11/10: Difference between revisions

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~~{{Wikipedia| Neutral interval #Second }}~~

'''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 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 one could make~~ the ~~argument that it 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 (so~~ that [[~~100/99~~|~~S10~~]] ~~is tempered out). Meanwhile, when tuned just or near-just, it functions~~ as ~~almost exactly~~ a ~~third of~~ [[~~4/3~~]]~~, a very xenmelodic role corresponding to tempering out (~~[[~~100~~/99]]~~)/(~~[[~~121/120~~]]~~) =~~ [[~~4000~~/~~3993~~|~~S10/S11~~]] ~~= (~~[[4~~/3|12/9~~]]~~)/(~~[[11/10]]~~)<sup>3</sup>~~.

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 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 ==

11/10 ~~may be treated implicitly~~ as a ~~comma in JI scales~~ that ~~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~~ [[~~echidna~~]], a [[~~hedgehog~~]] ~~lookalike that detempers~~ [[100/99]] ~~and~~ [[121/120]] and ~~which efficiently and accurately finds~~ [[11-limit]] and (no-13's) [[17-limit]] harmonies.

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.

~~== Trivia ==~~

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.

~~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.

== See also ==

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-{{Wikipedia| Neutral interval #Second }}
+'''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 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 one could make the argument that it 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 (so that [[100/99|S10]] is tempered out). Meanwhile, when tuned just or near-just, it functions as almost exactly a third of [[4/3]], a very xenmelodic role corresponding to tempering out ([[100/99]])/([[121/120]]) = [[4000/3993|S10/S11]] = ([[4/3|12/9]])/([[11/10]])<sup>3</sup>.
+/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 ==
-/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.
+/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 ==
-/10 may be treated implicitly as a comma in JI scales that 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 [[echidna]], a [[hedgehog]] lookalike that detempers [[100/99]] and [[121/120]] and which efficiently and accurately finds [[11-limit]] and (no-13's) [[17-limit]] harmonies.
+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.
-== Trivia ==
+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.
-Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50&nbsp;Hz) is exactly 44/5, or three octaves above an 11/10.
 == See also ==