# 11 tone equal temperament

11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents. It is the fifth prime edo, after 2edo, 3edo, 5edo, and 7edo.

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".

# Tuning

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 545.45 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.

# Subgroup

11edo provides the same tuning on the 2*11 subgroup 2.9.15.7.11 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.

# Intervals

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\11edo = 0¢ 2\11 = 218¢ 5\11 = 545 9\11 = 982 11\11 = 1200
difference 0 +14¢ -6¢ +13¢
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 solfege

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.

# 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".

Degree Size in Solfege Approximate Ratios* Sagittal

notation

(22edo subset)

Up/down notation

with major wider

than minor

Up/down notation

with major narrower

than minor

TDW

Machine

notation

Cents pions 7mus
0 0.00 do 1/1 A P1 A P1 A Q\P#
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.E916) 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.1716) 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
11 1200.00 1272 1536 (60016) do 2/1 A P8 A P8 A Q\P#
• in 2.7.9.11.15.17 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).

# MOS 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 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

## Commas

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.

## 11edo Zine

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.

# Compositions

File:11EDO-improv.mp3 First Piece Ever by George Secor, 1970. Apparently the first piece ever written for 11edo.

Cool My Head by David Hamill, 2010

Hyperimprovisations Nuggetwarp (I II III) by Jacob Barton, 2009

She Is My Lilac-Hued Obsession on City of the Asleep, Map of an Internal Landscape (2009)

The Turquoise Dabo Girl play by Bill Sethares (spectrally bent synth ens.)

Prelude11ET by Aaron Andrew Hunt (neo-Baroque)

The Stuffed Ones by Christopher Bailey (keyboards concréte) goopy ellie ziggy towelbear

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.

Orange Clips on Sausages play by Andrew Heathwaite

Blue Gel play by Andrew Heathwaite

Eleven Birds (video and music) ('audio only ) by Chris Vaisvil

## Videos

The Stuffed Ones: Goopy, Ziggy, Ellie, Towelbear by zipzappoozoo