11/10: Difference between revisions
Wikispaces>spt3125 **Imported revision 513194580 - Original comment: ** |
m Text replacement - " {{Interval_Edo_Approximation | " to "{{Interval edo approximation|" |
||
| (45 intermediate revisions by 19 users not shown) | |||
| Line 1: | Line 1: | ||
{{Infobox Interval | |||
| Name = large undecimal neutral second, undecimal submajor second | |||
| Color name = 1og2, logu 2nd | |||
| Sound = jid_11_10_pluck_adu_dr220.mp3 | |||
}} | |||
'''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 | 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]]. | ||
{{Interval edo approximation|11/10}} | |||
== 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. 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]]). | |||
11/10 | |||
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. | |||
== See also == | |||
* [[20/11]] – its [[octave complement]] | |||
* [[15/11]] – its [[fifth complement]] | |||
* [[40/33]] – its [[fourth complement]] | |||
* [[Gallery of just intervals]] | |||
* [[List of superparticular intervals]] | |||
[[Category:Second]] | |||
[[Category:Neutral second]] | |||
[[Category:Submajor second]] | |||
[[Category:Over-5 intervals]] | |||
[[Category:Equable heptatonic]] | |||
Latest revision as of 13:07, 3 November 2025
| Interval information |
undecimal submajor second
reduced
[sound info]
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 4c sharp of an octave-reduced stack of 9 generators in BPS.
| Edo | Step size | Cents (¢) | Absolute error (¢) | Relative error (%) |
|---|---|---|---|---|
| 7 | 1\7 | 171.43 | +6.42 | +3.75 |
| 14 | 2\14 | 171.43 | +6.42 | +7.50 |
| 15 | 2\15 | 160.00 | -5.00 | -6.26 |
| 22 | 3\22 | 163.64 | -1.37 | -2.51 |
| 29 | 4\29 | 165.52 | +0.51 | +1.24 |
| 36 | 5\36 | 166.67 | +1.66 | +4.99 |
| 37 | 5\37 | 162.16 | -2.84 | -8.76 |
| 43 | 6\43 | 167.44 | +2.44 | +8.73 |
| 44 | 6\44 | 163.64 | -1.37 | -5.02 |
| 51 | 7\51 | 164.71 | -0.30 | -1.27 |
| 58 | 8\58 | 165.52 | +0.51 | +2.48 |
| 65 | 9\65 | 166.15 | +1.15 | +6.23 |
| 66 | 9\66 | 163.64 | -1.37 | -7.52 |
| 72 | 10\72 | 166.67 | +1.66 | +9.97 |
| 73 | 10\73 | 164.38 | -0.62 | -3.78 |
| 80 | 11\80 | 165.00 | -0.00 | -0.03 |
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. 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).
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.