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13/11
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289.20972 cents
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, 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: Gallery of Just Intonation Intervals, gentle chords, List of root-3rd-P5 triads in JI

The Noble Mediant (earliest description of 13:11 as the "Neo-Gothic" minor third)