11/10: Difference between revisions

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{{Infobox Interval

~~| Icon =~~

| Name = large undecimal neutral second, undecimal submajor second

~~| Ratio = 11/10~~

~~| Monzo = -1 0 -1 0 1~~

~~| Cents = 165.00423~~

| Name = large neutral second

| Color name = 1og2, logu 2nd

| Sound = jid_11_10_pluck_adu_dr220.mp3

}}

'''11/10''', the '''large neutral second''', is ~~an interval favored by Ptolemy~~. ~~Three~~ of ~~them are less than~~ [[4/3]] ~~by the wizardharry comma~~, 4000/3993.

'''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.

~~Coincidentally~~, the ~~interval between~~ the ~~most common tuning frequency (A440)~~ and the ~~second most common AC electrical frequency~~ (~~50hz~~) is ~~exactly 44~~/5, or three ~~octaves above~~ an 11/10.

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]].

== 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 ==

* [[~~Gallery of Just Intervals~~]]

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

* [[20/11]] - its [[~~inverse interval~~]]

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

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

* [[Gallery of just intervals]]

* [[List of superparticular intervals]]

~~[[Category:11-limit]]~~

~~[[Category:Interval]]~~

~~[[Category:Ratio]]~~

~~[[Category:Just interval]]~~

~~[[Category:Superparticular]]~~

[[Category:Second]]

[[Category:Neutral second]]

[[Category:~~todo~~:~~expand~~]]

[[Category:Submajor second]]

[[Category:Over-5 intervals]]

[[Category:Equable heptatonic]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Icon =
+| Name = large undecimal neutral second, undecimal submajor second
-| Ratio = 11/10
-| Monzo = -1 0 -1 0 1
-| Cents = 165.00423
-| Name = large neutral second
 | Color name = 1og2, logu 2nd
 | Sound = jid_11_10_pluck_adu_dr220.mp3
 }}
-'''11/10''', the '''large neutral second''', is an interval favored by Ptolemy. Three of them are less than [[4/3]] by the wizardharry comma, 4000/3993.
+'''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.
-Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50hz) is exactly 44/5, or three octaves above an 11/10.
+/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. 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 ==
-* [[Gallery of Just Intervals]]
+* [[20/11]] – its [[octave complement]]
-* [[20/11]] - its [[inverse interval]]
+* [[15/11]] – its [[fifth complement]]
+* [[40/33]] – its [[fourth complement]]
+* [[Gallery of just intervals]]
+* [[List of superparticular intervals]]
-[[Category:11-limit]]
-[[Category:Interval]]
-[[Category:Ratio]]
-[[Category:Just interval]]
-[[Category:Superparticular]]
 [[Category:Second]]
+[[Category:Neutral second]]
-[[Category:todo:expand]]
+[[Category:Submajor second]]
+[[Category:Over-5 intervals]]
+[[Category:Equable heptatonic]]