# 13/11

(Redirected from 13 11)

13/11 |0 0 0 0 -1 1>

289.20972 cents

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 interval in between add mediant (13/11) intervals in between add mediants (20/17 and 19/16) intervals in between 7/6 266.9¢ 6/5 315.6¢ << 36:35 48.7¢ >> 7/6 266.9¢ 13/11 289.2¢ 6/5 315.6¢ << 78:77 22.3¢ >> << 66:65 26.4¢ >> 7/6 266.9¢ 20/17 281.4¢ 13/11 289.2¢ 19/16 297.5¢ 6/5 315.6¢ << 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.

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