11/10: Difference between revisions

Line 4:

| Sound = jid_11_10_pluck_adu_dr220.mp3

}}

~~{{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, however, one could make the argument that 11/10 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 due to [[100/99]] being tempered out.

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]].

Meanwhile, 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]] or [[tempered out]], but is also very close in size to a stack consisting of an [[apotome]] and [[33/32]], leading to the [[schisma]] being fudged or tempered out.

Assuming you go with either of the aforementioned options, keeping 11/10 distinct from 12/11 ensures that 11/10 has a way of bridging [[quartertone]]-based chords with more typical [[5-limit]] and [[Pythagorean tuning|Pythagorean]] chords as a sort of step between notes, however, if you temper out [[121/120]], expect this ability to vanish.

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]]. In any [[Octave equivalence|octave-repeating]] tuning, a good approximation of 11/10 indicates a good approximation of 11/5. So, it could be argued that 11/10 is a high priority for any octave-repeating 11-limit tuning.

== 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 about 4c sharp of the [[chroma]] of [[BPS]].

== Temperaments ==

11/10 ~~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~~ temperaments in the [[stearnsmic clan]] such as [[pogo]], [[supers]], or [[echidna]], all of which detemper [[100/99]] and [[121/120]] and efficiently and accurately find [[11-limit]] and (no-13's) [[17-limit]] harmonies.

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. With a half-octave period, a generator of 11/10 leads to temperaments in the [[stearnsmic clan]] such as [[pogo]], [[supers]], or [[echidna]], all of which detemper [[100/99]] and [[121/120]] and efficiently and accurately find [[11-limit]] and (no-13's) [[17-limit]] harmonies.

~~=== Exotemperaments ===~~

11/10 is tempered out in the [[patent val]]s of [[edo]]s 1, 2, 3 and 5. An example rank-2 [[exotemperament]] that treats it as a comma to be tempered out is [[very low accuracy temperaments #Antietam|antietam]].

~~== Trivia ==~~

Using sqrt(11/10) as a generator leads to low-complexity [[Nautilus]] with one period to the octave and high-accuracy [[Harry]] with two periods; 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 ==

@@ Line 4: / Line 4: @@
 | Sound = jid_11_10_pluck_adu_dr220.mp3
 }}
-{{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, however, one could make the argument that 11/10 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 due to [[100/99]] being tempered out.
+/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]].
-Meanwhile, 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]] or [[tempered out]], but is also very close in size to a stack consisting of an [[apotome]] and [[33/32]], leading to the [[schisma]] being fudged or tempered out.
-Assuming you go with either of the aforementioned options, keeping 11/10 distinct from 12/11 ensures that 11/10 has a way of bridging [[quartertone]]-based chords with more typical [[5-limit]] and [[Pythagorean tuning|Pythagorean]] chords as a sort of step between notes, however, if you temper out [[121/120]], expect this ability to vanish.
-/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]]. In any [[Octave equivalence|octave-repeating]] tuning, a good approximation of 11/10 indicates a good approximation of 11/5. So, it could be argued that 11/10 is a high priority for any octave-repeating 11-limit tuning.
 == 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 about 4c sharp of the [[chroma]] of [[BPS]].
 == Temperaments ==
-/10 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 temperaments in the [[stearnsmic clan]] such as [[pogo]], [[supers]], or [[echidna]], all of which detemper [[100/99]] and [[121/120]] and efficiently and accurately find [[11-limit]] and (no-13's) [[17-limit]] harmonies.
+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. With a half-octave period, a generator of 11/10 leads to temperaments in the [[stearnsmic clan]] such as [[pogo]], [[supers]], or [[echidna]], all of which detemper [[100/99]] and [[121/120]] and efficiently and accurately find [[11-limit]] and (no-13's) [[17-limit]] harmonies.
-=== Exotemperaments ===
-/10 is tempered out in the [[patent val]]s of [[edo]]s 1, 2, 3 and 5. An example rank-2 [[exotemperament]] that treats it as a comma to be tempered out is [[very low accuracy temperaments #Antietam|antietam]].
-== Trivia ==
+Using sqrt(11/10) as a generator leads to low-complexity [[Nautilus]] with one period to the octave and high-accuracy [[Harry]] with two periods; 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 ==