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, 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 ==
== 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 ==
== 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 ==
== See also ==

Revision as of 18:49, 2 April 2025

Interval information
Ratio 11/10
Factorization 2-1 × 5-1 × 11
Monzo [-1 0 -1 0 1⟩
Size in cents 165.0042¢
Names large undecimal neutral second,
undecimal submajor second
Color name 1og2, logu 2nd
FJS name [math]\displaystyle{ \text{m2}^{11}_{5} }[/math]
Special properties superparticular,
reduced
Tenney norm (log2 nd) 6.78136
Weil norm (log2 max(n, d)) 6.91886
Wilson norm (sopfr(nd)) 18

[sound info]
Open this interval in xen-calc

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 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 chords as a step between notes.

11/10 is the octave-reduced form of 11/5, one of the three most concordant 11-limit intervals within the entire 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 about 4c sharp of the chroma of BPS.

Temperaments

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.

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.

See also

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