11/10: Difference between revisions

'''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 [[Fudging|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 tuning|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 [[4/1|first two octaves]] along with [[11/4]] and [[11/3]].

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]].

Using 11/10 as a generator and tempering out 4000/3993 (as previously mentioned) leads to temperaments that look like [[porcupine]] but whose harmonies have higher accuracy. A [[semi-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 observe [[100/99]][[~]][[121/120]] and find accurate [[11-limit]] and no-13's [[17-limit]] harmonies. The three temperaments have the same mapping in the 2.3.7.11/5.17 subgroup and only differ on the mapping of 5 and 11. Of these, echidna's mapping of the no-13's 17-limit is the simplest. The complexity of pogo rewards you with increased accuracy, being a weak schismic extension, and supers shows an intermediate path between them.  

Using sqrt(11/10) ([[21/20]]~[[22/21]]) 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.

 

== See also ==

* [[20/11]] – its [[octave complement]]

* [[15/11]] – its [[fifth complement]]

* [[40/33]] – its [[fourth complement]]

* [[List of superparticular intervals]]

 

[[Category:Submajor second]]

[[Category:Over-5 intervals]]

[[Category:Equable heptatonic]]