11edo

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← 10edo 11edo 12edo →
Prime factorization 11 (prime)
Step size 109.091 ¢ 
Fifth 6\11 (654.545 ¢)
Semitones (A1:m2) -2:3 (-218.2 ¢ : 327.3 ¢)
Dual sharp fifth 7\11 (763.636 ¢)
Dual flat fifth 6\11 (654.545 ¢)
Dual major 2nd 2\11 (218.182 ¢)
(semiconvergent)
Consistency limit 3
Distinct consistency limit 3

11 equal divisions of the octave (11edo), or 11-tone equal temperament (11-TET, 11ET) when viewed from a regular temperament perspective, is the tuning that divides the octave into eleven equal steps of about 109.09 cents. It is the fifth prime edo, after 2edo, 3edo, 5edo, and 7edo.

Theory

Approximation of odd harmonics in 11edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -47.4 +50.0 +13.0 +14.3 -5.9 +32.2 +2.6 +4.1 +29.8 -34.4 +26.3
Relative (%) -43.5 +45.9 +11.9 +13.1 -5.4 +29.5 +2.4 +3.8 +27.3 -31.5 +24.1
Steps
(reduced) 		17
(6) 		26
(4) 		31
(9) 		35
(2) 		38
(5) 		41
(8) 		43
(10) 		45
(1) 		47
(3) 		48
(4) 		50
(6)

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] 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 concordant. 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 545.45 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 2.9.15.7.11.17 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 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a wtn.

Intervals and notation

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".

The 11edo solfege in the table is derived from 22edo solfege.

# Cents Solfege Approximate Ratios* Sagittal
notation
(22edo subset) 		Up/down notation
with major wider
than minor 		Up/down notation
with major narrower
than minor 		Smitonic
(3rd-gen)
notation 		TDW
Machine
notation 		Pseudo-Diatonic Category Audio
0 0.00 do 1/1 A P1 A P1 A A Q, P# Unison
1 109.09 ra 15/14, 16/15, 17/16, 18/17 AII\ or B!!/ ^1, m2 ^A, B ^1, M2 ^A, B A#, Bb Q#, Rb Minor second
2 218.18 re 8/7, 9/8, 17/15 B ~2, m3 ^B, Cb ~2, M3 ^B, C# B R Major second
3 327.27 me 6/5, 11/9, 17/14 C/I or BII\ or D\!!/ M2, ~3 B#, vC m2, ~3 Bb, vC C R#, Sb Minor third
4 436.36 mo 9/7, 14/11, 22/17 D\! or C/II\ M3, v4 C, vD m3, v4 C, vD C#, Db S Major third/Minor fourth
5 545.45 fu 11/8, 15/11 D/I or E\!!/ P4, v5 D, vE P4, v5 D, vE D S#, Tb Major fourth
6 654.55 su 16/11, 22/15 E\! or D/II\ ^4, P5 ^D, E ^4, P5 ^D, E D#, Eb T Minor fifth
7 763.64 lo 14/9, 11/7, 17/11 F ^5, m6 ^E, Fb ^5, M6 ^E, F# E T#, Ub Major fifth/Minor sixth
8 872.73 la 5/3, 18/11, 28/17 FII\ or G!!/ ~6, m7 vF, Gb ~6, M7 vF, G# F U Major sixth
9 981.82 ta 7/4, 16/9, 30/17 G M6, ~7 F, vG m6, ~7 F, vG F#, Gb U#, Pb Minor seventh
10 1090.91 ti 15/8, 17/9, 28/15, 32/17 GII\ or A!!/ M7, v8 G, vAv m7, v8 G, vAv G P, Qb Major seventh
11 1200.00 do 2/1 A P8 A P8 A A Q, P# Octave
  • in 2.7.9.11.15.17 subgroup

11edo in Sagittal notation:

Sagittal11EDO.jpg

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

Pentatonic 5th-generated: D * * E G * * A C * * D (Sensoid generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

Octatonic 5th-generated: A B * C D E * F G * H A (Sensoid generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

Nonatonic 5th-generated: A B * C D E F G * H J A (Joanatonic generator = narrow 3/2 = 6\11 = perfect 6th)

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

Heptatonic 3rd-generated: D * E F * G A * B C * D (Smitonic generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

Hexatonic 2nd-generated: R * S * T * U * P Q * R (Machinoid generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

Regular temperament properties

Commas

11edo tempers out the following commas. (Note: This assumes val ⟨11 17 26 31 38 41].)

Prime
limit 		Ratio[1] Monzo Cents Color name Name(s)
5 135/128 [-7 3 1⟩ 92.18 Layobi Major chroma
5 (16 digits) [-25 7 6⟩ 31.57 Lala-tribiyo Ampersand's comma
5 (42 digits) [-68 18 17⟩ 2.52 Quinla-seyo Vavoom
7 (18 digits) [-10 7 8 -7⟩ 22.41 Lasepru-aquadbiyo Blackjackisma
7 1029/1024 [-10 1 0 3⟩ 8.43 Latrizo Gamelisma
7 225/224 [-5 2 2 -1⟩ 7.71 Ruyoyo Marvel comma
7 16875/16807 [0 3 4 -5⟩ 6.99 Quinru-aquadyo Mirkwai
7 2401/2400 [-5 -1 -2 4⟩ 0.72 Bizozogu Breedsma
11 121/120 [-3 -1 -1 0 2⟩ 14.37 Lologu Biyatisma
11 65536/65219 [16 0 0 -2 -3⟩ 8.39 Satrilu-aruru Orgonisma
  1. ↑ Ratios longer than 10 digits are presented by placeholders with informative hints

JI approximation

Harmonic 8 9 11 14 16
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\11 = 0¢ 2\11 = 218¢ 5\11 = 545 9\11 = 982 11\11 = 1200
Difference 0 +14¢ -6¢ +13¢ 0¢
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
Difference +14¢ -20¢ +18¢ -13¢

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

11edo_approx_2-7-9-11-15-17_2ndsave.png

Scales

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.

2\11 generates 2 2 2 2 3, a 1L 4s scale named machine[5]; and 2 2 2 2 2 1, a 5L 1s scale named machine[6].

3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a 4L 3s scale named orgone[7].

4\11 generates 4 4 3; 1 3 1 3 3, a 3L 2s scale; and 1 1 2 1 1 2 1 2, a 3L 5s scale.

5\11 generates joan scales 5 5 1; 1 4 1 4 1, a 2L 3s scale; 1 1 3 1 1 3 1, a 2L 5s scale; and 1 1 1 2 1 1 1 2 1, a 2L 7s scale.

See 11edo Modes

Pathological modes

2 1 1 1 2 1 1 1 1 2L 7s MOS

3 1 1 1 1 1 1 1 1 1L 8s MOS

2 1 1 1 1 1 1 1 1 1 1L 9s MOS

Instruments

11edo ukulele:

11-edo-ukulele.JPG

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.

Introductory Materials

Music

See also: Category:11edo tracks

Videos

See also

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