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'''13/11'''
{{Infobox Interval
|0 0 0 0 -1 1>
| Name = tridecimal minor third, major minthmic minor third
| Color name = 3o1u3, tholu 3rd
| Sound = jid_13_11_pluck_adu_dr220.mp3
}}


289.20972 cents
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]].


[[File:jid_13_11_pluck_adu_dr220.mp3]] [[:File:jid_13_11_pluck_adu_dr220.mp3|sound sample]]
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.


In [[13-limit|13-limit]] [[Just_intonation|Just Intonation]], 13/11 is the tridecimal minor third (or [[Neo-Gothic|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.
{| class="wikitable center-all"
 
|-
13/11 is the classic [[mediant|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|19-limit]] JI, about 297.5¢). (See the diagram below.)
! Subminor and minor third
 
| 7/6 <br> 266.9{{c}}
{| class="wikitable"
| colspan="7" |
| 6/5 <br> 315.6{{c}}
|-
! Interval in between
|
| colspan="3" | &lt;&lt;
| [[36/35|36:35]] <br> 48.7{{c}}
| colspan="3" | &gt;&gt;
|
|-
|-
! | subminor and minor third
!
| style="text-align:center;" | 7/6
| colspan="9" |  
 
266.9¢
| style="text-align:center;" |  
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" | 6/5
 
315.6¢
|-
|-
! | interval in between
! Add mediant (13/11)
| style="text-align:center;" |
| 7/6 <br> 266.9{{c}}
| style="text-align:center;" | &lt;&lt;
| colspan="3" |
| style="text-align:center;" |
| 13/11 <br> 289.2{{c}}
| style="text-align:center;" |  
| colspan="3" |
| style="text-align:center;" | [[36/35|36:35]]
| 6/5 <br> 315.6{{c}}
 
48.7¢
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" | &gt;&gt;
| |
|-
|-
! | add mediant (13/11)
! Intervals in between
| style="text-align:center;" | 7/6
|  
 
| &lt;&lt;
266.9¢
| [[78/77|78:77]] <br> 22.3{{c}}
| style="text-align:center;" |  
| &gt;&gt;
| style="text-align:center;" |
|  
| style="text-align:center;" |
| &lt;&lt;
| style="text-align:center;" | 13/11
| [[66/65|66:65]] <br> 26.4{{c}}
 
| &gt;&gt;
289.2¢
|
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" | 6/5
 
315.6¢
|-
|-
! | intervals in between
!
| style="text-align:center;" |
| colspan="9" |  
| style="text-align:center;" | &lt;&lt;
| style="text-align:center;" | [[78/77|78:77]]
 
22.3¢
| style="text-align:center;" | &gt;&gt;
| |
| style="text-align:center;" | &lt;&lt;
| style="text-align:center;" | [[66/65|66:65]]
 
26.4¢
| style="text-align:center;" | &gt;&gt;
| |  
|-
|-
! | add mediants (20/17 and 19/16)
! Add mediants (20/17 and 19/16)
| style="text-align:center;" | 7/6
| 7/6 <br> 266.9{{c}}
 
|  
266.9¢
| [[20/17]] <br> 281.4{{c}}
| style="text-align:center;" |  
|  
| style="text-align:center;" | 20/17
| '''13/11''' <br> '''289.2{{c}}'''
 
|  
281.4¢
| [[19/16]] <br> 297.5{{c}}
| style="text-align:center;" |  
|  
| style="text-align:center;" | '''13/11'''
| 6/5 <br> 315.6{{c}}
 
'''289.2¢'''
| style="text-align:center;" |  
| style="text-align:center;" | 19/16
 
297.5¢
| style="text-align:center;" |  
| style="text-align:center;" | 6/5
 
315.6¢
|-
|-
! | intervals in between
! Intervals in between
| style="text-align:center;" |  
|  
| style="text-align:center;" | &lt;&lt; [[120/119|120:119]] &gt;&gt;
| &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}}
|
|}


14.5¢
== Approximation ==
| style="text-align:center;" |
This interval is well approximated by [[17edo|4\17]] (282.353 cents), and even better, by [[29edo|7\29]] (289.655 cents).
| style="text-align:center;" | &lt;&lt; [[221/220|221:220]] &gt;&gt;


7.9¢
{{Interval edo approximation|13/11}}
| style="text-align:center;" |
| style="text-align:center;" | &lt;&lt; [[209/208|209:208]] &gt;&gt;


8.3¢
== See also ==
| style="text-align:center;" |
* [[22/13]] – its [[octave complement]]
| style="text-align:center;" | &lt;&lt; [[96/95|96:95]] &gt;&gt;
* [[33/26]] – its [[fifth complement]]
 
* [[44/39]] – its [[fourth complement]]
18.1¢
* [[Ed13/11]]
| style="text-align:center;" |
* [[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


13/11 is also [[352/351|352/351]] (about 4.9¢) narrower than [[32/27|32/27]], the minor third in Pythagorean ([[3-limit|3-limit]]) tuning.
== External links ==
* [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


See: [[Gallery_of_Just_Intervals|Gallery of Just Intonation Intervals]], [[gentle_chords|gentle chords]], [[List_of_root-3rd-P5_triads_in_JI|List of root-3rd-P5 triads in JI]]
== Notes ==
<references group="note"/>


[http://dkeenan.com/Music/NobleMediant.txt The Noble Mediant] (earliest description of 13:11 as the "Neo-Gothic" minor third)
[[Category:Third]]
[[Category:minor_third]]
[[Category:Minor third]]
[[Category:tredecimal]]
[[Category:Over-11 intervals]]
[[Category:Major minthmic]]
[[Category:Gentle]]
[[Category:Neo-gothic]]
[[Category:Taxicab-2 intervals]]
Retrieved from "https://en.xen.wiki/w/13/11"