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→21st century: Add Bryan Deister's ''11edo groove'' (2026)
 
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{{Japanese version page| [[11平均律|日本語]] }}
{{interwiki
| de = 11-EDO
| en = 11edo
| es =
| ja = 11平均律
}}
{{Infobox ET}}
{{ED intro}}
== Theory ==
Compared to 12edo, the intervals of 11edo are stretched:


__FORCETOC__
* The "minor second" at 109.09 cents, functions melodically 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.
{{Harmonics in equal|11}}


=11 tone equal temperament=
11edo does not approximate many small prime harmonics well, only providing good approximations to 7/4 and 11/8. However, 11edo can be treated as a subset of 22edo, and take 22edo's [[6/5]], [[9/7]], and [[16/15]] via direct approximation.


11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s 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.


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|22edo]], a tuning known as "elevenplus".
11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only [[5832/5831|0.3 cents]]. It may therefore be worth considering this JI tuning as an alternative to 11edo.


=Tuning=
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:


<ul><li>The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.</li><li>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.</li><li>The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."</li><li>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.</li><li>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.</li></ul>
[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]


=Subgroup=
== Intervals and Notation ==
11edo provides the same tuning on the [[k*N_subgroups|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.
=== Ups and downs 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".


=Intervals=
The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].


{| class="wikitable"
{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and downs notation|Up/down notation]] <br> with major wider <br> than minor
! colspan="2" | Up/down notation <br> with major narrower <br> than minor
! [[Smitonic]]<br>(3rd-gen)<br>notation
! [[Tútim Dennsuul Wafiil|TDW]] <br> [[Machinoid|Machine]] <br> notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
|-
! | Harmonic
| 5
! | 8
| 545.45
! |  
| '''fu'''
! | 9
| [[11/8]], [[15/11]]
! |  
| P4, v5
! | 11
| D, vE
! |  
| P4, v5
! | 14
| D, vE
! |  
| D
! | 16
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
|-
! | JI interval from 1/1
| 6
| | 1/1 = 0 cents
| 654.55
| |
| '''su'''
| | 9/8 = 204
| [[16/11]], [[22/15]]
| |  
| ^4, P5
| | 11/8 = 551
| ^D, E
| |
| ^4, P5
| | 7/4 = 969
| ^D, E
| |
| D#, Eb
| | 2/1 = 1200
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
|-
! | nearest 11edo interval
| 7
| | 0\11edo = 0¢
| 763.64
| |
| '''lo'''
| | 2\11 = 218¢
| [[14/9]], [[11/7]], [[17/11]]
| |  
| ^5, m6
| | 5\11 = 545
| ^E, Fb
| |  
| ^5, M6
| | 9\11 = 982
| ^E, F#
| |
| E
| | 11\11 = 1200
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
|-
! | difference
| 8
| | 0
| 872.73
| |  
| '''la'''
| | +14¢
| [[5/3]], [[18/11]], [[28/17]]
| |
| ~6, m7
| | -6¢
| vF, Gb
| |
| ~6, M7
| | +13¢
| vF, G#
| |
| F
| | 0¢
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
|-
! | JI interval between
| 9
| |  
| 981.82
| | 9:8 = 204¢
| '''ta'''
| |  
| [[7/4]], [[16/9]], [[30/17]]
| | 11:9 = 347
| M6, ~7
| |
| F, vG
| | 14:11 = 418
| m6, ~7
| |
| F, vG
| | 8:7 = 231
| F#, Gb
| |  
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
|-
! | nearest 11edo interval
| 10
| |  
| 1090.91
| | 2\11 = 218¢
| '''ti'''
| |  
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| | 3\11 = 327
| M7, v8
| |
| G, vAv
| | 4\11 = 436
| m7, v8
| |
| G, vAv
| | 2\11 = 218
| G
| |  
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
|-
! | difference
| 11
| |  
| 1200.00
| | +14¢
| '''do'''
| |  
| 2/1
| | -20¢
| P8
| |
| A
| | +18¢
| P8
| |
| A
| | -13¢
| A
| |  
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
|}
<nowiki>* in 2.7.9.11.15.17 subgroup</nowiki>


11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:
The ups and downs notations above 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.
 
'''<u>Pentatonic 5th-generated:</u>''' '''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)
 
'''<u>Octatonic 5th-generated:</u>''' '''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...
 
'''<u>Nonatonic 5th-generated:</u> 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...
 
'''<u>Heptatonic 3rd-generated:</u> D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)


[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]
D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D


==11edo solfege==
P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8
An 11edo solfege system can easily be applied from the [[22edo_Solfege|22edo solfege]] system.


A chromatic scale would thus be sung: '''do ra re me mo fu su lo la ta ti do'''.
genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...


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


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.
'''<u>Hexatonic 2nd-generated:</u> R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)


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


{| class="wikitable"
P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7
|-
! | Degree
! | Size in


[[cent|Cents]]
genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...
! | Solfege
! | Approximate Ratios*
! | Sagittal


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


(22edo subset)
===Sagittal notation===
! colspan="2" | [[Ups_and_Downs_Notation|Up/down notation]]
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====


with major wider
<imagemap>
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
</imagemap>


than minor
====Revo flavor====
! colspan="2" | Up/down notation


with major narrower
<imagemap>
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
</imagemap>


than minor
== Regular temperament properties ==
! | [[Tútim_Dennsuul_Wafiil|TDW]]
=== Uniform maps ===
{{Uniform map|edo=11}}


Machine
=== Commas ===
11et [[tempering out|tempers out]] the following [[comma]]s. This assumes val {{val| 11 17 26 31 38 41 }}.


notation
{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
|-
| style="text-align:center;" | 0
! [[Harmonic limit|Prime<br>limit]]
| style="text-align:right;" | 0.00
! [[Ratio]]<ref group=note>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
| style="text-align:center;" | '''do'''
! [[Monzo]]
| style="text-align:center;" | 1/1
! [[Cent]]s
| style="text-align:center;" | A
! [[Color name]]
| style="text-align:center;" | P1
! Name(s)
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|-
|-
| style="text-align:center;" | 1
| 3
| style="text-align:right;" | 109.09
| [[177147/131072]]
| style="text-align:center;" | '''ra'''
| {{Monzo| -17 11 }}
| style="text-align:center;" | [[15/14|15/14]], [[16/15|16/15]], [[17/16|17/16]], [[18/17|18/17]]
| 521.50
| style="text-align:center;" | AII\ or B!!/
| sasawa 3rd
| style="text-align:center;" | ^1, m2
| Pythagorean augmented third
| style="text-align:center;" | A^, B
| style="text-align:center;" | ^1, M2
| style="text-align:center;" | A^, B
| style="text-align:center;" | Q#\Rb
|-
|-
| style="text-align:center;" | 2
| 5
| style="text-align:right;" | 218.18
| [[135/128]]
| style="text-align:center;" | '''re'''
| {{Monzo| -7 3 1 }}
| style="text-align:center;" | [[8/7|8/7]], [[9/8|9/8]], [[17/15|17/15]]
| 92.18
| style="text-align:center;" | B
| Layobi
| style="text-align:center;" | ~2, m3
| Major chroma
| style="text-align:center;" | B^, Cb
| style="text-align:center;" | ~2, M3
| style="text-align:center;" | B^, C#
| style="text-align:center;" | R
|-
|-
| style="text-align:center;" | 3
| 5
| style="text-align:right;" | 327.27
| [[144/125]]
| style="text-align:center;" | '''me'''
| {{Monzo| 4 2 -3 }}
| style="text-align:center;" | [[6/5|6/5]], [[11/9|11/9]], [[17/14|17/14]]
| 244.97
| style="text-align:center;" | C/I or BII\ or D\!!/
| Trigu
| style="text-align:center;" | M2, ~3
| University comma
| style="text-align:center;" | B#, Cv
| style="text-align:center;" | m2, ~3
| style="text-align:center;" | Bb, Cv
| style="text-align:center;" | R#\Sb
|-
|-
| style="text-align:center;" | 4
| 5
| style="text-align:right;" | 436.36
| [[34171875/33554432|(16 digits)]]
| style="text-align:center;" | '''mo'''
| {{Monzo| -25 7 6 }}
| style="text-align:center;" | [[9/7|9/7]], [[14/11|14/11]], [[22/17|22/17]]
| 31.57
| style="text-align:center;" | D\! or C/II\
| Lala-tribiyo
| style="text-align:center;" | M3, v4
| [[Ampersand comma]]
| style="text-align:center;" | C, Dv
| style="text-align:center;" | m3, v4
| style="text-align:center;" | C, Dv
| style="text-align:center;" | S
|-
|-
| style="text-align:center;" | 5
| 5
| style="text-align:right;" | 545.45
| <abbr title="295578376007080078125/295147905179352825856">(42 digits)</abbr>
| style="text-align:center;" | '''fu'''
| {{Monzo| -68 18 17 }}
| style="text-align:center;" | [[11/8|11/8]], [[15/11|15/11]]
| 2.52
| style="text-align:center;" | D/I or E\!!/
| Quinla-seyo
| style="text-align:center;" | P4, v5
| [[Vavoom comma]]
| style="text-align:center;" | D, Ev
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, Ev
| style="text-align:center;" | S#\Tb
|-
|-
| style="text-align:center;" | 6
| 7
| style="text-align:right;" | 654.55
| <abbr title="854296875/843308032">(18 digits)</abbr>
| style="text-align:center;" | '''su'''
| {{Monzo| -10 7 8 -7 }}
| style="text-align:center;" | [[16/11|16/11]], [[22/15|22/15]]
| 22.41
| style="text-align:center;" | E\! or D/II\
| Lasepru-aquadbiyo
| style="text-align:center;" | ^4, P5
| [[Blackjackisma]]
| style="text-align:center;" | D^, E
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | D^, E
| style="text-align:center;" | T
|-
|-
| style="text-align:center;" | 7
| 7
| style="text-align:right;" | 763.64
| [[1029/1024]]
| style="text-align:center;" | '''lo'''
| {{Monzo| -10 1 0 3 }}
| style="text-align:center;" | [[11/7|11/7]], [[14/9|14/9]], [[17/11|17/11]]
| 8.43
| style="text-align:center;" | F
| Latrizo
| style="text-align:center;" | ^5, m6
| Gamelisma
| style="text-align:center;" | E^, Fb
| style="text-align:center;" | ^5, M6
| style="text-align:center;" | E^, F#
| style="text-align:center;" | T#\Ub
|-
|-
| style="text-align:center;" | 8
| 7
| style="text-align:right;" | 872.73
| [[225/224]]
| style="text-align:center;" | '''la'''
| {{Monzo| -5 2 2 -1 }}
| style="text-align:center;" | [[5/3|5/3]], [[18/11|18/11]], [[28/17|28/17]]
| 7.71
| style="text-align:center;" | FII\ or G!!/
| Ruyoyo
| style="text-align:center;" | ~6, m7
| Marvel comma
| style="text-align:center;" | Fv, Gb
| style="text-align:center;" | ~6, M7
| style="text-align:center;" | Fv, G#
| style="text-align:center;" | U
|-
|-
| style="text-align:center;" | 9
| 7
| style="text-align:right;" | 981.82
| [[16875/16807]]
| style="text-align:center;" | '''ta'''
| {{Monzo| 0 3 4 -5 }}
| style="text-align:center;" | [[7/4|7/4]], [[16/9|16/9]], [[30/17|30/17]]
| 6.99
| style="text-align:center;" | G
| Quinru-aquadyo
| style="text-align:center;" | M6, ~7
| Mirkwai comma
| style="text-align:center;" | F, Gv
| style="text-align:center;" | m6, ~7
| style="text-align:center;" | F, Gv
| style="text-align:center;" | U#\Pb
|-
|-
| style="text-align:center;" | 10
| 7
| style="text-align:right;" | 1090.91
| [[2401/2400]]
| style="text-align:center;" | '''ti'''
| {{Monzo| -5 -1 -2 4 }}
| style="text-align:center;" | [[15/8|15/8]], [[17/9|17/9]], [[28/15|28/15]], [[32/17|32/17]]
| 0.72
| style="text-align:center;" | GII\ or A!!/
| Bizozogu
| style="text-align:center;" | M7, v8
| Breedsma
| style="text-align:center;" | G, Av
| style="text-align:center;" | m7, v8
| style="text-align:center;" | G, Av
| style="text-align:center;" | P\Qb
|-
|-
| style="text-align:center;" | 11
| 11
| style="text-align:right;" | 1200.00
| [[121/120]]
| style="text-align:center;" | '''do'''
| {{Monzo| -3 -1 -1 0 2 }}
| style="text-align:center;" | 2/1
| 14.37
| style="text-align:center;" | A
| Lologu
| style="text-align:center;" | P8
| Biyatisma
| style="text-align:center;" | A
|-
| style="text-align:center;" | P8
| 11
| style="text-align:center;" | A
| [[65536/65219]]
| style="text-align:center;" | Q\P#
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
|}
*in 2.7.9.11.15.17 subgroup


11edo in [[Sagittal_notation|Sagittal notation]]:
== Approximation to JI ==
{| class="wikitable"
|-
! 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:
 
[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]
 
== Octave stretch or compression ==
11edo has about equally bad sharp and flat mappings of [[prime]]s 3 and 5. The 7 and 13 are quite sharp, but the 11 is a little flat. To use it as a 2.7.11.13 tuning, slight [[octave shrinking]] is advisable. Examples of slightly compressed versions of 11edo include (least to most compressed) [[ed6|28ed6]], [[ed12|39ed12]], [[zpi|30zpi]], [[equal tuning|35ed9]] and [[ed7|31ed7]].
 
To use its primes 3 or 5, extreme octave shrinking can be used, at the cost of making the octaves sound significantly weaker. [[equal tuning|37ed10]] is a very compressed version of 11edo.
 
== Scales ==
{{Main|11edo modes}}
 
=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|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.
 
== Instruments ==
'''11edo ukulele'''
 
[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]
 
'''Ensembles'''
 
In February 2011, [http://oddmusicuc.wordpress.com/ 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.


[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]
'''Lumatone'''


For alternative notations, see [[Ups_and_Downs_Notation#Summary of EDO notation-"Supersharp" EDOs|Ups and Downs Notation -"Supersharp" EDOs]] (pentatonic, octotonic and nonatonic fifth-generated) and [[Ups_and_Downs_Notation#Natural Generators|Ups and Downs Notation - Natural Generators]] (heptatonic third-generated).
[[Lumatone mapping for 11edo|Lumatone mappings for 11edo]] are available.


=MOS Scales=
== Introductory Materials ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|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.
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])


2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
== Music ==
{{Catrel|11edo tracks}}
=== 11 equal divisions of the octave (11edo proper) ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)


3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.


4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)


5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)


See [[11edo_Modes|11edo Modes]]
; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&index=4 "Towelbear"]


==Commas==
; [[Jacob Barton]]
11 EDO tempers out the following commas. (Note: This assumes val &lt; [http://tel.wikispaces.com/11_17_26_31_38_41 11 17 26 31 38 41] |.)
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]


{| class="wikitable"
; [[City of the Asleep]]
|-
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)
! | Rational
 
! | Monzo
; [[Jason Conklin]]
! | Size ([[cent|Cents]])
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]
! | Name 1
 
! | Name 2
; [[Bryan Deister]]
! | Name 3
* https://www.youtube.com/shorts/dsROnKnMnq8 ''11edo groove''] (2026)
|-
 
| style="text-align:center;" | [[135/128|135/128]]
; [[E8 Heterotic]]
| | | -7 3 1 &gt;
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)
| style="text-align:right;" | 92.18
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| | | -25 7 6 &gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 1776337/1773750
| | | -68 18 17 &gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" | Vavoom
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 9859966/9733137
| | | -10 7 8 -7 &gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 1029/1024
| | | -10 1 0 3 &gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | [[225/224|225/224]]
| | | -5 2 2 -1 &gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" |
|-
| style="text-align:center;" | 16875/16807
| | | 0 3 4 -5 &gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 2401/2400
| | | -5 -1 -2 4 &gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 121/120
| | | -3 -1 -1 0 2 &gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" | Biyatisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 65536/65219
| | | 16 0 0 -2 -3 &gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" | Orgonisma
| style="text-align:center;" |
| style="text-align:center;" |
|}


=11edo Instant Ensemble=
; [[Francium]]
In February 2011, [http://oddmusicuc.wordpress.com/ 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.
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* "Sleep Slope" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/2GnUioPrMlJmMaacE6DK7i Spotify] | [https://francium223.bandcamp.com/track/sleep-slope Bandcamp] | [https://www.youtube.com/watch?v=hM0BAC_YZnQ YouTube]


==11edo Zine==
; [[David Hamill]]
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|11edo Zine]].
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)


=Compositions=
; [[Andrew Heathwaite]]
<span style=""><span style=""><span style="">''[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]''</span></span></span> by [[George_Secor|George Secor]], 1970. Apparently the first piece ever written for 11edo.
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]


<span style=""><span style=""><span style="">''[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head]''</span></span></span> by [[David_Hamill|David Hamill]], 2010
; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)


Hyperimprovisations Nuggetwarp ([http://soundclick.com/share.cfm?id=10267904 I] [http://soundclick.com/share.cfm?id=10267905 II] [http://soundclick.com/share.cfm?id=10267906 III]) by [[Jacob_Barton|Jacob Barton]], 2009
; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}


<span style=""><span style=""><span style="">''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession]''</span></span></span> on City of the Asleep, [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]


[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] <span style=""><span style=""><span style="">''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]''</span></span></span> by [[Bill_Sethares|Bill Sethares]] (spectrally bent synth ens.)
; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}


<span style=""><span style=""><span style="">''[http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]''</span></span></span> by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]] (neo-Baroque)
; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)


[http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher_Bailey|Christopher Bailey]] (keyboards concréte) <span style=""><span style=""><span style="">''[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]''</span></span></span> <span style=""><span style=""><span style="">''[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]''</span></span></span> <span style=""><span style=""><span style="">''[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]''</span></span></span> <span style=""><span style=""><span style="">''[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]''</span></span></span>
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude & Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)


<span style=""><span style=""><span style="">''[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns]''</span></span></span> by Dr. Ozan Yarman
; [[Joseph Monzo]]
* [https://www.youtube.com/shorts/JMrFUKfqfeY ''Monzo, 2026-0608: 11edo, 11/8 time, piano, musescore3''] (2026)


[http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by X. J. Scott
; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning
* "Search Party" from ''No Fun House'' (2025) – [https://open.spotify.com/track/7CMiwDuuRuFpB0skfwT2Ap Spotify] | [https://mundoworld.bandcamp.com/track/search-party Bandcamp] | [https://www.youtube.com/watch?v=hLs6MjuousI YouTube]


[http://soundclick.com/share?songid=8839070 conversation is] <span style=""><span style=""><span style="">''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]''</span></span></span> by [[Andrew_Heathwaite|Andrew Heathwaite]].
; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)


Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)


[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&songID=933772 Orange Clips on Sausages] <span style=""><span style=""><span style="">''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]''</span></span></span> by Andrew Heathwaite
; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)


[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&songID=834492 Blue Gel] <span style=""><span style=""><span style="">''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]''</span></span></span> by Andrew Heathwaite
; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)


<span style=""><span style=""><span style="">''[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo]''</span></span></span> by [[Chris_Vaisvil|Chris Vaisvil]]
; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)


<span style=""><span style=""><span style="">''[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]''</span></span></span> by [[John_Lyle_Smith|John Lyle Smith]]
; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}


<span style=""><span style=""><span style=""><span style="">''[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor]''</span></span></span></span> by [[Chris_Vaisvil|Chris Vaisvil]]
; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}


<span style=""><span style="">''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]''</span></span> by [[Randy_Winchester|Randy Winchester]]
; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)


''[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]'' by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])


''[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo]'' by [[Jon_Lyle_Smith|Jon Lyle Smith]]
; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)


''[http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]'' by [[Aaron_Krister_Johnson|Aaron Krister Johnson]]
=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* ''11 Tone March'' (2023/2024)
** [https://www.youtube.com/shorts/K2QVvaRUXIQ <nowiki>[short clip]</nowiki>] (2023, with Lumatone view)
** [https://www.youtube.com/watch?v=z0lWcguNsNs <nowiki>[full version]</nowiki>] (2024, with tuning specification in video description)


''<span style="">[http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) (</span>''''[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]'' '') by [[Chris_Vaisvil|Chris Vaisvil]]''
== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]


''<span style="">[http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by [[Chris_Vaisvil|Chris Vaisvil]]</span>''
* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
* [https://www.youtube.com/watch?v=4WlTPfRDPCY untitled1], computer


==Videos==
== See also ==
The Stuffed Ones: <span style=""><span style=""><span style="">''[http://www.youtube.com/watch?v=NU0VvGRelUQ&feature=related Goopy]''</span></span></span>, <span style=""><span style=""><span style="">''[http://www.youtube.com/watch?v=4D9wDl_oxHE&feature=related Ziggy]''</span></span></span>, <span style=""><span style=""><span style="">''[http://www.youtube.com/watch?v=53IiHdXfJwI&feature=related Ellie]''</span></span></span>, <span style=""><span style=""><span style="">''[http://www.youtube.com/watch?v=4sZqpRcB-lk&feature=related Towelbear]''</span></span></span> by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.


<ul><li><span style=""><span style=""><span style="">''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''</span></span></span>, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube</li></ul>
== Notes ==
==Instruments==
<references group=note/>


[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]      [[Category:11-tone]]
[[Category:Listen]]
[[Category:11edo]]
{{Todo|add rank 2 temperaments table}}
[[Category:edo]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:prime_edo]]
[[Category:scale]]
[[Category:subgroup]]
[[Category:todo:unify_precision]]

Latest revision as of 07:30, 23 September 2026

← 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 (abbreviated 11edo or 11ed2), also called 11-tone equal temperament (11tet) or 11 equal temperament (11et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 11 equal parts of about 109 ¢ each. Each step represents a frequency ratio of 21/11, or the 11th root of 2.

Theory

Compared to 12edo, the intervals of 11edo are stretched:

  • The "minor second" at 109.09 cents, functions melodically 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.
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)

11edo does not approximate many small prime harmonics well, only providing good approximations to 7/4 and 11/8. However, 11edo can be treated as a subset of 22edo, and take 22edo's 6/5, 9/7, and 16/15 via direct approximation.

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 has a good approximation of 9/7, hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.

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".

A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.

Intervals and Notation

Ups and downs 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* 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 P1 A P1 A A Q, P# Unison
1 109.09 ra 15/14, 16/15, 17/16, 18/17 ^1, m2 ^A, B ^1, M2 ^A, B A#, Bb Q#, Rb Minor second
2 218.18 re 8/7, 9/8, 17/15 ~2, m3 ^B, Cb ~2, M3 ^B, C# B R Major second
3 327.27 me 6/5, 11/9, 17/14 M2, ~3 B#, vC m2, ~3 Bb, vC C R#, Sb Minor third
4 436.36 mo 9/7, 14/11, 22/17 M3, v4 C, vD m3, v4 C, vD C#, Db S Major third/Minor fourth
5 545.45 fu 11/8, 15/11 P4, v5 D, vE P4, v5 D, vE D S#, Tb Major fourth
6 654.55 su 16/11, 22/15 ^4, P5 ^D, E ^4, P5 ^D, E D#, Eb T Minor fifth
7 763.64 lo 14/9, 11/7, 17/11 ^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 ~6, m7 vF, Gb ~6, M7 vF, G# F U Major sixth
9 981.82 ta 7/4, 16/9, 30/17 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 M7, v8 G, vAv m7, v8 G, vAv G P, Qb Major seventh
11 1200.00 do 2/1 P8 A P8 A A Q, P# Octave

* in 2.7.9.11.15.17 subgroup

The ups and downs notations above 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...

Sagittal notation

This notation is a subset of the notations for EDOs 22, 44, and 66.

Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation22-EDO notation

Revo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation22-EDO notation

Regular temperament properties

Uniform maps

13-limit uniform maps between 10.8 and 11.2
Min. size Max. size Wart notation Map
10.6744 10.8399 11cdeef ⟨11 17 25 30 37 40]
10.8399 10.8643 11cdf ⟨11 17 25 30 38 40]
10.8643 10.9446 11cf ⟨11 17 25 31 38 40]
10.9446 10.9823 11c ⟨11 17 25 31 38 41]
10.9823 11.0413 11 ⟨11 17 26 31 38 41]
11.0413 11.1290 11b ⟨11 18 26 31 38 41]
11.1290 11.2149 11be ⟨11 18 26 31 39 41]

Commas

11et tempers out the following commas. This assumes val ⟨11 17 26 31 38 41].

Prime
limit 		Ratio[note 1] Monzo Cents Color name Name(s)
3 177147/131072 [-17 11⟩ 521.50 sasawa 3rd Pythagorean augmented third
5 135/128 [-7 3 1⟩ 92.18 Layobi Major chroma
5 144/125 [4 2 -3⟩ 244.97 Trigu University comma
5 (16 digits) [-25 7 6⟩ 31.57 Lala-tribiyo Ampersand comma
5 (42 digits) [-68 18 17⟩ 2.52 Quinla-seyo Vavoom comma
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 comma
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

Approximation to JI

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

Octave stretch or compression

11edo has about equally bad sharp and flat mappings of primes 3 and 5. The 7 and 13 are quite sharp, but the 11 is a little flat. To use it as a 2.7.11.13 tuning, slight octave shrinking is advisable. Examples of slightly compressed versions of 11edo include (least to most compressed) 28ed6, 39ed12, 30zpi, 35ed9 and 31ed7.

To use its primes 3 or 5, extreme octave shrinking can be used, at the cost of making the octaves sound significantly weaker. 37ed10 is a very compressed version of 11edo.

Scales

Main article: 11edo modes

MOS scales

Main article: List of 11edo 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.

Instruments

11edo ukulele

11-edo-ukulele.JPG

Ensembles

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.

Lumatone

Lumatone mappings for 11edo are available.

Introductory Materials

Music

See also: Category:11edo tracks

11 equal divisions of the octave (11edo proper)

Modern renderings

Arthur Schutt

20th century

George Secor
Bill Sethares

21st century

Abnormality
Christopher Bailey
Jacob Barton
City of the Asleep
Jason Conklin
Bryan Deister
E8 Heterotic
Francium
David Hamill
Andrew Heathwaite
Hideya
Aaron Andrew Hunt
Alexandru Ianu
Aaron Krister Johnson
groundfault
Claudi Meneghin
Joseph Monzo
Mundoworld
No Clue Music
NullPointerException Music
Phanomium
X. J. Scott
Sevish
Jon Lyle Smith
Chris Vaisvil
Randy Winchester
Ozan Yarman
Yeah Gore

Unequal Derivatives of 11edo

Bryan Deister
  • 11 Tone March (2023/2024)

Videos

See also

  • 11edo Zine — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.

Notes

  1. ↑ Ratios longer than 10 digits are presented by placeholders with informative hints
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