13/11
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- This revision was by author spt3125 and made on 2014-06-07 22:38:53 UTC.
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Original Wikitext content:
**13/11** |0 0 0 0 -1 1> 289.20972 cents [[media type="file" key="jid_13_11_pluck_adu_dr220.mp3"]] [[file:xenharmonic/jid_13_11_pluck_adu_dr220.mp3|sound sample]] In [[13-limit]] [[Just Intonation]], 13/11 is the tridecimal minor third (or [[Neo-Gothic]] minor third), measuring about 289.2¢. It is the difference between the 11th and 13th harmonics. The (octave-reduced) 11th harmonic ([[11_8|11/8]], about 551.3¢) and 13th harmonic ([[13_8|13/8]], about 840.5¢) are both quite xenharmonic and demand new interval categories, while 13/11 can be likened unto some kind of relatively complex minor third. It can even function as such in a 13-limit Neo-Gothic minor triad of 22:26:33, with a [[3_2|3/2]] perfect fifth between 33 and 22. Compare this to 22:26:32 (11:13:16), which has the much more dissonant [[16_11|16/11]] as the outside interval in place of 3/2. The latter triad sounds more like a xenharmonic version of a diminished triad, and could not be confused with simpler diminished triads such as 5:6:7. 13/11 is the classic [[mediant]] between the simpler and more familiar ratios [[6_5|6/5]] and [[7_6|7/6]], as it can be given as (6+7)/(5+6). This puts in between the latter ratios, slightly closer to 7/6. More complex minor thirds can be generated by taking the mediant between 13/11 and 7/6 (which yields (13+7)/(11+6) = [[20_17|20/17]], the septendecimal subminor third, about 281.4¢) and between 13/11 and 6/5 (which yields (13+6)/(11+5) = [[19_16|19/16]], the overtone minor third of [[19-limit]] JI, about 297.5¢). (See the diagram below.) ||~ subminor and minor third ||= 7/6 266.9¢ ||= ||= ||= ||= ||= ||= ||= ||= 6/5 315.6¢ || ||~ interval in between ||= ||= << ||= ||= ||= [[36_35|36:35]] 48.7¢ ||= ||= ||= >> || || ||~ add mediant (13/11) ||= 7/6 266.9¢ ||= ||= ||= ||= 13/11 289.2¢ ||= ||= ||= ||= 6/5 315.6¢ || ||~ intervals in between ||= ||= << ||= [[78_77|78:77]] 22.3¢ ||= >> || ||= << ||= [[66_65|66:65]] 26.4¢ ||= >> || || ||~ add mediants (20/17 and 19/16) ||= 7/6 266.9¢ ||= ||= 20/17 281.4¢ ||= ||= **13/11** **289.2¢** ||= ||= 19/16 297.5¢ ||= ||= 6/5 315.6¢ || ||~ intervals in between ||= ||= << [[120_119|120:119]] >> 14.5¢ ||= ||= << [[221_220|221:220]] >> 7.9¢ ||= ||= << [[209_208|209:208]] >> 8.3¢ ||= ||= << [[96_95|96:95]] >> 18.1¢ ||= || 13/11 is also [[352_351|352/351]] (about 4.9¢) narrower than [[32_27|32/27]], the minor third in Pythagorean ([[3-limit]]) tuning. See: [[Gallery of Just Intervals|Gallery of Just Intonation Intervals]], [[gentle chords]], [[List of root-3rd-P5 triads in JI]] [[http://dkeenan.com/Music/NobleMediant.txt|The Noble Mediant]] (earliest description of 13:11 as the "Neo-Gothic" minor third)
Original HTML content:
<html><head><title>13_11</title></head><body><strong>13/11</strong><br />
|0 0 0 0 -1 1><br />
289.20972 cents<br />
<!-- ws:start:WikiTextMediaRule:0:<img src="http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_13_11_pluck_adu_dr220.mp3?h=20&w=240" class="WikiMedia WikiMediaFile" id="wikitext@@media@@type=&quot;file&quot; key=&quot;jid_13_11_pluck_adu_dr220.mp3&quot;" title="Local Media File"height="20" width="240"/> --><embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_13_11_pluck_adu_dr220.mp3?file_extension=mp3&autostart=false&repeat=false&showdigits=true&showfsbutton=false&width=240&height=20"></embed><!-- ws:end:WikiTextMediaRule:0 --> <a href="http://xenharmonic.wikispaces.com/file/view/jid_13_11_pluck_adu_dr220.mp3/513213592/jid_13_11_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_13_11_pluck_adu_dr220.mp3/513213592/jid_13_11_pluck_adu_dr220.mp3');">sound sample</a><br />
<br />
In <a class="wiki_link" href="/13-limit">13-limit</a> <a class="wiki_link" href="/Just%20Intonation">Just Intonation</a>, 13/11 is the tridecimal minor third (or <a class="wiki_link" href="/Neo-Gothic">Neo-Gothic</a> minor third), measuring about 289.2¢. It is the difference between the 11th and 13th harmonics. The (octave-reduced) 11th harmonic (<a class="wiki_link" href="/11_8">11/8</a>, about 551.3¢) and 13th harmonic (<a class="wiki_link" href="/13_8">13/8</a>, about 840.5¢) are both quite xenharmonic and demand new interval categories, while 13/11 can be likened unto some kind of relatively complex minor third. It can even function as such in a 13-limit Neo-Gothic minor triad of 22:26:33, with a <a class="wiki_link" href="/3_2">3/2</a> perfect fifth between 33 and 22. Compare this to 22:26:32 (11:13:16), which has the much more dissonant <a class="wiki_link" href="/16_11">16/11</a> as the outside interval in place of 3/2. The latter triad sounds more like a xenharmonic version of a diminished triad, and could not be confused with simpler diminished triads such as 5:6:7.<br />
<br />
13/11 is the classic <a class="wiki_link" href="/mediant">mediant</a> between the simpler and more familiar ratios <a class="wiki_link" href="/6_5">6/5</a> and <a class="wiki_link" href="/7_6">7/6</a>, as it can be given as (6+7)/(5+6). This puts in between the latter ratios, slightly closer to 7/6. More complex minor thirds can be generated by taking the mediant between 13/11 and 7/6 (which yields (13+7)/(11+6) = <a class="wiki_link" href="/20_17">20/17</a>, the septendecimal subminor third, about 281.4¢) and between 13/11 and 6/5 (which yields (13+6)/(11+5) = <a class="wiki_link" href="/19_16">19/16</a>, the overtone minor third of <a class="wiki_link" href="/19-limit">19-limit</a> JI, about 297.5¢). (See the diagram below.)<br />
<br />
<table class="wiki_table">
<tr>
<th>subminor and minor third<br />
</th>
<td style="text-align: center;">7/6<br />
266.9¢<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">6/5<br />
315.6¢<br />
</td>
</tr>
<tr>
<th>interval in between<br />
</th>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><<<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/36_35">36:35</a><br />
48.7¢<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">>><br />
</td>
<td><br />
</td>
</tr>
<tr>
<th>add mediant (13/11)<br />
</th>
<td style="text-align: center;">7/6<br />
266.9¢<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">13/11<br />
289.2¢<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">6/5<br />
315.6¢<br />
</td>
</tr>
<tr>
<th>intervals in between<br />
</th>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><<<br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/78_77">78:77</a><br />
22.3¢<br />
</td>
<td style="text-align: center;">>><br />
</td>
<td><br />
</td>
<td style="text-align: center;"><<<br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/66_65">66:65</a><br />
26.4¢<br />
</td>
<td style="text-align: center;">>><br />
</td>
<td><br />
</td>
</tr>
<tr>
<th>add mediants (20/17 and 19/16)<br />
</th>
<td style="text-align: center;">7/6<br />
266.9¢<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">20/17<br />
281.4¢<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><strong>13/11</strong><br />
<strong>289.2¢</strong><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">19/16<br />
297.5¢<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">6/5<br />
315.6¢<br />
</td>
</tr>
<tr>
<th>intervals in between<br />
</th>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><< <a class="wiki_link" href="/120_119">120:119</a> >><br />
14.5¢<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><< <a class="wiki_link" href="/221_220">221:220</a> >><br />
7.9¢<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><< <a class="wiki_link" href="/209_208">209:208</a> >><br />
8.3¢<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><< <a class="wiki_link" href="/96_95">96:95</a> >><br />
18.1¢<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
</table>
<br />
<br />
13/11 is also <a class="wiki_link" href="/352_351">352/351</a> (about 4.9¢) narrower than <a class="wiki_link" href="/32_27">32/27</a>, the minor third in Pythagorean (<a class="wiki_link" href="/3-limit">3-limit</a>) tuning.<br />
<br />
See: <a class="wiki_link" href="/Gallery%20of%20Just%20Intervals">Gallery of Just Intonation Intervals</a>, <a class="wiki_link" href="/gentle%20chords">gentle chords</a>, <a class="wiki_link" href="/List%20of%20root-3rd-P5%20triads%20in%20JI">List of root-3rd-P5 triads in JI</a><br />
<br />
<a class="wiki_link_ext" href="http://dkeenan.com/Music/NobleMediant.txt" rel="nofollow">The Noble Mediant</a> (earliest description of 13:11 as the "Neo-Gothic" minor third)</body></html>