11/10: Difference between revisions
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{{Infobox Interval | {{Infobox Interval | ||
| Name = large undecimal neutral second, undecimal submajor second | |||
| Name = large undecimal neutral second, | |||
| Color name = 1og2, logu 2nd | | Color name = 1og2, logu 2nd | ||
| Sound = jid_11_10_pluck_adu_dr220.mp3 | | Sound = jid_11_10_pluck_adu_dr220.mp3 | ||
}} | }} | ||
'''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''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an | |||
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]]. | |||
== 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]]. | |||
{{Interval edo approximation|11/10}} | |||
== Temperaments == | |||
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 == | == See also == | ||
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* [[List of superparticular intervals]] | * [[List of superparticular intervals]] | ||
[[Category:Second]] | [[Category:Second]] | ||
[[Category:Neutral second]] | [[Category:Neutral second]] | ||
[[Category:Submajor second]] | [[Category:Submajor second]] | ||
[[Category:Over-5]] | [[Category:Over-5 intervals]] | ||
[[Category: | [[Category:Equable heptatonic]] | ||