33/13

33/13 |0 1 0 0 1 -1>

1612.7452814608 cents

In 13-limit Just Intonation, 33/13 is the tridecimal major tenth (or Neo-Gothic major tenth), measuring about 1612.7¢. It is the difference between the 13th and 33th 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 33/13 can be likened unto some kind of relatively complex major tenth. 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.

33/13 is the classic mediant between the simpler and more familiar ratios 15/6 and 18/7, as it can be given as (15+18)/(6+7). This puts in between the latter ratios, slightly closer to 18/7. More complex minor thirds can be generated by taking the mediant between 33/13 and 18/7 (which yields (33+18)/(13+7) = 51/20, the septendecimal supermajor tenth, about 1620.6¢) and between 33/13 and 15/6 (which yields (33+15)/(13+6) = 48/19, the undertone major tenth of 19-limit JI, about 1604.4¢).

33/13 is also 352/351 (about 4.9¢) wider than 81/32, the major tenth in Pythagorean (3-limit) tuning.

See: Gallery of Just Intonation Intervals, List of root-3rd-P5 triads in JI

The Noble Mediant (earliest description of 33:13 as the "Neo-Gothic" major tenth)