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 {{w|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~~]] 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~~ [[~~4000~~/~~3993~~]].

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 ~~is treated~~ as a ~~comma in edos 1, 2, 3, 5,~~ and ~~some very low accuracy~~ temperaments ~~such as~~ [[~~Very low accuracy temperaments #Antietam|antietam~~]].

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.

~~It 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~~ [[~~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.

~~== 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 {{w|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]] 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 [[4000/3993]].
+/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 is treated as a comma in edos 1, 2, 3, 5, and some very low accuracy temperaments such as [[Very low accuracy temperaments #Antietam|antietam]].
+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.
-It 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 [[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.
-== 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 ==