13/11: Difference between revisions

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{{Infobox Interval
{{Infobox Interval
| Icon =
| Name = tridecimal minor third, major minthmic minor third
| Ratio = 13/11
| Color name = 3o1u3, tholu 3rd
| Monzo = 0 0 0 0 -1 1
| Cents = 289.20972
| Name = tridecimal minor third, <br> Neo-Gothic minor third
| Color name =  
| Sound = jid_13_11_pluck_adu_dr220.mp3
| Sound = jid_13_11_pluck_adu_dr220.mp3
}}'''13/11'''
}}


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 [[harmonic]]s. The (octave-reduced) 11th harmonic ([[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]] perfect fifth between 33 and 22. 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.
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|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 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]], 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.


{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
! subminor and minor third
! Subminor and minor third
| 7/6 <br> 266.9¢
| 7/6 <br> 266.9{{c}}
| colspan="7" |
| colspan="7" |
| 6/5 <br> 315.6¢
| 6/5 <br> 315.6{{c}}
|-
|-
! interval in between
! Interval in between
|
|
| colspan="3" | &lt;&lt;
| colspan="3" | &lt;&lt;
| [[36/35|36:35]] <br> 48.7¢
| [[36/35|36:35]] <br> 48.7{{c}}
| colspan="3" | &gt;&gt;
| colspan="3" | &gt;&gt;
|  
|  
Line 30: Line 26:
| colspan="9" |  
| colspan="9" |  
|-
|-
! add mediant (13/11)
! Add mediant (13/11)
| 7/6 <br> 266.9¢
| 7/6 <br> 266.9{{c}}
| colspan="3" |
| colspan="3" |
| 13/11 <br> 289.2¢
| 13/11 <br> 289.2{{c}}
| colspan="3" |
| colspan="3" |
| 6/5 <br> 315.6¢
| 6/5 <br> 315.6{{c}}
|-
|-
! intervals in between
! Intervals in between
|  
|  
| &lt;&lt;
| &lt;&lt;
| [[78/77|78:77]] <br> 22.3¢
| [[78/77|78:77]] <br> 22.3{{c}}
| &gt;&gt;
| &gt;&gt;
|  
|  
| &lt;&lt;
| &lt;&lt;
| [[66/65|66:65]] <br> 26.4¢
| [[66/65|66:65]] <br> 26.4{{c}}
| &gt;&gt;
| &gt;&gt;
|   
|   
Line 51: Line 47:
| colspan="9" |  
| colspan="9" |  
|-
|-
! add mediants (20/17 and 19/16)
! Add mediants (20/17 and 19/16)
| 7/6 <br> 266.9¢
| 7/6 <br> 266.9{{c}}
|  
|  
| [[20/17]] <br> 281.4¢
| [[20/17]] <br> 281.4{{c}}
|  
|  
| '''13/11''' <br> '''289.2¢'''
| '''13/11''' <br> '''289.2{{c}}'''
|  
|  
| [[19/16]] <br> 297.5¢
| [[19/16]] <br> 297.5{{c}}
|  
|  
| 6/5 <br> 315.6¢
| 6/5 <br> 315.6{{c}}
|-
|-
! intervals in between
! Intervals in between
|  
|  
| &lt;&lt; [[120/119|120:119]] &gt;&gt; <br> 14.5¢
| &lt;&lt; [[120/119|120:119]] &gt;&gt; <br> 14.5{{c}}
|  
|  
| &lt;&lt; [[221/220|221:220]] &gt;&gt; <br> 7.9¢
| &lt;&lt; [[221/220|221:220]] &gt;&gt; <br> 7.9{{c}}
|  
|  
| &lt;&lt; [[209/208|209:208]] &gt;&gt; <br> 8.3¢
| &lt;&lt; [[209/208|209:208]] &gt;&gt; <br> 8.3{{c}}
|  
|  
| &lt;&lt; [[96/95|96:95]] &gt;&gt; <br> 18.1¢
| &lt;&lt; [[96/95|96:95]] &gt;&gt; <br> 18.1{{c}}
|  
|  
|}
|}


13/11 is also [[352/351]] (about 4.9¢) narrower than [[32/27]], the minor third in Pythagorean ([[3-limit]]) tuning.
== Approximation ==
This interval is well approximated by [[17edo|4\17]] (282.353 cents), and even better, by [[29edo|7\29]] (289.655 cents).  
 
{{Interval edo approximation|13/11}}


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


== External links ==
== 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


* [http://dkeenan.com/Music/NobleMediant.txt The Noble Mediant] (earliest description of 13:11 as the "Neo-Gothic" minor third)
== Notes ==
<references group="note"/>


[[Category:Third]]
[[Category:Minor third]]
[[Category:Minor third]]
[[Category:13-limit]]
[[Category:Over-11 intervals]]
[[Category:Third]]
[[Category:Major minthmic]]
[[Category:Listen]]
[[Category:Gentle]]
[[Category:Interval ratio]]
[[Category:Neo-gothic]]
[[Category:Taxicab-2 intervals]]

Latest revision as of 09:26, 16 January 2026

Interval information
Ratio 13/11
Subgroup monzo 11.13 [-1 1⟩
Size in cents 289.2097¢
Names tridecimal minor third,
major minthmic minor third
Color name 3o1u3, tholu 3rd
FJS name [math]\displaystyle{ \text{m3}^{13}_{11} }[/math]
Special properties reduced
Tenney norm (log2 nd) 7.15987
Weil norm (log2 max(n, d)) 7.40088
Wilson norm (sopfr(nd)) 24

[sound info]
Open this interval in xen-calc

In 13-limit just intonation, 13/11 is a tridecimal minor third, specifically the major minthmic minor third, measuring about 289.2 cents. It is the difference between the 11th and 13th harmonics. The octave-reduced 11th harmonic (11/8, about 551.3 ¢) and 13th harmonic (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 is a major minthma (352/351) narrower than the Pythagorean minor third (32/27). It is the simplest 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[note 1]. 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 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 ¢) 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 ¢). See the diagram below.

Subminor and minor third 7/6
266.9 ¢ 		6/5
315.6 ¢
Interval in between << 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
22.3 ¢ 		>> << 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 >>
14.5 ¢ 		<< 221:220 >>
7.9 ¢ 		<< 209:208 >>
8.3 ¢ 		<< 96:95 >>
18.1 ¢

Approximation

This interval is well approximated by 4\17 (282.353 cents), and even better, by 7\29 (289.655 cents).


Edo approximations for 13/11 (289.21 ¢)
≤ 80edo, relative error ≤ 10%
Edo Step size Cents (¢) Absolute error (¢) Relative error (%)
4 1\4 300.00 +10.79 +3.60
8 2\8 300.00 +10.79 +7.19
17 4\17 282.35 -6.86 -9.71
21 5\21 285.71 -3.50 -6.12
25 6\25 288.00 -1.21 -2.52
29 7\29 289.66 +0.45 +1.08
33 8\33 290.91 +1.70 +4.67
37 9\37 291.89 +2.68 +8.27
46 11\46 286.96 -2.25 -8.64
50 12\50 288.00 -1.21 -5.04
54 13\54 288.89 -0.32 -1.44
58 14\58 289.66 +0.45 +2.15
62 15\62 290.32 +1.11 +5.75
66 16\66 290.91 +1.70 +9.35
75 18\75 288.00 -1.21 -7.56
79 19\79 288.61 -0.60 -3.96

See also

External links

  • The Noble Mediant by Margo Schulter and David Keenan, the earliest description of 13/11 as the "neo-Gothic" minor third

Notes

  1. ↑ This is a minor minthmic chord where 13/11 and 14/11 sum to a perfect fifth. Shown here is the simplest JI representation.
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