11/10
| Interval information |
undecimal submajor second
reduced
(Shannon, [math]\displaystyle{ \sqrt{nd} }[/math])
[sound info]
11/10, the large undecimal neutral second or undecimal submajor second, is the simplest submajor second. It is 15 cents sharp of 12/11 and 17 cents flat of 10/9. 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, but is also very close in size to a stack consisting of an apotome and 33/32, leading to the schisma being fudged. Keeping 11/10 distinct from 12/11 ensures that 11/10 bridges quartertone-based chords with more typical 5-limit and Pythagorean chords as a step between notes.
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.
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. 11/10 is also approximated within 2 cents by 22edo, and is 4c sharp of an octave-reduced stack of 9 generators in BPS.
Temperaments
Using 11/10 as a generator tempering out 4000/3993 (as previously mentioned) leads to scales that look like porcupine but whose harmonies can more accurately be explained. A half-octave period is exceptionally natural when 11/10 is a generator, because by virtue of making the (extremely accurate) approximation of the half-octave by 99/70, 9/7 is found as the period-complement of the generator. Taking this approach, this gives us temperaments in the stearnsmic clan such as pogo, supers, or echidna, all of which detemper 100/99 ~ 121/120 and accurately find 11-limit and (no-13's) 17-limit harmonies. Of these, echidna's mapping of the no-13's 17-limit is the simplest, though all three have the same mapping of the 2.3.7.11/10.17 subgroup so that they only differ on the mapping of 5 and 11. The complexity of 5 and 11 in pogo are used to increase accuracy, being a weak schismic extension. That leaves supers as the odd one out; if you are using an edo tuning for it, 58edo supports echidna while 94edo supports pogo, so it seems to exist as a portable alternate way of finding primes 5 and 11 across systems, unless you use the 152edo tuning, which requires using the second-best mapping of 13 (the 152f val).
Using sqrt(11/10) (22/21~21/20) as a generator leads to the low-complexity Nautilus with one period to the octave, and if you use two periods to the octave with this generator you get the high-accuracy temperament Harry; using cbrt(11/10) as a generator leads to Escapade with one period to the octave.