11/10
| Interval information |
undecimal submajor second
reduced
(Shannon, [math]\sqrt{nd}[/math])
[sound info]
11/10, the large undecimal neutral second or undecimal submajor second, is an interval favored by Ptolemy. Depending on who you ask, this interval, on its own, is either considerably more or considerably less exotic than 12/11 or a number of other simple 11-limit intervals.
If tempered sharp, however, one could make the argument that 11/10 functions a bit more like a narrowed 10/9 in light of its usage in such a capacity in systems like 41edo and 63edo where 11/10 and 10/9 are tempered together due to 100/99 being tempered out.
Meanwhile, when tuned just or near-just, it not only has the very exotic melodic role of being almost exactly a third of 4/3, leading to 4000/3993 being fudged or tempered out, but is also very close in size to a stack consisting of an apotome and 33/32, leading to the schisma being fudged or tempered out.
Assuming you go with either of the aforementioned options, keeping 11/10 distinct from 12/11 ensures that 11/10 has a way of bridging quartertone-based chords with more typical 5-limit and Pythagorean chords as a sort of step between notes, however, if you temper out 121/120, expect this ability to vanish.
11/10 is the octave-reduced form of 11/5, one of the three most concordant 11-limit intervals within the entire first two octaves along with 11/4 and 11/3. In any octave-repeating tuning, a good approximation of 11/10 indicates a good approximation of 11/5. So, it could be argued that 11/10 is a high priority for any octave-repeating 11-limit tuning.
Approximation
11/10 is approximated extremely precisely by 80edo and its multiples, with a chain of 80 11/10's failing to close at the octave by a mere third of a cent, close enough that you could theoretically tune an instrument to 80edo by ear using it if you had the patience.
Temperaments
11/10 may be treated implicitly as a comma in JI scales that for example do not find 11/8 and 5/4 above the same degree, but usually it makes much more sense to use it as a generator, such as the aforementioned very accurate strategy of making it a third of 4/3, leading to scales that look like porcupine but whose harmonies can more accurately be explained in a number of ways depending partially on the exact tempering used. If you use a half-octave period you get temperaments in the stearnsmic clan such as pogo, supers, or echidna, all of which detemper 100/99 and 121/120 and efficiently and accurately find 11-limit and (no-13's) 17-limit harmonies.
Exotemperaments
11/10 is tempered out in the patent vals of edos 1, 2, 3 and 5. An example rank-2 exotemperament that treats it as a comma to be tempered out is antietam.
Trivia
Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50 Hz) is exactly 44/5, or three octaves above an 11/10.