11/10: Difference between revisions

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| Sound = jid_11_10_pluck_adu_dr220.mp3

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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~~. ~~Despite its being tempered together with~~ [[12/11]] ~~in systems like~~ [[~~24edo~~]], ~~this interval~~, ~~on its own~~, is ~~considerably less exotic than 12/11~~ in ~~terms of sound when compared~~ to a ~~number~~ of ~~other~~ [[~~11-limit~~]] ~~intervals- in fact~~, ~~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~~ [[~~22edo~~]] ~~and~~ [[~~41edo~~]] ~~where~~ 11/10 and 10/~~9 are tempered together~~.

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~~]]~~. If it is used as a generator instead~~, ~~it produces~~ [[~~porcupine~~]], ~~although it~~ is ~~slightly sharper than~~ the ~~optimal~~ tuning for ~~porcupine~~ and ~~does not fit~~ the ~~80edo patent~~ val ~~mapping~~.

Using 11/10 as a generator tempering out 4000/3993 (as previously mentioned) leads to scales that look like [[porcupine]] but whose harmonies can more accurately be explained. A half-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 detemper [[100/99]] ~ [[121/120]] and accurately find [[11-limit]] and (no-13's) [[17-limit]] harmonies. Of these, echidna's mapping of the no-13's 17-limit is the simplest, though all three have the same mapping of the 2.3.7.11/10.17 subgroup so that they only differ on the mapping of 5 and 11. The complexity of 5 and 11 in pogo are used to increase accuracy, being a weak schismic extension. That leaves supers as the odd one out; if you are using an edo tuning for it, 58edo supports echidna while 94edo supports pogo, so it seems to exist as a portable alternate way of finding primes 5 and 11 across systems, unless you use the 152edo tuning, which requires using the second-best mapping of 13 (the 152f [[val]]).

~~== Trivia ==~~

Using sqrt(11/10) (22/21[[~]]21/20) 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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[[Category:Neutral second]]

[[Category:Submajor second]]

[[Category:Over-5]]

[[Category:Over-5 intervals]]

[[Category:Equable heptatonic]]

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 | Sound = jid_11_10_pluck_adu_dr220.mp3
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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.  Despite its being tempered together with [[12/11]] in systems like [[24edo]], this interval, on its own, is considerably less exotic than 12/11 in terms of sound when compared to a number of other [[11-limit]] intervals- in fact, 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 [[22edo]] and [[41edo]] where 11/10 and 10/9 are tempered together.
+/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]]. If it is used as a generator instead, it produces [[porcupine]], although it is slightly sharper than the optimal tuning for porcupine and does not fit the 80edo patent val mapping.
+Using 11/10 as a generator tempering out 4000/3993 (as previously mentioned) leads to scales that look like [[porcupine]] but whose harmonies can more accurately be explained. A half-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 detemper [[100/99]] ~ [[121/120]] and accurately find [[11-limit]] and (no-13's) [[17-limit]] harmonies. Of these, echidna's mapping of the no-13's 17-limit is the simplest, though all three have the same mapping of the 2.3.7.11/10.17 subgroup so that they only differ on the mapping of 5 and 11. The complexity of 5 and 11 in pogo are used to increase accuracy, being a weak schismic extension. That leaves supers as the odd one out; if you are using an edo tuning for it, 58edo supports echidna while 94edo supports pogo, so it seems to exist as a portable alternate way of finding primes 5 and 11 across systems, unless you use the 152edo tuning, which requires using the second-best mapping of 13 (the 152f [[val]]).
-== Trivia ==
+Using sqrt(11/10) (22/21[[~]]21/20) 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 ==
@@ Line 26: / Line 26: @@
 [[Category:Neutral second]]
 [[Category:Submajor second]]
-[[Category:Over-5]]
+[[Category:Over-5 intervals]]
 [[Category:Equable heptatonic]]