11edo: Difference between revisions

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{{interwiki

| de =

| de = 11-EDO

| en = 11edo

| es =

| ja = 11平均律

}}~~__FORCETOC__~~

}}

=~~11 tone equal temperament~~=

== 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.

~~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 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.

~~=Tuning=~~

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.

~~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 545.455 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.~~</li></ul>~~

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.

~~=Subgroup=~~

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]]".

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~~.

~~=Intervals=~~

[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

{| class="wikitable"

== 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|22edo solfege]].

{| class="wikitable center-all right-2 left-4"

! #

! [[Cent]]s

! Solfege

! Approximate Ratios*

! colspan="2" | [[Ups and downs notation|Up/down notation]] with major wider than minor

! colspan="2" | Up/down notation with major narrower than minor

! [[Smitonic]] (3rd-gen) notation

! [[Tútim Dennsuul Wafiil|TDW]] [[Machinoid|Machine]] notation

! Pseudo-Diatonic Category

!Audio

|-

| 0

| 0.00

| '''do'''

| 1/1

| P1

| A

| P1

| 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>

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

~~11edo also may be considered a~~ 2.7.9.11.15.~~17 subgroup temperament~~. ~~See diagram~~:

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

~~[[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]]~~

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

~~==11edo solfege==~~

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

~~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: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

=~~Notation~~=

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

~~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.~~

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

~~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".

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

~~{| class="wikitable"~~

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

|-

~~! rowspan="2" | Degree~~

~~! colspan="3" | Size in~~

~~! rowspan="2" | Solfege~~

! rowspan="2" | Approximate Ratios*

~~! rowspan="2" | Sagittal~~

~~notation~~

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

~~(22edo subset)~~

===Sagittal notation===

~~! colspan~~=~~"2" rowspan~~=~~"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~~

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" rowspan~~=~~"2" | Up/down notation~~

with ~~major narrower~~

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 ==

~~! rowspan~~=~~"2"~~ | ~~[[Tútim_Dennsuul_Wafiil|TDW]]~~

=== Uniform maps ===

~~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"

|-

![[~~cent~~|~~Cents~~]]

! [[Harmonic limit|Prime limit]]

!~~pions~~

! [[Ratio]]<ref group=note>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>

!~~7mus~~

! [[Monzo]]

! [[Cent]]s

! [[Color name]]

! Name(s)

|-

| ~~colspan="4" style="text-align:center;" | 0~~

| 3

~~| style="text-align:center;" | '''do'''~~

| [[177147/131072]]

| ~~style="text-align:center;" | 1~~/1

| {{Monzo| -17 11 }}

~~| style="text-align:center;" | A~~

| 521.50

| ~~style="text-align:center;" | P1~~

| sasawa 3rd

| ~~style="text~~-~~align:center;" | A~~

| Pythagorean augmented third

| ~~style="text-align:center;" | P1~~

| ~~style="text-align:center;" | A~~

| ~~style="text-align:center;" | Q\P#~~

|-

| ~~style="text-align:center;" | 1~~

| 5

| ~~style="text-align:right;" | 109.09~~

| [[135/128]]

~~|115.64~~

| {{Monzo| -7 3 1 }}

~~|139.64 (8B.A316)~~

| 92.18

~~| style="text-align:center;" | '''ra'''~~

| Layobi

~~| style="text-align:center;" | [[15/14|15/14]],~~ [[~~16/15|16~~/~~15]], [[17/16|17/16]], [[18/17|18/17~~]]

| Major chroma

| ~~style="text-align:center;" | AII\ or B!!/~~

| ~~style="text~~-~~align:center;" | ^~~1~~, m2~~

| ~~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~~

| [[144/125]]

~~|231.27~~

| {{Monzo| 4 2 -3 }}

|~~279.27 (117.4616)~~

| 244.97

~~| style="text-align:center;" | '''re'''~~

| Trigu

~~| style="text-align:center;" | [[8/7|8/7]], [[9/8|9/8]],~~ [[~~17/15|17~~/15]]

| University comma

| ~~style="text-align:center;"~~ | B

~~| style="text-align:center;" | ~~~2~~, m3~~

~~| 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~~

| [[34171875/33554432|(16 digits)]]

~~|346.91~~

| {{Monzo| -25 7 6 }}

~~|418.91~~ (~~1A2.E88~~16~~~~)

| 31.57

~~| style="text-align:center;" | '''me'''~~

| Lala-tribiyo

~~| style="text-align:center;" | [[6/5|6/5]~~]~~, [[11/9|11/9~~]~~], [[17/14|17/14]]~~

| [[Ampersand comma]]

~~| style="text-align:center;" | C/I or BII\ or D\!!/~~

| ~~style="text-align:center;"~~ | ~~M2, ~3~~

~~| style="text~~-~~align:center;" | B#, Cv~~

| ~~style="text-align:center;" | m2, ~3~~

| ~~style="text~~-~~align:center;" | Bp, Cv~~

| ~~style="text-align:center;" | R#\Sb~~

|-

| ~~style="text-align:center;" | 4~~

| 5

| ~~style~~="~~text-align:right;~~" ~~| 436.36~~

| <abbr title="295578376007080078125/295147905179352825856">(42 digits)</abbr>

~~|462.545~~

| {{Monzo| -68 18 17 }}

~~|558.545~~ (~~22E.8C16~~</~~sub~~>)

| 2.52

| ~~style="text~~-~~align:center;"~~ | ~~'''mo'''~~

| Quinla-seyo

| ~~style="text~~-~~align:center;" | [[9/7~~|~~9/7]],~~ [[~~14/11|14/11~~]]~~, [[22/17|22/17]]~~

| [[Vavoom comma]]

~~| style="text-align:center;" | D\! or C/II\~~

~~| style="text-align:center;" | M3, v4~~

~~| 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~~

| 7

| ~~style~~="~~text-align:right;~~" ~~| 545.455~~

| <abbr title="854296875/843308032">(18 digits)</abbr>

~~|578.~~18

| {{Monzo| -10 7 8 -7 }}

~~|698.18 (2BA.2F16~~</~~sub~~>)

| 22.41

| ~~style="text-align:center;" | '''fu'''~~

| Lasepru-aquadbiyo

| ~~style="text~~-~~align:center;" | [[11/~~8~~|11/8]], [[15/11|15/11]]~~

| [[Blackjackisma]]

~~| style="text~~-~~align:center;" | D/I or E\!!/~~

| ~~style="text-align:center;" | P4, v5~~

| ~~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;" | 654.545~~

| [[1029/1024]]

~~|693.82~~

| {{Monzo| -10 1 0 3 }}

|~~837.82 (345.D116)~~

| 8.43

~~| style="text-align:center;" | '''su'''~~

| Latrizo

~~| style="text-align:center;" | [[16/11|16/11]],~~ [[~~22/15|22~~/15]]

| Gamelisma

| ~~style="text-align:center;" | E\! or D/II\~~

| ~~style="text~~-~~align:center;" | ^4, P5~~

| ~~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~~

| [[225/224]]

~~|809.455~~

| {{Monzo| -5 2 2 -1 }}

~~|977.455 (3D1.7416)~~

| 7.71

~~| style="text-align:center;" | '''lo'''~~

| Ruyoyo

~~| style="text-align:center;"~~ | [[~~11/7|11~~/~~7]], [[14/9|14/9]], [[17/11|17/11~~]]

| Marvel comma

| ~~style="text-align:center;"~~ | F

~~| style="text~~-~~align:center;" | ^~~5~~, m6~~

~~| 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~~

| [[16875/16807]]

~~|925.09~~

| {{Monzo| 0 3 4 -5 }}

~~|1117.09 (45D.17816)~~

| 6.99

~~| style="text-align:center;" | '''la'''~~

| Quinru-aquadyo

~~| style="text-align:center;"~~ | [[5/~~3|5/3]], [[18/11|18/11]], [[28/17|28/17~~]]

| Mirkwai comma

| ~~style="text-align:center;"~~ | ~~FII\ or G!!/~~

~~| style="text~~-~~align:center;" | ~6, m7~~

~~| 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~~

| [[2401/2400]]

~~|1040.73~~

| {{Monzo| -5 -1 -2 4 }}

~~|1256.73 (4E8.BA16)~~

| 0.72

~~| style="text-align:center;" | '''ta'''~~

| Bizozogu

~~| style="text-align:center;"~~ | [[7/~~4|7/4~~]]~~, [[16/9~~|~~16/9]], [[30/17~~|~~30/17]]~~

| Breedsma

~~| style="text~~-~~align:center;" | G~~

~~| style="text~~-~~align:center;" | M6, ~7~~

~~| 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~~

| 11

~~| style="text-align:right;" | 1090.91~~

| [[121/120]]

~~|1156.36~~

| {{Monzo| -3 -1 -1 0 2 }}

~~|1396.36 (574.5D16)~~

| 14.37

~~| style="text-align:center;" | '''ti'''~~

| Lologu

~~| style="text-align:center;"~~ | [[~~15/8|15~~/8]]~~, [[17/9|17/9]], [[28/15~~|~~28/15]], [[32/17|32/17]]~~

| Biyatisma

| ~~style="text~~-~~align:center;" | GII\ or A!!/~~

~~| style="text~~-~~align:center;" | M7, v8~~

~~| 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~~

| [[65536/65219]]

|~~1272~~

| {{Monzo| 16 0 0 -2 -3 }}

|~~1536 (600~~16~~)~~

| 8.39

~~| style="text-align:center;" | '''do'''~~

| Satrilu-aruru

~~| style="text~~-~~align:center;" |~~ 2/1

| Orgonisma

~~| style="text~~-~~align:center;" | A~~

| ~~style="text-align:center;" | P8~~

~~| style="text-align:center;" | A~~

~~| style="text-align:center;" | P8~~

| ~~style="text~~-~~align:center;" | A~~

| ~~style="text-align:center;" | Q\P#~~

|}

*in 2.7.9.11.15.17 subgroup

~~11edo in [[Sagittal_notation|Sagittal notation]]:~~

~~[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]~~

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).

~~=MOS Scales=~~

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.

~~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].~~

~~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].~~

~~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.~~

~~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.~~

~~See [[11edo_Modes|11edo Modes]]~~

~~==Commas==~~

~~11 EDO tempers out the following commas. (Note: This assumes val < 11 17 26 31 38 41 |.)~~

== Approximation to JI ==

{| class="wikitable"

|-

! ~~| Rational~~

! Harmonic

! ~~| Monzo~~

! 8

! ~~| Size ([[cent|Cents]])~~

!

! ~~| Name 1~~

! 9

! ~~| Name 2~~

!

! ~~| Name 3~~

! 11

|-

!

~~| style="text-align:center;" | [[135/128|135/128]]~~

! 14

~~| |<nowiki> | -7 3 1 </nowiki>>~~

!

~~| style="text-align:right;" | 92.18~~

! 16

~~| style="text-align:center;" | Major Chroma~~

~~| style="text-align:center;" | Major Limma~~

~~| style="text-align:center;" | Pelogic Comma~~

|-

~~| style="text-align:center;" | 9931568/9752117~~

~~| |<nowiki> | -25 7 6 </nowiki>>~~

~~| style="text-align:right;" | 31.57~~

~~| style="text-align:center;" | Ampersand's Comma~~

~~| style="text-align:center;" |~~

|-

~~| style="text-align:center;" | 1776337/1773750~~

~~| |<nowiki> | -68 18 17 </nowiki>>~~

~~| style="text-align:right;" | 2.52~~

~~| style="text-align:center;" | Vavoom~~

~~| style="text-align:center;" |~~

|-

~~| style="text-align:center;" | 9859966/9733137~~

~~| |<nowiki> | -10 7 8 -7 </nowiki>>~~

~~| style="text-align:right;" | 22.41~~

~~| style="text-align:center;" | Blackjackisma~~

~~| style="text-align:center;" |~~

|-

~~| style="text-align:center;" | 1029~~/~~1024~~

! JI interval from 1/1

| ~~|<nowiki> | -10~~ 1 0 ~~3 </nowiki>>~~

| 1/1 = 0 cents

| ~~style~~=~~"text-align:right;"~~ | ~~8.43~~

|

| ~~style~~=~~"text-align:center;"~~ | ~~Gamelisma~~

| 9/8 = 204

| ~~style~~=~~"text-align:center;"~~ |

|

| ~~style~~=~~"text-align:center;" |~~

| 11/8 = 551

|

| 7/4 = 969

|

| 2/1 = 1200

|-

| ~~style~~=~~"text-align:center;" | [[225/224~~|~~225/224]]~~

! Nearest 11edo interval

| ~~|<nowiki> | -5~~ 2 ~~2 -1 </nowiki>>~~

| 0\11 = 0¢

| ~~style="text-align:right;" | 7.71~~

|

| ~~style~~=~~"text-align:center;"~~ | ~~Septimal Kleisma~~

| 2\11 = 218¢

| ~~style~~=~~"text-align:center;"~~ | ~~Marvel Comma~~

|

| ~~style~~=~~"text-align:center;" |~~

| 5\11 = 545

|

| 9\11 = 982

|

| 11\11 = 1200

|-

| ~~style="text-align:center;"~~ | ~~16875/16807~~

! Difference

| |~~<nowiki>~~ | ~~0 3 4~~ -~~5 </nowiki>>~~

| 0

| ~~style="text-align:right;" | 6.99~~

|

| ~~style="text-align:center;" | Mirkwai~~

| +14¢

~~| style="text-align:center;"~~ |

|

| ~~style="text-align:center;" |~~

| -6¢

|

| +13¢

|

| 0¢

|-

~~| style="text-align:center;" | 2401/2400~~

! JI interval between

| ~~|<nowiki> | -5 -1 -2 4 </nowiki>>~~

|

| ~~style~~=~~"text-align:right;"~~ | ~~0.72~~

| 9:8 = 204¢

| ~~style~~=~~"text-align:center;"~~ | ~~Breedsma~~

|

| ~~style~~=~~"text-align:center;"~~ |

| 11:9 = 347

| ~~style~~=~~"text-align:center;"~~ |

|

| 14:11 = 418

|

| 8:7 = 231

|

|-

| ~~style="text-align:center;" | 121/120~~

! Nearest 11edo interval

| ~~|<nowiki> | -3 -1 -1 0~~ 2 ~~</nowiki>>~~

|

| ~~style="text-align:right;" | 14.37~~

| 2\11 = 218¢

| ~~style~~=~~"text-align:center;"~~ | ~~Biyatisma~~

|

| ~~style~~=~~"text-align:center;"~~ |

| 3\11 = 327

| ~~style~~=~~"text-align:center;"~~ |

|

| 4\11 = 436

|

| 2\11 = 218

|

|-

| ~~style="text-align:center;"~~ | ~~65536/65219~~

! Difference

| |~~<nowiki> | 16 0 0~~ -~~2 -3 </nowiki>>~~

|

| ~~style="text-align:right;"~~ | ~~8.39~~

| +14¢

| ~~style="text-align:center;" | Orgonisma~~

|

| ~~style="text~~-~~align:center;" |~~

| -20¢

~~| style="text-align:center;"~~ |

|

| +18¢

|

| -13¢

|

|}

=11edo ~~Instant Ensemble=~~

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

~~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~~.

=~~=11edo Zine==~~

[[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]]

~~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~~]].

=~~Compositions~~=

== Octave stretch or compression ==

[[~~File:11EDO-improv.mp3~~]]

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]].

~~''[http://xenharmony~~.~~wikispaces~~.~~com/space/showimage/11EDO-improv~~.~~mp3 First Piece Ever~~]~~'' by~~ [[~~George_Secor~~|~~George Secor~~]], ~~1970. Apparently the first piece ever written for 11edo~~.

~~''[http://www.focalchords.com/audio/Cool_My_Head_11EDO~~.~~mp3 Cool My Head]'' by~~ [[~~David_Hamill~~|~~David Hamill~~]]~~, 2010~~

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.

~~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~~

== Scales ==

~~''~~[~~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]'' on City of the Asleep~~, ~~[http://cityoftheasleep~~.~~com/music Map~~ of ~~an Internal Landscape]~~ (~~2009~~)

=== 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.

~~[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] ~~''~~[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]~~''~~ by [[Bill_Sethares|Bill Sethares]] (spectrally bent synth ens.)~~

== Instruments ==

'''11edo ukulele'''

~~''~~[~~http~~:~~//www~~.h-pi.~~com/mp3/Prelude11ET~~.~~mp3 Prelude11ET]'' by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt~~]] ~~(neo-Baroque)~~

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

~~[http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher_Bailey|Christopher Bailey]] (keyboards concréte) ~~''~~[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]~~''~~ ~~''[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]'' ''[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]'' ''[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]''

'''Ensembles'''

~~''~~[http://~~www~~.~~ozanyarman~~.com/~~files/music/Icicle_Caverns.mp3 Icicle Caverns~~]~~'' by Dr. Ozan Yarman~~

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.

~~[http://soundclick.com/share.cfm?id=955383 Angkor Wat~~, ~~September 1066] by X. J. Scott~~

[http://soundclick.com/share?songid=8839070 conversation is] ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]'' by [[Andrew_Heathwaite|Andrew Heathwaite]].

~~Text is~~ a ~~sentence borrowed from a paper by Larry Richards, set to an 11~~-~~tone row. For guitar and voice.~~

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&songID=933772 Orange Clips on Sausages] ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]'' by Andrew Heathwaite

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&songID=834492 Blue Gel] ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]'' by Andrew Heathwaite

~~''[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at~~ 11edo~~]'' by [[Chris_Vaisvil|Chris Vaisvil]]~~

~~''[http://micro.soonlabel.com/jon-lyle-smith/Jaunt~~.~~mp3 Jaunt]'' by [[John_Lyle_Smith|John Lyle Smith]]~~

~~''[http~~:~~//micro.soonlabel~~.~~com/11-ET/20110902_prepared_seagull_metamorphis~~.~~mp3 The Metamorphosis of Gregor]'' by [[Chris_Vaisvil|Chris Vaisvil]]~~

~~~~''~~[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]~~''~~ by [[Randy_Winchester|Randy Winchester]]~~

'''Lumatone'''

''[~~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]~~

[[Lumatone mapping for 11edo|Lumatone mappings for 11edo]] are available.

''[~~http~~:~~//archive.org/download/CounterpointIn11edo/CounterpointIn11edo~~.mp3 ~~Counterpoint in~~ 11edo]'~~' by~~ [[~~Jon_Lyle_Smith|Jon Lyle Smith~~]]

== Introductory Materials ==

* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

~~''[http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]'' by [[Aaron_Krister_Johnson~~|~~Aaron Krister Johnson]]~~

== Music ==

~~''~~[~~http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ('<nowiki/>'''~~[~~http://micro.soonlabel.com/11-ET/20120928-piano-~~11edo~~-eleven-birds.mp3 audio only]'''''<nowiki~~/~~>''' '') by [[Chris_Vaisvil|Chris Vaisvil~~]]~~'''''~~

See [[11edo/Music]].

''~~~~[~~http~~://~~soundcloud~~.com/~~vaisvil~~/~~the~~-~~execution-of-12-equal The Execution of 12 Equal~~] 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]

~~==Videos==~~

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

~~The Stuffed Ones: ''~~[~~http~~://www.youtube.com/watch?v=~~NU0VvGRelUQ&feature=related Goopy~~]''~~~~, ~~''~~[~~http://www.~~youtube~~.com/watch?v=4D9wDl_oxHE&feature=related Ziggy]'', ''~~[~~http~~://www.youtube.com/watch?v=~~53IiHdXfJwI&feature=related Ellie~~]~~''~~, ~~''[http://www.youtube.com/watch?v=4sZqpRcB-lk&feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]~~

* [https://www.youtube.com/watch?v=4WlTPfRDPCY untitled1], computer

~~<ul><li>''[http://www.youtube.com/watch?v~~=~~AhPjsCoMy-Q 11-equal Improvisation]'',~~ [[~~Mike_Battaglia_FAQ|Mike Battaglia~~]] ~~- youtube</li></ul>~~

== See also ==

~~==Instruments==~~

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

~~[[File:11-edo-ukulele.JPG|alt~~=~~11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]~~

== Notes ==

~~[[Category:11-tone]]~~

~~[[Category:11edo]]~~

~~[[Category:Edo]]~~

[[Category:Listen]]

~~[[Category:Macrotonal]]~~

~~[[Category:Prime EDO]]~~

~~[[Category:Scale]]~~

~~[[Category:Subgroup]]~~

~~[[Category:todo:unify_precision]]~~

@@ Line 1: / Line 1: @@
 {{interwiki
-| de =
+| de = 11-EDO
 | en = 11edo
 | es =
 | ja = 11平均律
-}}__FORCETOC__
+}}
-=11 tone equal temperament=
+{{Infobox ET}}
+{{ED intro}}
+== Theory ==
+Compared to 12edo, the intervals of 11edo are stretched:
--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]].
+* 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}}
-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".
+edo 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.
-=Tuning=
+edo 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.
-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 545.455 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.</li></ul>
+edo 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.
-=Subgroup=
+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]]".
-edo 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.
-=Intervals=
+[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]
-{| class="wikitable"
+== Intervals and Notation ==
+=== Ups and downs notation ===
+edo 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|22edo solfege]].
+{| 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>
+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
-edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:
+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)
+D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D
-[[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]]
+P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8
-==11edo solfege==
+genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...
-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: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...
-=Notation=
+'''<u>Hexatonic 2nd-generated:</u> R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)
-edo 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.
+R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R
-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".
+P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7
-{| class="wikitable"
+genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...
-|-
-! rowspan="2" | Degree
-! colspan="3" | Size in
-! rowspan="2" | Solfege
-! rowspan="2" | Approximate Ratios*
-! rowspan="2" | Sagittal
-notation
+genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...
-(22edo subset)
+===Sagittal notation===
-! colspan="2" rowspan="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" rowspan="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 ==
-! rowspan="2" | [[Tútim_Dennsuul_Wafiil|TDW]]
+=== Uniform maps ===
+{{Uniform map|edo=11}}
-Machine
+=== Commas ===
+et [[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"
 |-
-![[cent|Cents]]
+! [[Harmonic limit|Prime<br>limit]]
-!pions
+! [[Ratio]]<ref group=note>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
-!7mus
+! [[Monzo]]
+! [[Cent]]s
+! [[Color name]]
+! Name(s)
 |-
-| colspan="4" style="text-align:center;" | 0
+| 3
-| style="text-align:center;" | '''do'''
+| [[177147/131072]]
-| style="text-align:center;" | 1/1
+| {{Monzo| -17 11 }}
-| style="text-align:center;" | A
+| 521.50
-| style="text-align:center;" | P1
+| sasawa 3rd
-| style="text-align:center;" | A
+| Pythagorean augmented third
-| style="text-align:center;" | P1
-| style="text-align:center;" | A
-| style="text-align:center;" | Q\P#
 |-
-| style="text-align:center;" | 1
+| 5
-| style="text-align:right;" | 109.09
+| [[135/128]]
-|115.64
+| {{Monzo| -7 3 1 }}
-|139.64 (8B.A3<sub>16</sub>)
+| 92.18
-| style="text-align:center;" | '''ra'''
+| Layobi
-| style="text-align:center;" | [[15/14|15/14]], [[16/15|16/15]], [[17/16|17/16]], [[18/17|18/17]]
+| Major chroma
-| style="text-align:center;" | AII\ or B!!/
-| style="text-align:center;" | ^1, m2
-| 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
+| [[144/125]]
-|231.27
+| {{Monzo| 4 2 -3 }}
-|279.27 (117.46<sub>16</sub>)
+| 244.97
-| style="text-align:center;" | '''re'''
+| Trigu
-| style="text-align:center;" | [[8/7|8/7]], [[9/8|9/8]], [[17/15|17/15]]
+| University comma
-| style="text-align:center;" | B
-| style="text-align:center;" | ~2, m3
-| 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
+| [[34171875/33554432|(16 digits)]]
-|346.91
+| {{Monzo| -25 7 6 }}
-|418.91 (1A2.E88<sub>16</sub>)
+| 31.57
-| style="text-align:center;" | '''me'''
+| Lala-tribiyo
-| style="text-align:center;" | [[6/5|6/5]], [[11/9|11/9]], [[17/14|17/14]]
+| [[Ampersand comma]]
-| style="text-align:center;" | C/I or BII\ or D\!!/
-| style="text-align:center;" | M2, ~3
-| style="text-align:center;" | B#, Cv
-| style="text-align:center;" | m2, ~3
-| style="text-align:center;" | Bp, Cv
-| style="text-align:center;" | R#\Sb
 |-
-| style="text-align:center;" | 4
+| 5
-| style="text-align:right;" | 436.36
+| <abbr title="295578376007080078125/295147905179352825856">(42 digits)</abbr>
-|462.545
+| {{Monzo| -68 18 17 }}
-|558.545 (22E.8C<sub>16</sub>)
+| 2.52
-| style="text-align:center;" | '''mo'''
+| Quinla-seyo
-| style="text-align:center;" | [[9/7|9/7]], [[14/11|14/11]], [[22/17|22/17]]
+| [[Vavoom comma]]
-| style="text-align:center;" | D\! or C/II\
-| style="text-align:center;" | M3, v4
-| 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
+| 7
-| style="text-align:right;" | 545.455
+| <abbr title="854296875/843308032">(18 digits)</abbr>
-|578.18
+| {{Monzo| -10 7 8 -7 }}
-|698.18 (2BA.2F<sub>16</sub>)
+| 22.41
-| style="text-align:center;" | '''fu'''
+| Lasepru-aquadbiyo
-| style="text-align:center;" | [[11/8|11/8]], [[15/11|15/11]]
+| [[Blackjackisma]]
-| style="text-align:center;" | D/I or E\!!/
-| style="text-align:center;" | P4, v5
-| 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;" | 654.545
+| [[1029/1024]]
-|693.82
+| {{Monzo| -10 1 0 3 }}
-|837.82 (345.D1<sub>16</sub>)
+| 8.43
-| style="text-align:center;" | '''su'''
+| Latrizo
-| style="text-align:center;" | [[16/11|16/11]], [[22/15|22/15]]
+| Gamelisma
-| style="text-align:center;" | E\! or D/II\
-| style="text-align:center;" | ^4, P5
-| 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
+| [[225/224]]
-|809.455
+| {{Monzo| -5 2 2 -1 }}
-|977.455 (3D1.74<sub>16</sub>)
+| 7.71
-| style="text-align:center;" | '''lo'''
+| Ruyoyo
-| style="text-align:center;" | [[11/7|11/7]], [[14/9|14/9]], [[17/11|17/11]]
+| Marvel comma
-| style="text-align:center;" | F
-| style="text-align:center;" | ^5, m6
-| 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
+| [[16875/16807]]
-|925.09
+| {{Monzo| 0 3 4 -5 }}
-|1117.09 (45D.178<sub>16</sub>)
+| 6.99
-| style="text-align:center;" | '''la'''
+| Quinru-aquadyo
-| style="text-align:center;" | [[5/3|5/3]], [[18/11|18/11]], [[28/17|28/17]]
+| Mirkwai comma
-| style="text-align:center;" | FII\ or G!!/
-| style="text-align:center;" | ~6, m7
-| 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
+| [[2401/2400]]
-|1040.73
+| {{Monzo| -5 -1 -2 4 }}
-|1256.73 (4E8.BA<sub>16</sub>)
+| 0.72
-| style="text-align:center;" | '''ta'''
+| Bizozogu
-| style="text-align:center;" | [[7/4|7/4]], [[16/9|16/9]], [[30/17|30/17]]
+| Breedsma
-| style="text-align:center;" | G
-| style="text-align:center;" | M6, ~7
-| 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
+| 11
-| style="text-align:right;" | 1090.91
+| [[121/120]]
-|1156.36
+| {{Monzo| -3 -1 -1 0 2 }}
-|1396.36 (574.5D<sub>16</sub>)
+| 14.37
-| style="text-align:center;" | '''ti'''
+| Lologu
-| style="text-align:center;" | [[15/8|15/8]], [[17/9|17/9]], [[28/15|28/15]], [[32/17|32/17]]
+| Biyatisma
-| style="text-align:center;" | GII\ or A!!/
-| style="text-align:center;" | M7, v8
-| 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
+| [[65536/65219]]
-|1272
+| {{Monzo| 16 0 0 -2 -3 }}
-|1536 (600<sub>16</sub>)
+| 8.39
-| style="text-align:center;" | '''do'''
+| Satrilu-aruru
-| style="text-align:center;" | 2/1
+| Orgonisma
-| style="text-align:center;" | A
-| style="text-align:center;" | P8
-| style="text-align:center;" | A
-| style="text-align:center;" | P8
-| style="text-align:center;" | A
-| style="text-align:center;" | Q\P#
 |}
-*in 2.7.9.11.15.17 subgroup
-edo in [[Sagittal_notation|Sagittal notation]]:
-[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]
-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).
-=MOS Scales=
-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.
-\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].
-\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
-\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.
-\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.
-See [[11edo_Modes|11edo Modes]]
-==Commas==
-EDO tempers out the following commas. (Note: This assumes val &lt; 11 17 26 31 38 41 |.)
+== Approximation to JI ==
 {| class="wikitable"
 |-
-! | Rational
+! Harmonic
-! | Monzo
+! 8
-! | Size ([[cent|Cents]])
+!
-! | Name 1
+! 9
-! | Name 2
+!
-! | Name 3
+! 11
-|-
+!
-| style="text-align:center;" | [[135/128|135/128]]
+! 14
-| |<nowiki> | -7 3 1 </nowiki>&gt;
+!
-| style="text-align:right;" | 92.18
+! 16
-| style="text-align:center;" | Major Chroma
-| style="text-align:center;" | Major Limma
-| style="text-align:center;" | Pelogic Comma
-|-
-| style="text-align:center;" | 9931568/9752117
-| |<nowiki> | -25 7 6 </nowiki>&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
-| |<nowiki> | -68 18 17 </nowiki>&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
-| |<nowiki> | -10 7 8 -7 </nowiki>&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
+! JI interval from 1/1
-| |<nowiki> | -10 1 0 3 </nowiki>&gt;
+| 1/1 = 0 cents
-| style="text-align:right;" | 8.43
+|
-| style="text-align:center;" | Gamelisma
+| 9/8 = 204
-| style="text-align:center;" |
+|
-| style="text-align:center;" |
+| 11/8 = 551
+|
+| 7/4 = 969
+|
+| 2/1 = 1200
 |-
-| style="text-align:center;" | [[225/224|225/224]]
+! Nearest 11edo interval
-| |<nowiki> | -5 2 2 -1 </nowiki>&gt;
+| 0\11 = 0¢
-| style="text-align:right;" | 7.71
+|
-| style="text-align:center;" | Septimal Kleisma
+| 2\11 = 218¢
-| style="text-align:center;" | Marvel Comma
+|
-| style="text-align:center;" |
+| 5\11 = 545
+|
+| 9\11 = 982
+|
+| 11\11 = 1200
 |-
-| style="text-align:center;" | 16875/16807
+! Difference
-| |<nowiki> | 0 3 4 -5 </nowiki>&gt;
+| 0
-| style="text-align:right;" | 6.99
+|
-| style="text-align:center;" | Mirkwai
+| +14¢
-| style="text-align:center;" |
+|
-| style="text-align:center;" |
+| -6¢
+|
+| +13¢
+|
+| 0¢
 |-
-| style="text-align:center;" | 2401/2400
+! JI interval between
-| |<nowiki> | -5 -1 -2 4 </nowiki>&gt;
+|
-| style="text-align:right;" | 0.72
+| 9:8 = 204¢
-| style="text-align:center;" | Breedsma
+|
-| style="text-align:center;" |
+| 11:9 = 347
-| style="text-align:center;" |
+|
+| 14:11 = 418
+|
+| 8:7 = 231
+|
 |-
-| style="text-align:center;" | 121/120
+! Nearest 11edo interval
-| |<nowiki> | -3 -1 -1 0 2 </nowiki>&gt;
+|
-| style="text-align:right;" | 14.37
+| 2\11 = 218¢
-| style="text-align:center;" | Biyatisma
+|
-| style="text-align:center;" |
+| 3\11 = 327
-| style="text-align:center;" |
+|
+| 4\11 = 436
+|
+| 2\11 = 218
+|
 |-
-| style="text-align:center;" | 65536/65219
+! Difference
-| |<nowiki> | 16 0 0 -2 -3 </nowiki>&gt;
+|
-| style="text-align:right;" | 8.39
+| +14¢
-| style="text-align:center;" | Orgonisma
+|
-| style="text-align:center;" |
+| -20¢
-| style="text-align:center;" |
+|
+| +18¢
+|
+| -13¢
+|
 |}
-=11edo Instant Ensemble=
+edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:
-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.
-==11edo Zine==
+[[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]]
-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]].
-=Compositions=
+== Octave stretch or compression ==
-[[File:11EDO-improv.mp3]]
+edo 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]].
-<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.
-<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
+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.
-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
+== Scales ==
+{{Main|11edo modes}}
-<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)
+=== 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.
-[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.)
+== Instruments ==
+'''11edo ukulele'''
-<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)
+[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]
-[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>
+'''Ensembles'''
-<span style=""><span style=""><span style="">''[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns]''</span></span></span> by Dr. Ozan Yarman
+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.
-[http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by X. J. Scott
-[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]].
-Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
-[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
-[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
-<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]]
-<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]]
-<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]]
-<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]]
+'''Lumatone'''
-''[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]
+[[Lumatone mapping for 11edo|Lumatone mappings for 11edo]] are available.
-''[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo]'' by [[Jon_Lyle_Smith|Jon Lyle Smith]]
+== Introductory Materials ==
+* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])
-''[http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]'' by [[Aaron_Krister_Johnson|Aaron Krister Johnson]]
+== Music ==
+{{Catrel|11edo tracks}}
-''<span style="">[http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) (</span>'<nowiki/>'''[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]'''''<nowiki/>''' '') by [[Chris_Vaisvil|Chris Vaisvil]]'''''
+See [[11edo/Music]].
-''<span style="">[http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by [[Chris_Vaisvil|Chris Vaisvil]]</span>''
+== 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]
-==Videos==
+* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
-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]
+* [https://www.youtube.com/watch?v=4WlTPfRDPCY untitled1], computer
-<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>
+== See also ==
-==Instruments==
+* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
-[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]
+== Notes ==
+<references group=note/>
-[[Category:11-tone]]
-[[Category:11edo]]
-[[Category:Edo]]
 [[Category:Listen]]
-[[Category:Macrotonal]]
+{{Todo|add rank 2 temperaments table}}
-[[Category:Prime EDO]]
-[[Category:Scale]]
-[[Category:Subgroup]]
-[[Category:todo:unify_precision]]

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