11 tone equal temperament
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone on the whites. The superfluous Ab can be made a note of 22edo, a tuning known as "elevenplus".
Compared to 12edo, the intervals of 11edo are stretched:
- The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
- The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
- The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
- The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
- The "perfect fourth," at 545.455 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.
11edo provides the same tuning on the 2*11 subgroup 126.96.36.199.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.
|JI interval from 1/1||1/1 = 0 cents||9/8 = 204||11/8 = 551||7/4 = 969||2/1 = 1200|
|nearest 11edo interval||0\11edo = 0¢||2\11 = 218¢||5\11 = 545||9\11 = 982||11\11 = 1200|
|JI interval between||9:8 = 204¢||11:9 = 347||14:11 = 418||8:7 = 231|
|nearest 11edo interval||2\11 = 218¢||3\11 = 327||4\11 = 436||2\11 = 218|
11edo also may be considered a 188.8.131.52.15.17 subgroup temperament. See diagram:
An 11edo solfege system can easily be applied from the 22edo solfege system.
A chromatic scale would thus be sung: do ra re me mo fu su lo la ta ti do.
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the melodic meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.
The second approach preserves the harmonic meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".
|Degree||Size in||Solfege||Approximate Ratios*||Sagittal
with major wider
with major narrower
|1||109.09||115.64||139.64 (8B.A316)||ra||15/14, 16/15, 17/16, 18/17||AII\ or B!!/||^1, m2||A^, B||^1, M2||A^, B||Q#\Rb|
|2||218.18||231.27||279.27 (117.4616)||re||8/7, 9/8, 17/15||B||~2, m3||B^, Cb||~2, M3||B^, C\\||R|
|3||327.27||346.91||418.91 (1A2.E8816)||me||6/5, 11/9, 17/14||C/I or BII\ or D\!!/||M2, ~3||B#, Cv||m2, ~3||Bp, Cv||R#\Sb|
|4||436.36||462.545||558.545 (22E.8C16)||mo||9/7, 14/11, 22/17||D\! or C/II\||M3, v4||C, Dv||m3, v4||C, Dv||S|
|5||545.455||578.18||698.18 (2BA.2F16)||fu||11/8, 15/11||D/I or E\!!/||P4, v5||D, Ev||P4, v5||D, Ev||S#\Tb|
|6||654.545||693.82||837.82 (345.D116)||su||16/11, 22/15||E\! or D/II\||^4, P5||D^, E||^4, P5||D^, E||T|
|7||763.64||809.455||977.455 (3D1.7416)||lo||11/7, 14/9, 17/11||F||^5, m6||E^, Fb||^5, M6||E^, F\\||T#\Ub|
|8||872.73||925.09||1117.09 (45D.17816)||la||5/3, 18/11, 28/17||FII\ or G!!/||~6, m7||Fv, Gb||~6, M7||Fv, G\\||U|
|9||981.82||1040.73||1256.73 (4E8.BA16)||ta||7/4, 16/9, 30/17||G||M6, ~7||F, Gv||m6, ~7||F, Gv||U#\Pb|
|10||1090.91||1156.36||1396.36 (574.5D16)||ti||15/8, 17/9, 28/15, 32/17||GII\ or A!!/||M7, v8||G, Av||m7, v8||G, Av||P\Qb|
- in 184.108.40.206.15.17 subgroup
11edo in Sagittal notation:
For alternative notations, see Ups and Downs Notation -"Supersharp" EDOs (pentatonic, octotonic and nonatonic fifth-generated) and Ups and Downs Notation - Natural Generators (heptatonic third-generated).
Although 11edo has one fewer interval in the octave than 12edo, in terms of moment-of-symmetry scales, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.
See 11edo Modes
11 EDO tempers out the following commas. (Note: This assumes val < 11 17 26 31 38 41 |.)
|Rational||Monzo||Size (Cents)||Name 1||Name 2||Name 3|
|135/128||| -7 3 1 >||92.18||Major Chroma||Major Limma||Pelogic Comma|
|9931568/9752117||| -25 7 6 >||31.57||Ampersand's Comma|
|1776337/1773750||| -68 18 17 >||2.52||Vavoom|
|9859966/9733137||| -10 7 8 -7 >||22.41||Blackjackisma|
|1029/1024||| -10 1 0 3 >||8.43||Gamelisma|
|225/224||| -5 2 2 -1 >||7.71||Septimal Kleisma||Marvel Comma|
|16875/16807||| 0 3 4 -5 >||6.99||Mirkwai|
|2401/2400||| -5 -1 -2 4 >||0.72||Breedsma|
|121/120||| -3 -1 -1 0 2 >||14.37||Biyatisma|
|65536/65219||| 16 0 0 -2 -3 >||8.39||Orgonisma|
11edo Instant Ensemble
In February 2011, Oddmusic U-C, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See 11edo Zine.
Icicle Caverns by Dr. Ozan Yarman
Angkor Wat, September 1066 by X. J. Scott
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.