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13/11: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox Interval
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| Name = tridecimal minor third, major minthmic minor third
: This revision was by author [[User:spt3125|spt3125]] and made on <tt>2014-06-07 22:38:53 UTC</tt>.<br>
| Color name = 3o1u3, tholu 3rd
: The original revision id was <tt>513213838</tt>.<br>
| Sound = jid_13_11_pluck_adu_dr220.mp3
: 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>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">**13/11**
|0 0 0 0 -1 1&gt;
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.
In [[13-limit]] [[just intonation]], '''13/11''' is a '''tridecimal minor third''', specifically the '''major minthmic minor third''', measuring about 289.2 [[cent]]s. It is the difference between the [[11/1|11th]] and [[13/1|13th]] [[harmonic]]s. The octave-reduced 11th harmonic ([[11/8]], about 551.3{{c}}) and 13th harmonic ([[13/8]], about 840.5{{c}}) are both quite xenharmonic and demand new interval categories, while 13/11 can be likened unto some kind of relatively complex minor third – it is a [[352/351|major minthma (352/351)]] narrower than the [[32/27|Pythagorean minor third (32/27)]]. It is the simplest [[neogothic major and minor|neogothic minor third]], and can function as such in a 13-limit neogothic minor triad of [[22:26:33]], with a [[3/2]] perfect fifth between 33 and 22<ref group="note">This is a [[minor minthmic chords|minor minthmic chord]] where 13/11 and [[14/11]] sum to a perfect fifth. Shown here is the simplest JI representation. </ref>. Compare this to 22:26:32 ([[11:13:16]]), which has the much more dissonant [[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.)
13/11 is the classic [[mediant]] between the simpler and more familiar ratios [[6/5]] and [[7/6]], as it can be given as (6 + 7)/(5 + 6). This puts it 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]], the septendecimal subminor third, about 281.4{{c}}) and between 13/11 and 6/5 (which yields (13 + 6)/(11 + 5) = [[19/16]], the overtone minor third of [[19-limit]] JI, about 297.5{{c}}). See the diagram below.


||~ subminor and minor third ||= 7/6
{| class="wikitable center-all"
266.9¢ ||=   ||=  ||=  ||=  ||=  ||=  ||=  ||= 6/5
|-
315.6¢ ||
! Subminor and minor third
||~ interval in between ||=   ||= &lt;&lt; ||=  ||=  ||= [[36_35|36:35]]
| 7/6 <br> 266.9{{c}}
48.7¢ ||=   ||=  ||= &gt;&gt; ||   ||
| colspan="7" |
||~ add mediant (13/11) ||= 7/6
| 6/5 <br> 315.6{{c}}
266.9¢ ||=   ||=  ||=  ||= 13/11
|-
289.2¢ ||=   ||=  ||=  ||= 6/5
! Interval in between
315.6¢ ||
|
||~ intervals in between ||=  ||= &lt;&lt; ||= [[78_77|78:77]]
| colspan="3" | &lt;&lt;
22.3¢ ||= &gt;&gt; ||   ||= &lt;&lt; ||= [[66_65|66:65]]
| [[36/35|36:35]] <br> 48.7{{c}}
26.4¢ ||= &gt;&gt; ||   ||
| colspan="3" | &gt;&gt;
||~ add mediants (20/17 and 19/16) ||= 7/6
|  
266.9¢ ||=  ||= 20/17
|-
281.4¢ ||=  ||= **13/11**
!
**289.2¢** ||=  ||= 19/16
| colspan="9" |  
297.5¢ ||=  ||= 6/5
|-
315.6¢ ||
! Add mediant (13/11)
||~ intervals in between ||=  ||= &lt;&lt; [[120_119|120:119]] &gt;&gt;
| 7/6 <br> 266.9{{c}}
14.5¢ ||=  ||= &lt;&lt; [[221_220|221:220]] &gt;&gt;
| colspan="3" |
7.9¢ ||=  ||= &lt;&lt; [[209_208|209:208]] &gt;&gt;
| 13/11 <br> 289.2{{c}}
8.3¢ ||=  ||= &lt;&lt; [[96_95|96:95]] &gt;&gt;
| colspan="3" |
18.1¢ ||=  ||
| 6/5 <br> 315.6{{c}}
|-
! Intervals in between
|  
| &lt;&lt;
| [[78/77|78:77]] <br> 22.3{{c}}
| &gt;&gt;
|  
| &lt;&lt;
| [[66/65|66:65]] <br> 26.4{{c}}
| &gt;&gt;
|
|-
!
| colspan="9" |  
|-
! Add mediants (20/17 and 19/16)
| 7/6 <br> 266.9{{c}}
|  
| [[20/17]] <br> 281.4{{c}}
|  
| '''13/11''' <br> '''289.2{{c}}'''
|  
| [[19/16]] <br> 297.5{{c}}
|  
| 6/5 <br> 315.6{{c}}
|-
! Intervals in between
|  
| &lt;&lt; [[120/119|120:119]] &gt;&gt; <br> 14.5{{c}}
|  
| &lt;&lt; [[221/220|221:220]] &gt;&gt; <br> 7.9{{c}}
|  
| &lt;&lt; [[209/208|209:208]] &gt;&gt; <br> 8.3{{c}}
|  
| &lt;&lt; [[96/95|96:95]] &gt;&gt; <br> 18.1{{c}}
|  
|}


== Approximation ==
This interval is well approximated by [[17edo|4\17]] (282.353 cents), and even better, by [[29edo|7\29]] (289.655 cents).


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.
{{Interval edo approximation|13/11}}


See: [[Gallery of Just Intervals|Gallery of Just Intonation Intervals]], [[gentle chords]], [[List of root-3rd-P5 triads in JI]]
== See also ==
* [[22/13]] – its [[octave complement]]
* [[33/26]] – its [[fifth complement]]
* [[44/39]] – its [[fourth complement]]
* [[Ed13/11]]
* [[Gallery of just intervals]]
* [[Gentle chords]]
* [[List of root-3rd-P5 triads in JI]]
* [[:File:Ji-13-11-csound-foscil-220hz.mp3]] – another sound example


[[http://dkeenan.com/Music/NobleMediant.txt|The Noble Mediant]] (earliest description of 13:11 as the "Neo-Gothic" minor third)</pre></div>
== External links ==
<h4>Original HTML content:</h4>
* [http://dkeenan.com/Music/NobleMediant.txt ''The Noble Mediant''] by Margo Schulter and David Keenan, the earliest description of 13/11 as the "neo-Gothic" minor third
<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;13_11&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;strong&gt;13/11&lt;/strong&gt;&lt;br /&gt;
|0 0 0 0 -1 1&amp;gt;&lt;br /&gt;
289.20972 cents&lt;br /&gt;
&lt;!-- ws:start:WikiTextMediaRule:0:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_13_11_pluck_adu_dr220.mp3?h=20&amp;amp;w=240&amp;quot; class=&amp;quot;WikiMedia WikiMediaFile&amp;quot; id=&amp;quot;wikitext@@media@@type=&amp;amp;quot;file&amp;amp;quot; key=&amp;amp;quot;jid_13_11_pluck_adu_dr220.mp3&amp;amp;quot;&amp;quot; title=&amp;quot;Local Media File&amp;quot;height=&amp;quot;20&amp;quot; width=&amp;quot;240&amp;quot;/&amp;gt; --&gt;&lt;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&amp;autostart=false&amp;repeat=false&amp;showdigits=true&amp;showfsbutton=false&amp;width=240&amp;height=20"&gt;&lt;/embed&gt;&lt;!-- ws:end:WikiTextMediaRule:0 --&gt; &lt;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');"&gt;sound sample&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
In &lt;a class="wiki_link" href="/13-limit"&gt;13-limit&lt;/a&gt; &lt;a class="wiki_link" href="/Just%20Intonation"&gt;Just Intonation&lt;/a&gt;, 13/11 is the tridecimal minor third (or &lt;a class="wiki_link" href="/Neo-Gothic"&gt;Neo-Gothic&lt;/a&gt; minor third), measuring about 289.2¢. It is the difference between the 11th and 13th harmonics. The (octave-reduced) 11th harmonic (&lt;a class="wiki_link" href="/11_8"&gt;11/8&lt;/a&gt;, about 551.3¢) and 13th harmonic (&lt;a class="wiki_link" href="/13_8"&gt;13/8&lt;/a&gt;, 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 &lt;a class="wiki_link" href="/3_2"&gt;3/2&lt;/a&gt; perfect fifth between 33 and 22. Compare this to 22:26:32 (11:13:16), which has the much more dissonant &lt;a class="wiki_link" href="/16_11"&gt;16/11&lt;/a&gt; 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.&lt;br /&gt;
&lt;br /&gt;
13/11 is the classic &lt;a class="wiki_link" href="/mediant"&gt;mediant&lt;/a&gt; between the simpler and more familiar ratios &lt;a class="wiki_link" href="/6_5"&gt;6/5&lt;/a&gt; and &lt;a class="wiki_link" href="/7_6"&gt;7/6&lt;/a&gt;, 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) = &lt;a class="wiki_link" href="/20_17"&gt;20/17&lt;/a&gt;, the septendecimal subminor third, about 281.4¢) and between 13/11 and 6/5 (which yields (13+6)/(11+5) = &lt;a class="wiki_link" href="/19_16"&gt;19/16&lt;/a&gt;, the overtone minor third of &lt;a class="wiki_link" href="/19-limit"&gt;19-limit&lt;/a&gt; JI, about 297.5¢). (See the diagram below.)&lt;br /&gt;
&lt;br /&gt;


== Notes ==
<references group="note"/>


&lt;table class="wiki_table"&gt;
[[Category:Third]]
    &lt;tr&gt;
[[Category:Minor third]]
        &lt;th&gt;subminor and minor third&lt;br /&gt;
[[Category:Over-11 intervals]]
&lt;/th&gt;
[[Category:Major minthmic]]
        &lt;td style="text-align: center;"&gt;7/6&lt;br /&gt;
[[Category:Gentle]]
266.9¢&lt;br /&gt;
[[Category:Neo-gothic]]
&lt;/td&gt;
[[Category:Taxicab-2 intervals]]
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;6/5&lt;br /&gt;
315.6¢&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;th&gt;interval in between&lt;br /&gt;
&lt;/th&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&amp;lt;&amp;lt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/36_35"&gt;36:35&lt;/a&gt;&lt;br /&gt;
48.7¢&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&amp;gt;&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;th&gt;add mediant (13/11)&lt;br /&gt;
&lt;/th&gt;
        &lt;td style="text-align: center;"&gt;7/6&lt;br /&gt;
266.9¢&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;13/11&lt;br /&gt;
289.2¢&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;6/5&lt;br /&gt;
315.6¢&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;th&gt;intervals in between&lt;br /&gt;
&lt;/th&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&amp;lt;&amp;lt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/78_77"&gt;78:77&lt;/a&gt;&lt;br /&gt;
22.3¢&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&amp;gt;&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&amp;lt;&amp;lt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/66_65"&gt;66:65&lt;/a&gt;&lt;br /&gt;
26.4¢&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&amp;gt;&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;th&gt;add mediants (20/17 and 19/16)&lt;br /&gt;
&lt;/th&gt;
        &lt;td style="text-align: center;"&gt;7/6&lt;br /&gt;
266.9¢&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;20/17&lt;br /&gt;
281.4¢&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;strong&gt;13/11&lt;/strong&gt;&lt;br /&gt;
&lt;strong&gt;289.2¢&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;19/16&lt;br /&gt;
297.5¢&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;6/5&lt;br /&gt;
315.6¢&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;th&gt;intervals in between&lt;br /&gt;
&lt;/th&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&amp;lt;&amp;lt; &lt;a class="wiki_link" href="/120_119"&gt;120:119&lt;/a&gt; &amp;gt;&amp;gt;&lt;br /&gt;
14.5¢&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&amp;lt;&amp;lt; &lt;a class="wiki_link" href="/221_220"&gt;221:220&lt;/a&gt; &amp;gt;&amp;gt;&lt;br /&gt;
7.9¢&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&amp;lt;&amp;lt; &lt;a class="wiki_link" href="/209_208"&gt;209:208&lt;/a&gt; &amp;gt;&amp;gt;&lt;br /&gt;
8.3¢&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&amp;lt;&amp;lt; &lt;a class="wiki_link" href="/96_95"&gt;96:95&lt;/a&gt; &amp;gt;&amp;gt;&lt;br /&gt;
18.1¢&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;br /&gt;
&lt;br /&gt;
13/11 is also &lt;a class="wiki_link" href="/352_351"&gt;352/351&lt;/a&gt; (about 4.9¢) narrower than &lt;a class="wiki_link" href="/32_27"&gt;32/27&lt;/a&gt;, the minor third in Pythagorean (&lt;a class="wiki_link" href="/3-limit"&gt;3-limit&lt;/a&gt;) tuning.&lt;br /&gt;
&lt;br /&gt;
See: &lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&gt;Gallery of Just Intonation Intervals&lt;/a&gt;, &lt;a class="wiki_link" href="/gentle%20chords"&gt;gentle chords&lt;/a&gt;, &lt;a class="wiki_link" href="/List%20of%20root-3rd-P5%20triads%20in%20JI"&gt;List of root-3rd-P5 triads in JI&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://dkeenan.com/Music/NobleMediant.txt" rel="nofollow"&gt;The Noble Mediant&lt;/a&gt; (earliest description of 13:11 as the &amp;quot;Neo-Gothic&amp;quot; minor third)&lt;/body&gt;&lt;/html&gt;</pre></div>
Retrieved from "https://en.xen.wiki/w/13/11"