; in macromeantone, it is exactly analogus to the </span><em><span style="background-color: #ffffff;">diatonic semitone</span></em><span style="background-color: #ffffff;"> which appears in the diatonic scale between, for instance, the major third and perfect fourth, and the major seventh and octave. </span><br /> | ||
<!-- ws:start:WikiTextHeadingRule:10:&lt;h4&gt; --><h4 id="toc5"><a name="Intervals--3\31 tritave - approx. 184.06¢- Whole tone or Large Minor Second-MOS Scales generated by 3\31:"></a><!-- ws:end:WikiTextHeadingRule:10 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h4&gt; --><h4 id="toc5"><a name="Intervals--3\31 tritave - approx. 184.06¢- Whole tone or Large Minor Second-MOS Scales generated by 3\31:"></a><!-- ws:end:WikiTextHeadingRule:10 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule:62:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Intervals-Selected just intervals by error-3\31 octave - approx. 116.13¢- Major Semitone or Diatonic Semitone or Large Major Second-MOS Scales generated by 3\31:&quot; title=&quot;Anchor: Intervals-Selected just intervals by error-3\31 octave - approx. 116.13¢- Major Semitone or Diatonic Semitone or Large Major Second-MOS Scales generated by 3\31:&quot;/&gt; --><a name="Intervals-Selected just intervals by error-3\31 octave - approx. 116.13¢- Major Semitone or Diatonic Semitone or Large Major Second-MOS Scales generated by 3\31:"></a><!-- ws:end:WikiTextAnchorRule:62 -->MOS Scales generated by 3\31:</span></h4> | ||
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<!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="Intervals--4\31 tritave - approx. 245.41¢ - Classical hemifourth or Neutral Second"></a><!-- ws:end:WikiTextHeadingRule:12 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="Intervals--4\31 tritave - approx. 245.41¢ - Classical hemifourth or Neutral Second"></a><!-- ws:end:WikiTextHeadingRule:12 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule:63:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Intervals-Selected just intervals by error-4\31 octave - approx. 154.84¢ - Neutral Tone or Neutral Second&quot; title=&quot;Anchor: Intervals-Selected just intervals by error-4\31 octave - approx. 154.84¢ - Neutral Tone or Neutral Second&quot;/&gt; --><a name="Intervals-Selected just intervals by error-4\31 octave - approx. 154.84¢ - Neutral Tone or Neutral Second"></a><!-- ws:end:WikiTextAnchorRule:63 -->4\31 tritave - approx. 245.41¢ - Classical hemifourth or Neutral Second</span></h3> | ||
<span style="background-color: #ffffff;">Exactly one half of the minor third and twice the minor semitone, 4\31 stands in for 15:13 (247.74¢). Although 31 is not extremely accurate with 5 or 13, it is notable that the inaccuracies of these harmonics cancel out so much, leaving the interval that distinguishes them (15/13) only about 2.3¢ off.</span><br /> | <span style="background-color: #ffffff;">Exactly one half of the minor third and twice the minor semitone, 4\31 stands in for 15:13 (247.74¢). Although 31 is not extremely accurate with 5 or 13, it is notable that the inaccuracies of these harmonics cancel out so much, leaving the interval that distinguishes them (15/13) only about 2.3¢ off.</span><br /> | ||
<!-- ws:start:WikiTextHeadingRule:14:&lt;h4&gt; --><h4 id="toc7"><a name="Intervals--4\31 tritave - approx. 245.41¢ - Classical hemifourth or Neutral Second-MOS Scales generated by 4\31:"></a><!-- ws:end:WikiTextHeadingRule:14 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h4&gt; --><h4 id="toc7"><a name="Intervals--4\31 tritave - approx. 245.41¢ - Classical hemifourth or Neutral Second-MOS Scales generated by 4\31:"></a><!-- ws:end:WikiTextHeadingRule:14 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule:64:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Intervals-Selected just intervals by error-4\31 octave - approx. 154.84¢ - Neutral Tone or Neutral Second-MOS Scales generated by 4\31:&quot; title=&quot;Anchor: Intervals-Selected just intervals by error-4\31 octave - approx. 154.84¢ - Neutral Tone or Neutral Second-MOS Scales generated by 4\31:&quot;/&gt; --><a name="Intervals-Selected just intervals by error-4\31 octave - approx. 154.84¢ - Neutral Tone or Neutral Second-MOS Scales generated by 4\31:"></a><!-- ws:end:WikiTextAnchorRule:64 -->MOS Scales generated by 4\31:</span></h4> | ||
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<!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="Intervals--5\31 tritave - approx. 306.77¢ - Sesquitone or Major Second"></a><!-- ws:end:WikiTextHeadingRule:16 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="Intervals--5\31 tritave - approx. 306.77¢ - Sesquitone or Major Second"></a><!-- ws:end:WikiTextHeadingRule:16 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule:65:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Intervals-Selected just intervals by error-5\31 octave - approx. 193.55¢ - Whole Tone or Major Second&quot; title=&quot;Anchor: Intervals-Selected just intervals by error-5\31 octave - approx. 193.55¢ - Whole Tone or Major Second&quot;/&gt; --><a name="Intervals-Selected just intervals by error-5\31 octave - approx. 193.55¢ - Whole Tone or Major Second"></a><!-- ws:end:WikiTextAnchorRule:65 -->5\31 tritave - approx. 306.77¢ - <span style="font-size: 1.1em;">Sesquitone or Major Second</span></span></h3> | ||
<span style="background-color: #ffffff;">At ~8.8 cents flat of a just 6:5, 5\31 is considered a &quot;sesquitone&quot;. Two of this sesquitone make a near-just 10:7 tritone. Because it is fairly close to no intelligibly small integer ratio but 6/5, 5\31 can function as a semi-stabilized harmonic ninth.</span><br /> | <span style="background-color: #ffffff;">At ~8.8 cents flat of a just 6:5, 5\31 is considered a &quot;sesquitone&quot;. Two of this sesquitone make a near-just 10:7 tritone. Because it is fairly close to no intelligibly small integer ratio but 6/5, 5\31 can function as a semi-stabilized harmonic ninth.</span><br /> | ||
<!-- ws:start:WikiTextHeadingRule:18:&lt;h4&gt; --><h4 id="toc9"><a name="Intervals--5\31 tritave - approx. 306.77¢ - Sesquitone or Major Second-MOS Scales generated by 5\31:"></a><!-- ws:end:WikiTextHeadingRule:18 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h4&gt; --><h4 id="toc9"><a name="Intervals--5\31 tritave - approx. 306.77¢ - Sesquitone or Major Second-MOS Scales generated by 5\31:"></a><!-- ws:end:WikiTextHeadingRule:18 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule:66:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Intervals-Selected just intervals by error-5\31 octave - approx. 193.55¢ - Whole Tone or Major Second-MOS Scales generated by 5\31:&quot; title=&quot;Anchor: Intervals-Selected just intervals by error-5\31 octave - approx. 193.55¢ - Whole Tone or Major Second-MOS Scales generated by 5\31:&quot;/&gt; --><a name="Intervals-Selected just intervals by error-5\31 octave - approx. 193.55¢ - Whole Tone or Major Second-MOS Scales generated by 5\31:"></a><!-- ws:end:WikiTextAnchorRule:66 -->MOS Scales generated by 5\31:</span></h4> | ||
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<!-- ws:start:WikiTextHeadingRule:20:&lt;h3&gt; --><h3 id="toc10"><a name="Intervals--6\31 tritave - approx. 368.12¢ - Supermajor Second"></a><!-- ws:end:WikiTextHeadingRule:20 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:20:&lt;h3&gt; --><h3 id="toc10"><a name="Intervals--6\31 tritave - approx. 368.12¢ - Supermajor Second"></a><!-- ws:end:WikiTextHeadingRule:20 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule:67:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Intervals-Selected just intervals by error-6\31 octave - approx. 232.26¢ - Supermajor Second&quot; title=&quot;Anchor: Intervals-Selected just intervals by error-6\31 octave - approx. 232.26¢ - Supermajor Second&quot;/&gt; --><a name="Intervals-Selected just intervals by error-6\31 octave - approx. 232.26¢ - Supermajor Second"></a><!-- ws:end:WikiTextAnchorRule:67 -->6\31 tritave - approx. 368.12¢ - Supermajor Second</span></h3> | ||
<span style="background-color: #ffffff;">Exactly one half of a narrow fourth, twice a tone, or thrice a two-third tone. In 17-limit tonal music, 6\31 closely represents </span><span style="background-color: #ffffff;">21:17 (365.825¢). In </span><span style="background-color: #ffffff;">macro</span><span style="background-color: #ffffff;">meantone, it is a diminished third, eg. C to Ebb.</span><br /> | <span style="background-color: #ffffff;">Exactly one half of a narrow fourth, twice a tone, or thrice a two-third tone. In 17-limit tonal music, 6\31 closely represents </span><span style="background-color: #ffffff;">21:17 (365.825¢). In </span><span style="background-color: #ffffff;">macro</span><span style="background-color: #ffffff;">meantone, it is a diminished third, eg. C to Ebb.</span><br /> | ||
<!-- ws:start:WikiTextHeadingRule:22:&lt;h4&gt; --><h4 id="toc11"><a name="Intervals--6\31 tritave - approx. 368.12¢ - Supermajor Second-MOS Scales generated by 6\31:"></a><!-- ws:end:WikiTextHeadingRule:22 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:22:&lt;h4&gt; --><h4 id="toc11"><a name="Intervals--6\31 tritave - approx. 368.12¢ - Supermajor Second-MOS Scales generated by 6\31:"></a><!-- ws:end:WikiTextHeadingRule:22 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule:68:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Intervals-Selected just intervals by error-6\31 octave - approx. 232.26¢ - Supermajor Second-MOS Scales generated by 6\31:&quot; title=&quot;Anchor: Intervals-Selected just intervals by error-6\31 octave - approx. 232.26¢ - Supermajor Second-MOS Scales generated by 6\31:&quot;/&gt; --><a name="Intervals-Selected just intervals by error-6\31 octave - approx. 232.26¢ - Supermajor Second-MOS Scales generated by 6\31:"></a><!-- ws:end:WikiTextAnchorRule:68 -->MOS Scales generated by 6\31:</span></h4> | ||
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<!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc12"><a name="Intervals--7\31 octave - approx. 429.47¢ - Subminor Third"></a><!-- ws:end:WikiTextHeadingRule:24 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc12"><a name="Intervals--7\31 octave - approx. 429.47¢ - Subminor Third"></a><!-- ws:end:WikiTextHeadingRule:24 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule:69:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Intervals-Selected just intervals by error-7\31 octave - approx. 270.97¢ - Subminor Third&quot; title=&quot;Anchor: Intervals-Selected just intervals by error-7\31 octave - approx. 270.97¢ - Subminor Third&quot;/&gt; --><a name="Intervals-Selected just intervals by error-7\31 octave - approx. 270.97¢ - Subminor Third"></a><!-- ws:end:WikiTextAnchorRule:69 -->7\31 octave - approx. 429.47¢ - Subminor Third</span></h3> | ||
<span style="background-color: #ffffff;">Exactly one half of a superfourth. In 7-limit tonal music, 7\31 stands in for 9:7 (435.08¢). In macromeantone temperament, it is an augmented 2nd, eg. C to D#.</span><br /> | <span style="background-color: #ffffff;">Exactly one half of a superfourth. In 7-limit tonal music, 7\31 stands in for 9:7 (435.08¢). In macromeantone temperament, it is an augmented 2nd, eg. C to D#.</span><br /> | ||
<!-- ws:start:WikiTextHeadingRule:26:&lt;h4&gt; --><h4 id="toc13"><a name="Intervals--7\31 octave - approx. 429.47¢ - Subminor Third-MOS Scales generated by 7\31:"></a><!-- ws:end:WikiTextHeadingRule:26 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:26:&lt;h4&gt; --><h4 id="toc13"><a name="Intervals--7\31 octave - approx. 429.47¢ - Subminor Third-MOS Scales generated by 7\31:"></a><!-- ws:end:WikiTextHeadingRule:26 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule:70:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Intervals-Selected just intervals by error-7\31 octave - approx. 270.97¢ - Subminor Third-MOS Scales generated by 7\31:&quot; title=&quot;Anchor: Intervals-Selected just intervals by error-7\31 octave - approx. 270.97¢ - Subminor Third-MOS Scales generated by 7\31:&quot;/&gt; --><a name="Intervals-Selected just intervals by error-7\31 octave - approx. 270.97¢ - Subminor Third-MOS Scales generated by 7\31:"></a><!-- ws:end:WikiTextAnchorRule:70 -->MOS Scales generated by 7\31:</span></h4> | ||
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<!-- ws:start:WikiTextHeadingRule:28:&lt;h3&gt; --><h3 id="toc14"><a name="Intervals--8\31 octave - approx. 490.83¢ - Minor Third"></a><!-- ws:end:WikiTextHeadingRule:28 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:28:&lt;h3&gt; --><h3 id="toc14"><a name="Intervals--8\31 octave - approx. 490.83¢ - Minor Third"></a><!-- ws:end:WikiTextHeadingRule:28 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule:71:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Intervals-Selected just intervals by error-8\31 octave - approx. 309.68¢ - Minor Third&quot; title=&quot;Anchor: Intervals-Selected just intervals by error-8\31 octave - approx. 309.68¢ - Minor Third&quot;/&gt; --><a name="Intervals-Selected just intervals by error-8\31 octave - approx. 309.68¢ - Minor Third"></a><!-- ws:end:WikiTextAnchorRule:71 -->8\31 octave - approx. 490.83¢ - Minor Third</span></h3> | ||
<span style="background-color: #ffffff;">A 4:3 ~1.3 syntonic comma flat. Exactly twice a neutral second, four times a minor semitone, and half of a large tritone.</span><br /> | <span style="background-color: #ffffff;">A 4:3 ~1.3 syntonic comma flat. Exactly twice a neutral second, four times a minor semitone, and half of a large tritone.</span><br /> | ||
<!-- ws:start:WikiTextHeadingRule:30:&lt;h4&gt; --><h4 id="toc15"><a name="Intervals--8\31 octave - approx. 490.83¢ - Minor Third-MOS Scales generated by 8\31:"></a><!-- ws:end:WikiTextHeadingRule:30 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:30:&lt;h4&gt; --><h4 id="toc15"><a name="Intervals--8\31 octave - approx. 490.83¢ - Minor Third-MOS Scales generated by 8\31:"></a><!-- ws:end:WikiTextHeadingRule:30 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule:72:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Intervals-Selected just intervals by error-8\31 octave - approx. 309.68¢ - Minor Third-MOS Scales generated by 8\31:&quot; title=&quot;Anchor: Intervals-Selected just intervals by error-8\31 octave - approx. 309.68¢ - Minor Third-MOS Scales generated by 8\31:&quot;/&gt; --><a name="Intervals-Selected just intervals by error-8\31 octave - approx. 309.68¢ - Minor Third-MOS Scales generated by 8\31:"></a><!-- ws:end:WikiTextAnchorRule:72 -->MOS Scales generated by 8\31:</span></h4> | ||
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<!-- ws:start:WikiTextHeadingRule:32:&lt;h3&gt; --><h3 id="toc16"><a name="Intervals--9\31 octave - approx. 348.39¢ - Neutral Third"></a><!-- ws:end:WikiTextHeadingRule:32 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:32:&lt;h3&gt; --><h3 id="toc16"><a name="Intervals--9\31 octave - approx. 348.39¢ - Neutral Third"></a><!-- ws:end:WikiTextHeadingRule:32 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule:73:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Intervals-Selected just intervals by error-9\31 octave - approx. 348.39¢ - Neutral Third&quot; title=&quot;Anchor: Intervals-Selected just intervals by error-9\31 octave - approx. 348.39¢ - Neutral Third&quot;/&gt; --><a name="Intervals-Selected just intervals by error-9\31 octave - approx. 348.39¢ - Neutral Third"></a><!-- ws:end:WikiTextAnchorRule:73 -->9\31 octave - approx. 348.39¢ - Neutral Third</span></h3> | ||
<span style="background-color: #ffffff;">A neutral 3rd, about 1¢ away from 11:9 (347.41¢). 9\31 is half a perfect fifth (making it a suitable generator for </span><a class="wiki_link" href="http://xenharmonic.wikispaces.com/mohajira">mohajira temperament</a><span style="background-color: #ffffff;">), and also thrice a major semitone. It is closer in quality to a minor third than a major third, but indeed, it is distinct. It is 11¢ shy of 16/13 (359.47¢), suggesting a </span><a class="wiki_link" href="http://xenharmonic.wikispaces.com/13-limit">13-limit</a><span style="background-color: #ffffff;"> interpretation for 31edo. However, its close proximity to 11/9 makes it hard to hear it as 16/13, which in JI has a different quality (and, as a neutral third, is more &quot;major-like&quot; than &quot;minor-like&quot;). Also, its inversion, 22\31 (851.61¢) is wide of the 13th harmonic by about 11¢, while all the lower harmonics are either near-just or narrow. This means the errors can accumulate, for instance, with 13/9 (636.62¢) represented by 17\31 (658.06¢), a good 21.4¢ sharp.</span><br /> | <span style="background-color: #ffffff;">A neutral 3rd, about 1¢ away from 11:9 (347.41¢). 9\31 is half a perfect fifth (making it a suitable generator for </span><a class="wiki_link" href="http://xenharmonic.wikispaces.com/mohajira">mohajira temperament</a><span style="background-color: #ffffff;">), and also thrice a major semitone. It is closer in quality to a minor third than a major third, but indeed, it is distinct. It is 11¢ shy of 16/13 (359.47¢), suggesting a </span><a class="wiki_link" href="http://xenharmonic.wikispaces.com/13-limit">13-limit</a><span style="background-color: #ffffff;"> interpretation for 31edo. However, its close proximity to 11/9 makes it hard to hear it as 16/13, which in JI has a different quality (and, as a neutral third, is more &quot;major-like&quot; than &quot;minor-like&quot;). Also, its inversion, 22\31 (851.61¢) is wide of the 13th harmonic by about 11¢, while all the lower harmonics are either near-just or narrow. This means the errors can accumulate, for instance, with 13/9 (636.62¢) represented by 17\31 (658.06¢), a good 21.4¢ sharp.</span><br /> | ||
<!-- ws:start:WikiTextHeadingRule:34:&lt;h4&gt; --><h4 id="toc17"><a name="Intervals--9\31 octave - approx. 348.39¢ - Neutral Third-MOS Scales generated by 9\31:"></a><!-- ws:end:WikiTextHeadingRule:34 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:34:&lt;h4&gt; --><h4 id="toc17"><a name="Intervals--9\31 octave - approx. 348.39¢ - Neutral Third-MOS Scales generated by 9\31:"></a><!-- ws:end:WikiTextHeadingRule:34 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule:74:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Intervals-Selected just intervals by error-9\31 octave - approx. 348.39¢ - Neutral Third-MOS Scales generated by 9\31:&quot; title=&quot;Anchor: Intervals-Selected just intervals by error-9\31 octave - approx. 348.39¢ - Neutral Third-MOS Scales generated by 9\31:&quot;/&gt; --><a name="Intervals-Selected just intervals by error-9\31 octave - approx. 348.39¢ - Neutral Third-MOS Scales generated by 9\31:"></a><!-- ws:end:WikiTextAnchorRule:74 -->MOS Scales generated by 9\31:</span></h4> | ||
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<!-- ws:start:WikiTextHeadingRule:36:&lt;h3&gt; --><h3 id="toc18"><a name="Intervals--10\31 octave - approx. 387.10¢ - Major Third"></a><!-- ws:end:WikiTextHeadingRule:36 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule:75:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Intervals-Selected just intervals by error-10\31 octave - approx. 387.10¢ - Major Third&quot; title=&quot;Anchor: Intervals-Selected just intervals by error-10\31 octave - approx. 387.10¢ - Major Third&quot;/&gt; --><a name="Intervals-Selected just intervals by error-10\31 octave - approx. 387.10¢ - Major Third"></a><!-- ws:end:WikiTextAnchorRule:75 -->10\31 octave - approx. 387.10¢ - Major Third</span></h3> | |||
<!-- ws:start:WikiTextHeadingRule:36:&lt;h3&gt; --><h3 id="toc18"><a name="Intervals--10\31 octave - approx. 387.10¢ - Major Third"></a><!-- ws:end:WikiTextHeadingRule:36 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule: | |||
<span style="background-color: #ffffff;">A near-just major 3rd (compare with 5:4 = 386.31¢). Has led to the characterization of 31-edo as &quot;smooth&quot;. Generates </span><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Wuerschmidt%20family">wurshmidt/worshmidt temperaments</a><span style="background-color: #ffffff;">.</span><br /> | <span style="background-color: #ffffff;">A near-just major 3rd (compare with 5:4 = 386.31¢). Has led to the characterization of 31-edo as &quot;smooth&quot;. Generates </span><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Wuerschmidt%20family">wurshmidt/worshmidt temperaments</a><span style="background-color: #ffffff;">.</span><br /> | ||
<!-- ws:start:WikiTextHeadingRule:38:&lt;h4&gt; --><h4 id="toc19"><a name="Intervals--10\31 octave - approx. 387.10¢ - Major Third-MOS Scales generated by 10\31:"></a><!-- ws:end:WikiTextHeadingRule:38 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:38:&lt;h4&gt; --><h4 id="toc19"><a name="Intervals--10\31 octave - approx. 387.10¢ - Major Third-MOS Scales generated by 10\31:"></a><!-- ws:end:WikiTextHeadingRule:38 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule:76:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Intervals-Selected just intervals by error-10\31 octave - approx. 387.10¢ - Major Third-MOS Scales generated by 10\31:&quot; title=&quot;Anchor: Intervals-Selected just intervals by error-10\31 octave - approx. 387.10¢ - Major Third-MOS Scales generated by 10\31:&quot;/&gt; --><a name="Intervals-Selected just intervals by error-10\31 octave - approx. 387.10¢ - Major Third-MOS Scales generated by 10\31:"></a><!-- ws:end:WikiTextAnchorRule:76 -->MOS Scales generated by 10\31:</span></h4> | ||
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<!-- ws:start:WikiTextHeadingRule:40:&lt;h3&gt; --><h3 id="toc20"><a name="Intervals--11\31 octave - approx. 425.806¢ - Supermajor Third"></a><!-- ws:end:WikiTextHeadingRule:40 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:40:&lt;h3&gt; --><h3 id="toc20"><a name="Intervals--11\31 octave - approx. 425.806¢ - Supermajor Third"></a><!-- ws:end:WikiTextHeadingRule:40 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule:77:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Intervals-Selected just intervals by error-11\31 octave - approx. 425.806¢ - Supermajor Third&quot; title=&quot;Anchor: Intervals-Selected just intervals by error-11\31 octave - approx. 425.806¢ - Supermajor Third&quot;/&gt; --><a name="Intervals-Selected just intervals by error-11\31 octave - approx. 425.806¢ - Supermajor Third"></a><!-- ws:end:WikiTextAnchorRule:77 -->11\31 octave - approx. 425.806¢ - Supermajor Third</span></h3> | ||
<span style="background-color: #ffffff;">11\31 functions as 14:11 (417.51¢), 23:18 (424.36¢), 32:25 (427.37¢), 9:7 (435.08¢) and others. In meantone temperament, it is a diminished fourth, eg. C to Fb. It is notable as closely approximating an interval of the </span><a class="wiki_link" href="http://xenharmonic.wikispaces.com/23-limit">23-limit</a><span style="background-color: #ffffff;">, suggesting the possibility of treating 16\31 (619.35¢) as a flat version of 23/16 (628.27¢). It is perhaps also notable for being close to 6\17, the bright major third of the ever-popular </span><a class="wiki_link" href="http://xenharmonic.wikispaces.com/17edo">17edo</a><span style="background-color: #ffffff;">. Generates </span><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Meantone%20family">squares temperament</a><span style="background-color: #ffffff;">.</span><br /> | <span style="background-color: #ffffff;">11\31 functions as 14:11 (417.51¢), 23:18 (424.36¢), 32:25 (427.37¢), 9:7 (435.08¢) and others. In meantone temperament, it is a diminished fourth, eg. C to Fb. It is notable as closely approximating an interval of the </span><a class="wiki_link" href="http://xenharmonic.wikispaces.com/23-limit">23-limit</a><span style="background-color: #ffffff;">, suggesting the possibility of treating 16\31 (619.35¢) as a flat version of 23/16 (628.27¢). It is perhaps also notable for being close to 6\17, the bright major third of the ever-popular </span><a class="wiki_link" href="http://xenharmonic.wikispaces.com/17edo">17edo</a><span style="background-color: #ffffff;">. Generates </span><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Meantone%20family">squares temperament</a><span style="background-color: #ffffff;">.</span><br /> | ||
<!-- ws:start:WikiTextHeadingRule:42:&lt;h4&gt; --><h4 id="toc21"><a name="Intervals--11\31 octave - approx. 425.806¢ - Supermajor Third-MOS Scales generated by 11\31:"></a><!-- ws:end:WikiTextHeadingRule:42 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:42:&lt;h4&gt; --><h4 id="toc21"><a name="Intervals--11\31 octave - approx. 425.806¢ - Supermajor Third-MOS Scales generated by 11\31:"></a><!-- ws:end:WikiTextHeadingRule:42 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule:78:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Intervals-Selected just intervals by error-11\31 octave - approx. 425.806¢ - Supermajor Third-MOS Scales generated by 11\31:&quot; title=&quot;Anchor: Intervals-Selected just intervals by error-11\31 octave - approx. 425.806¢ - Supermajor Third-MOS Scales generated by 11\31:&quot;/&gt; --><a name="Intervals-Selected just intervals by error-11\31 octave - approx. 425.806¢ - Supermajor Third-MOS Scales generated by 11\31:"></a><!-- ws:end:WikiTextAnchorRule:78 -->MOS Scales generated by 11\31:</span></h4> | ||
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<!-- ws:start:WikiTextHeadingRule:44:&lt;h3&gt; --><h3 id="toc22"><a name="Intervals--12\31 octave - approx. 464.52¢ - Narrow Fourth or Subfourth"></a><!-- ws:end:WikiTextHeadingRule:44 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:44:&lt;h3&gt; --><h3 id="toc22"><a name="Intervals--12\31 octave - approx. 464.52¢ - Narrow Fourth or Subfourth"></a><!-- ws:end:WikiTextHeadingRule:44 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule:79:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Intervals-Selected just intervals by error-12\31 octave - approx. 464.52¢ - Narrow Fourth or Subfourth&quot; title=&quot;Anchor: Intervals-Selected just intervals by error-12\31 octave - approx. 464.52¢ - Narrow Fourth or Subfourth&quot;/&gt; --><a name="Intervals-Selected just intervals by error-12\31 octave - approx. 464.52¢ - Narrow Fourth or Subfourth"></a><!-- ws:end:WikiTextAnchorRule:79 -->12\31 octave - approx. 464.52¢ - Narrow Fourth or Subfourth</span></h3> | ||
<span style="background-color: #ffffff;">Exactly twice a supermajor second, thrice a neutral second, or four times a minor second. In the 7-limit, 12\31 functions as 21:16 (470.78¢). It is also quite close to the </span><a class="wiki_link" href="http://xenharmonic.wikispaces.com/17-limit">17-limit</a><span style="background-color: #ffffff;">interval 17/13 (464.43¢), although 31edo does not offer up reasonable approximations of the 17th or 13th harmonics to help make this identity clear. This interval and its inversion 19\31 (735.48¢, a superfifth) are notable for being the only intervals in the 31edo octave larger than the 3\31 diatonic semitone (and smaller than its inversion, 28\31) that are not 11-limit consonances. Generates </span><a class="wiki_link" href="http://xenharmonic.wikispaces.com/semisept">semisept</a><span style="background-color: #ffffff;"> temperament.</span><br /> | <span style="background-color: #ffffff;">Exactly twice a supermajor second, thrice a neutral second, or four times a minor second. In the 7-limit, 12\31 functions as 21:16 (470.78¢). It is also quite close to the </span><a class="wiki_link" href="http://xenharmonic.wikispaces.com/17-limit">17-limit</a><span style="background-color: #ffffff;">interval 17/13 (464.43¢), although 31edo does not offer up reasonable approximations of the 17th or 13th harmonics to help make this identity clear. This interval and its inversion 19\31 (735.48¢, a superfifth) are notable for being the only intervals in the 31edo octave larger than the 3\31 diatonic semitone (and smaller than its inversion, 28\31) that are not 11-limit consonances. Generates </span><a class="wiki_link" href="http://xenharmonic.wikispaces.com/semisept">semisept</a><span style="background-color: #ffffff;"> temperament.</span><br /> | ||
<!-- ws:start:WikiTextHeadingRule:46:&lt;h4&gt; --><h4 id="toc23"><a name="Intervals--12\31 octave - approx. 464.52¢ - Narrow Fourth or Subfourth-MOS Scales generated by 12\31:"></a><!-- ws:end:WikiTextHeadingRule:46 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:46:&lt;h4&gt; --><h4 id="toc23"><a name="Intervals--12\31 octave - approx. 464.52¢ - Narrow Fourth or Subfourth-MOS Scales generated by 12\31:"></a><!-- ws:end:WikiTextHeadingRule:46 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule:80:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Intervals-Selected just intervals by error-12\31 octave - approx. 464.52¢ - Narrow Fourth or Subfourth-MOS Scales generated by 12\31:&quot; title=&quot;Anchor: Intervals-Selected just intervals by error-12\31 octave - approx. 464.52¢ - Narrow Fourth or Subfourth-MOS Scales generated by 12\31:&quot;/&gt; --><a name="Intervals-Selected just intervals by error-12\31 octave - approx. 464.52¢ - Narrow Fourth or Subfourth-MOS Scales generated by 12\31:"></a><!-- ws:end:WikiTextAnchorRule:80 -->MOS Scales generated by 12\31:</span></h4> | ||
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<!-- ws:start:WikiTextHeadingRule:48:&lt;h3&gt; --><h3 id="toc24"><a name="Intervals--13\31 octave - approx. 503.23¢ - Perfect Fourth"></a><!-- ws:end:WikiTextHeadingRule:48 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:48:&lt;h3&gt; --><h3 id="toc24"><a name="Intervals--13\31 octave - approx. 503.23¢ - Perfect Fourth"></a><!-- ws:end:WikiTextHeadingRule:48 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule:81:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Intervals-Selected just intervals by error-13\31 octave - approx. 503.23¢ - Perfect Fourth&quot; title=&quot;Anchor: Intervals-Selected just intervals by error-13\31 octave - approx. 503.23¢ - Perfect Fourth&quot;/&gt; --><a name="Intervals-Selected just intervals by error-13\31 octave - approx. 503.23¢ - Perfect Fourth"></a><!-- ws:end:WikiTextAnchorRule:81 -->13\31 octave - approx. 503.23¢ - Perfect Fourth</span></h3> | ||
<span style="background-color: #ffffff;">A slightly wide perfect fourth (compare to 4:3 = 498.04¢). As such, it functions marvelously as a generator for </span><a class="wiki_link" href="http://xenharmonic.wikispaces.com/meantone">meantone</a><span style="background-color: #ffffff;"> temperament.</span><br /> | <span style="background-color: #ffffff;">A slightly wide perfect fourth (compare to 4:3 = 498.04¢). As such, it functions marvelously as a generator for </span><a class="wiki_link" href="http://xenharmonic.wikispaces.com/meantone">meantone</a><span style="background-color: #ffffff;"> temperament.</span><br /> | ||
<!-- ws:start:WikiTextHeadingRule:50:&lt;h4&gt; --><h4 id="toc25"><a name="Intervals--13\31 octave - approx. 503.23¢ - Perfect Fourth-MOS Scales generated by 13\31:"></a><!-- ws:end:WikiTextHeadingRule:50 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:50:&lt;h4&gt; --><h4 id="toc25"><a name="Intervals--13\31 octave - approx. 503.23¢ - Perfect Fourth-MOS Scales generated by 13\31:"></a><!-- ws:end:WikiTextHeadingRule:50 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule:82:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Intervals-Selected just intervals by error-13\31 octave - approx. 503.23¢ - Perfect Fourth-MOS Scales generated by 13\31:&quot; title=&quot;Anchor: Intervals-Selected just intervals by error-13\31 octave - approx. 503.23¢ - Perfect Fourth-MOS Scales generated by 13\31:&quot;/&gt; --><a name="Intervals-Selected just intervals by error-13\31 octave - approx. 503.23¢ - Perfect Fourth-MOS Scales generated by 13\31:"></a><!-- ws:end:WikiTextAnchorRule:82 -->MOS Scales generated by 13\31:</span></h4> | ||
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<span style="background-color: #ffffff; font-size: 1.1em;"><strong>14\31 octave - approx. 541.94¢ - Superfourth</strong></span><br /> | |||
<span style="background-color: #ffffff;">Exactly twice a subminor third. Functions as both the 11:8 (551.32¢) and 15:11 (536.95¢) undecimal superfourths (121/120 is tempered out). Thus it makes possible a symmetrical tempered version of an 8:11:15 triad. As 11/8, 14\31 is about 9¢ flat; however, it fits nicely with the also-flat 9/8, allowing a near-just 11/9. Nonetheless, most 11-limit chords in 31edo have a somewhat unstable quality which distinguishes them from their just counterparts. Generates </span><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Starling%20temperaments">casablanca temperament</a><span style="background-color: #ffffff;">.</span><br /> | |||
<!-- 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 14\31:"></a><!-- ws:end:WikiTextHeadingRule:52 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule:83:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Intervals-Selected just intervals by error-14\31 octave - approx. 541.94¢ - Superfourth-MOS Scales generated by 14\31:&quot; title=&quot;Anchor: Intervals-Selected just intervals by error-14\31 octave - approx. 541.94¢ - Superfourth-MOS Scales generated by 14\31:&quot;/&gt; --><a name="Intervals-Selected just intervals by error-14\31 octave - approx. 541.94¢ - Superfourth-MOS Scales generated by 14\31:"></a><!-- ws:end:WikiTextAnchorRule:83 -->MOS Scales generated by 14\31:</span></h4> | |||
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<!-- ws:start:WikiTextHeadingRule:54:&lt;h3&gt; --><h3 id="toc27"><a name="Intervals--15\31 octave - approx. 580.65¢ - Small Tritone or Augmented Fourth or Subdiminished Fifth"></a><!-- ws:end:WikiTextHeadingRule:54 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule:84:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Intervals-Selected just intervals by error-15\31 octave - approx. 580.65¢ - Small Tritone or Augmented Fourth or Subdiminished Fifth&quot; title=&quot;Anchor: Intervals-Selected just intervals by error-15\31 octave - approx. 580.65¢ - Small Tritone or Augmented Fourth or Subdiminished Fifth&quot;/&gt; --><a name="Intervals-Selected just intervals by error-15\31 octave - approx. 580.65¢ - Small Tritone or Augmented Fourth or Subdiminished Fifth"></a><!-- ws:end:WikiTextAnchorRule:84 -->15\31 octave - approx. 580.65¢ - Small Tritone or Augmented Fourth or Subdiminished Fifth</span></h3> | |||
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<span style="background-color: #ffffff;">In 7-limit tonal music, functions quite well as 7:5 (582.51¢). Exactly thrice a whole tone. Generates </span><a class="wiki_link" href="http://xenharmonic.wikispaces.com/tritonic">tritonic</a><span style="background-color: #ffffff;"> temperament.</span><br /> | <span style="background-color: #ffffff;">In 7-limit tonal music, functions quite well as 7:5 (582.51¢). Exactly thrice a whole tone. Generates </span><a class="wiki_link" href="http://xenharmonic.wikispaces.com/tritonic">tritonic</a><span style="background-color: #ffffff;"> temperament.</span><br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:56:&lt;h4&gt; --><h4 id="toc28"><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:56 --><span style="background-color: #ffffff;"><!-- ws:start:WikiTextAnchorRule:85:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Intervals-Selected just intervals by error-15\31 octave - approx. 580.65¢ - Small Tritone or Augmented Fourth or Subdiminished Fifth-MOS Scales generated by 15\31:&quot; title=&quot;Anchor: Intervals-Selected just intervals by error-15\31 octave - approx. 580.65¢ - Small Tritone or Augmented Fourth or Subdiminished Fifth-MOS Scales generated by 15\31:&quot;/&gt; --><a name="Intervals-Selected just intervals by error-15\31 octave - approx. 580.65¢ - Small Tritone or Augmented Fourth or Subdiminished Fifth-MOS Scales generated by 15\31:"></a><!-- ws:end:WikiTextAnchorRule:85 -->MOS Scales generated by 15\31:</span></h4> | ||