13/11: Difference between revisions

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| Monzo = 0 0 0 0 -1 1
| Monzo = 0 0 0 0 -1 1
| Cents = 289.20972
| Cents = 289.20972
| Name = tridecimal minor third, <br> Neo-Gothic minor third
| Name = tridecimal minor third, <br>Neo-Gothic minor third
| Color name =  
| Color name =  
| FJS name = m3<sup>13</sup><sub>11</sub>
| Sound = jid_13_11_pluck_adu_dr220.mp3
| Sound = jid_13_11_pluck_adu_dr220.mp3
}}
}}


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 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]], 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.


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 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.)


{| class="wikitable center-all"
{| class="wikitable center-all"
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== See also ==
== See also ==
 
* [[22/13]] – its [[octave complement]]
* [[22/13]] - its [[octave complement]]
* [[33/26]] – its [[fifth complement]]
* [[33/26]] - its [[fifth complement]]
* [[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
* [[: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)
[[Category:13-limit]]
 
[[Category:Interval ratio]]
[[Category:Minor third]]
[[Category:Minor third]]
[[Category:13-limit]]
[[Category:Third]]
[[Category:Third]]
[[Category:Listen]]
[[Category:Listen]]
[[Category:Interval ratio]]
[[Category:Over-11]]
[[Category:Over-11]]
[[Category:Minthmic]]
[[Category:Gentle]]
[[Category:Neo-gothic]]

Revision as of 11:15, 20 September 2020

Interval information
Ratio 13/11
Subgroup monzo 11.13 [-1 1⟩
Size in cents 289.2097¢
Names tridecimal minor third,
Neo-Gothic minor third
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 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, 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 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.

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 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¢

13/11 is also 352/351 (about 4.9¢) narrower than 32/27, the minor third in Pythagorean (3-limit) tuning.

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
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