31edo: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>

: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2012-08-26 10:45:38 UTC</tt>.<br>

: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2012-08-26 11:00:21 UTC</tt>.<br>

: The original revision id was <tt>~~359972909~~</tt>.<br>

: The original revision id was <tt>359974327</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>

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===9\31 octave - approx. 348.39¢ - Neutral Third===

A neutral 3rd, about 1¢ away from 11:9 (347.41¢). 9\31 is half a perfect fifth (making it a suitable generator for [[mohajira|mohajira temperament]]), and also thrice a major semitone. It is closer in quality to a minor third than a major third, but indeed, it is distinct.

A neutral 3rd, about 1¢ away from 11:9 (347.41¢). 9\31 is half a perfect fifth (making it a suitable generator for [[mohajira|mohajira temperament]]), 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 [[13-limit]] 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 "major-like" than "minor-like"). 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.

====MOS Scales generated by 9\31:====

||~ number of tones ||~ MOS class ||~ 0 ||~ 1 ||~ 2 ||~ 3 ||~ 4 ||~ 5 ||~ 6 ||~ 7 ||~ 8 ||~ 9 ||~ 10 ||~ 11 ||~ 12 ||~ 13 ||~ 14 ||~ 15 ||~ 16 ||~ 17 ||~ 18 ||~ 19 ||~ 20 ||~ 21 ||~ 22 ||~ 23 ||~ 24 ||~ 25 ||~ 26 ||~ 27 ||~ 28 ||~ 29 ||~ 30 ||

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===11\31 octave - approx. 425.806¢ - Supermajor Third===

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. Generates [[Meantone family|squares temperament]].

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 [[23-limit]], 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 [[17edo]]. Generates [[Meantone family|squares temperament]].

====MOS Scales generated by 11\31:====

||~ number of tones ||~ MOS class ||~ 0 ||~ 1 ||~ 2 ||~ 3 ||~ 4 ||~ 5 ||~ 6 ||~ 7 ||~ 8 ||~ 9 ||~ 10 ||~ 11 ||~ 12 ||~ 13 ||~ 14 ||~ 15 ||~ 16 ||~ 17 ||~ 18 ||~ 19 ||~ 20 ||~ 21 ||~ 22 ||~ 23 ||~ 24 ||~ 25 ||~ 26 ||~ 27 ||~ 28 ||~ 29 ||~ 30 ||

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<br />

<h3 id="toc17"><a name="Intervals--9\31 octave - approx. 348.39¢ - Neutral Third"></a>9\31 octave - approx. 348.39¢ - Neutral Third</h3>

A neutral 3rd, about 1¢ away from 11:9 (347.41¢). 9\31 is half a perfect fifth (making it a suitable generator for <a class="wiki_link" href="/mohajira">mohajira temperament</a>), and also thrice a major semitone. It is closer in quality to a minor third than a major third, but indeed, it is distinct.<br />

A neutral 3rd, about 1¢ away from 11:9 (347.41¢). 9\31 is half a perfect fifth (making it a suitable generator for <a class="wiki_link" href="/mohajira">mohajira temperament</a>), and also thrice a major semitone. It is closer in quality to a minor third than a major third, but indeed, it is distinct. It is 11¢ shy of 16/13 (359.47¢), suggesting a <a class="wiki_link" href="/13-limit">13-limit</a> interpretation for 31edo. However, its close proximity to 11/9 makes it hard to hear it as 16/13, which in JI has a different quality (and, as a neutral third, is more &quot;major-like&quot; than &quot;minor-like&quot;). Also, its inversion, 22\31 (851.61¢) is wide of the 13th harmonic by about 11¢, while all the lower harmonics are either near-just or narrow. This means the errors can accumulate, for instance, with 13/9 (636.62¢) represented by 17\31 (658.06¢), a good 21.4¢ sharp.<br />

<h4 id="toc18"><a name="Intervals--9\31 octave - approx. 348.39¢ - Neutral Third-MOS Scales generated by 9\31:"></a>MOS Scales generated by 9\31:</h4>

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<br />

<h3 id="toc21"><a name="Intervals--11\31 octave - approx. 425.806¢ - Supermajor Third"></a>11\31 octave - approx. 425.806¢ - Supermajor Third</h3>

11\31 functions as 14:11 (417.51¢), 23:18 (424.36¢), 32:25 (427.37¢), 9:7 (435.08¢) and others. In meantone temperament, it is a diminished fourth, eg. C to Fb. Generates <a class="wiki_link" href="/Meantone%20family">squares temperament</a>.<br />

11\31 functions as 14:11 (417.51¢), 23:18 (424.36¢), 32:25 (427.37¢), 9:7 (435.08¢) and others. In meantone temperament, it is a diminished fourth, eg. C to Fb. It is notable as closely approximating an interval of the <a class="wiki_link" href="/23-limit">23-limit</a>, suggesting the possibility of treating 16\31 (619.35¢) as a flat version of 23/16 (628.27¢). It is perhaps also notable for being close to 6\17, the bright major third of the ever-popular <a class="wiki_link" href="/17edo">17edo</a>. Generates <a class="wiki_link" href="/Meantone%20family">squares temperament</a>.<br />

<h4 id="toc22"><a name="Intervals--11\31 octave - approx. 425.806¢ - Supermajor Third-MOS Scales generated by 11\31:"></a>MOS Scales generated by 11\31:</h4>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2012-08-26 10:45:38 UTC</tt>.<br>
+: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2012-08-26 11:00:21 UTC</tt>.<br>
-: The original revision id was <tt>359972909</tt>.<br>
+: The original revision id was <tt>359974327</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>
@@ Line 108: / Line 108: @@
 ===9\31 octave - approx. 348.39¢ - Neutral Third===
-A neutral 3rd, about 1¢ away from 11:9 (347.41¢). 9\31 is half a perfect fifth (making it a suitable generator for [[mohajira|mohajira temperament]]), and also thrice a major semitone. It is closer in quality to a minor third than a major third, but indeed, it is distinct.
+A neutral 3rd, about 1¢ away from 11:9 (347.41¢). 9\31 is half a perfect fifth (making it a suitable generator for [[mohajira|mohajira temperament]]), 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 [[13-limit]] 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 "major-like" than "minor-like"). 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.
 ====MOS Scales generated by 9\31:====
 ||~ number of tones ||~ MOS class ||~ 0 ||~ 1 ||~ 2 ||~ 3 ||~ 4 ||~ 5 ||~ 6 ||~ 7 ||~ 8 ||~ 9 ||~ 10 ||~ 11 ||~ 12 ||~ 13 ||~ 14 ||~ 15 ||~ 16 ||~ 17 ||~ 18 ||~ 19 ||~ 20 ||~ 21 ||~ 22 ||~ 23 ||~ 24 ||~ 25 ||~ 26 ||~ 27 ||~ 28 ||~ 29 ||~ 30 ||
@@ Line 134: / Line 134: @@
 ===11\31 octave - approx. 425.806¢ - Supermajor Third===
-\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. Generates [[Meantone family|squares temperament]].
+\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 [[23-limit]], 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 [[17edo]]. Generates [[Meantone family|squares temperament]].
 ====MOS Scales generated by 11\31:====
 ||~ number of tones ||~ MOS class ||~ 0 ||~ 1 ||~ 2 ||~ 3 ||~ 4 ||~ 5 ||~ 6 ||~ 7 ||~ 8 ||~ 9 ||~ 10 ||~ 11 ||~ 12 ||~ 13 ||~ 14 ||~ 15 ||~ 16 ||~ 17 ||~ 18 ||~ 19 ||~ 20 ||~ 21 ||~ 22 ||~ 23 ||~ 24 ||~ 25 ||~ 26 ||~ 27 ||~ 28 ||~ 29 ||~ 30 ||
@@ Line 3,231: / Line 3,231: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc17"&gt;&lt;a name="Intervals--9\31 octave - approx. 348.39¢ - Neutral Third"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;9\31 octave - approx. 348.39¢ - Neutral Third&lt;/h3&gt;
-  A neutral 3rd, about 1¢ away from 11:9 (347.41¢). 9\31 is half a perfect fifth (making it a suitable generator for &lt;a class="wiki_link" href="/mohajira"&gt;mohajira temperament&lt;/a&gt;), and also thrice a major semitone. It is closer in quality to a minor third than a major third, but indeed, it is distinct.&lt;br /&gt;
+  A neutral 3rd, about 1¢ away from 11:9 (347.41¢). 9\31 is half a perfect fifth (making it a suitable generator for &lt;a class="wiki_link" href="/mohajira"&gt;mohajira temperament&lt;/a&gt;), 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 &lt;a class="wiki_link" href="/13-limit"&gt;13-limit&lt;/a&gt; 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 &amp;quot;major-like&amp;quot; than &amp;quot;minor-like&amp;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.&lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc18"&gt;&lt;a name="Intervals--9\31 octave - approx. 348.39¢ - Neutral Third-MOS Scales generated by 9\31:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;MOS Scales generated by 9\31:&lt;/h4&gt;
@@ Line 4,406: / Line 4,406: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:42:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc21"&gt;&lt;a name="Intervals--11\31 octave - approx. 425.806¢ - Supermajor Third"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:42 --&gt;11\31 octave - approx. 425.806¢ - Supermajor Third&lt;/h3&gt;
-\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. Generates &lt;a class="wiki_link" href="/Meantone%20family"&gt;squares temperament&lt;/a&gt;.&lt;br /&gt;
+\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 &lt;a class="wiki_link" href="/23-limit"&gt;23-limit&lt;/a&gt;, 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 &lt;a class="wiki_link" href="/17edo"&gt;17edo&lt;/a&gt;. Generates &lt;a class="wiki_link" href="/Meantone%20family"&gt;squares temperament&lt;/a&gt;.&lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:44:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc22"&gt;&lt;a name="Intervals--11\31 octave - approx. 425.806¢ - Supermajor Third-MOS Scales generated by 11\31:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:44 --&gt;MOS Scales generated by 11\31:&lt;/h4&gt;