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| es =  
| es =  
| ja = 11平均律
| ja = 11平均律
}}__FORCETOC__
}}
{{Infobox ET
{{Infobox ET
| Prime factorization = 11 (prime)
| Prime factorization = 11 (prime)
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}}
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'''11 equal divisions of the octave''' ('''11EDO'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime EDO]], after [[2edo|2EDO]], [[3edo|3EDO]], [[5edo|5EDO]], and [[7edo|7EDO]].
'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].


== Theory ==
== Theory ==
{{Odd harmonics in edo|edo=11}}
{{primes 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]], a tuning known as "elevenplus".


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


Compared to 12EDO, the intervals of 11EDO are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.


<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 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.</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>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 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 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 tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.


11EDO is the largest EDO that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].
11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].


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


{| class="wikitable center-all right-1 right-2"
{| class="wikitable center-all right-1 right-2"
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! Solfege
! Solfege
! Approximate Ratios*
! Approximate Ratios*
! Sagittal <br> notation <br>(22EDO subset)
! Sagittal <br> notation <br>(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] <br> with major wider <br> than minor
! 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
! colspan="2" | Up/down notation <br> with major narrower <br> than minor
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*in 2.7.9.11.15.17 subgroup
*in 2.7.9.11.15.17 subgroup


11EDO in [[Sagittal notation]]:
11edo in [[Sagittal notation]]:


[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]
[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]
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== Commas ==
== Commas ==
11EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)


{| class="commatable wikitable center-1 center-2 right-4 center-5"
{| class="commatable wikitable center-1 center-2 right-4 center-5"
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| | 2/1 = 1200
| | 2/1 = 1200
|-
|-
! | nearest 11EDO interval
! | nearest 11edo interval
| | 0\11 = 0¢
| | 0\11 = 0¢
| |  
| |  
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| |  
| |  
|-
|-
! | nearest 11EDO interval
! | nearest 11edo interval
| |  
| |  
| | 2\11 = 218¢
| | 2\11 = 218¢
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|}
|}


11EDO also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:
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]]
[[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]]


== MOS Scales ==
== 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.
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]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
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5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11EDO Modes]]
See [[11edo_Modes|11edo Modes]]


== Pathological Modes ==
== Pathological Modes ==
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== Instruments ==
== Instruments ==
11EDO ukulele:
11edo ukulele:


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


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.
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.


== Music ==
== Music ==
{{See also|:Category:11edo tracks}}
{{See also|:Category:11edo tracks}}
* [[File:11edo-improv.mp3|link=Special:FilePath/11edo-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11EDO.
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{Dead link}} by [[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]], 2010
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
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** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [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)
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession She is My Lilac-Hued Obsession] by [[City of the Asleep]] (2007)
* [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]] (spectrally bent synth ens.)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]{{dead link}} by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]{{dead link}} by [[Aaron Andrew Hunt]] (neo-Baroque)
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &#124; SoundCloud] by Aaron Andrew Hunt
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones]{{dead link}} by [[Christopher Bailey]] (keyboards concréte):
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]  
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [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]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]{{dead link}} by [[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=933772 Orange Clips on Sausages]{{dead link}} [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://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&songID=834492 Blue Gel]{{dead link}} [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]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11EDO] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]{{dead link}} by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]{{dead link}} by [[Randy Winchester]]
* [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]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]{{dead link}} by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11EDO] by [[Jon Lyle Smith]]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11EDO]{{dead link}} by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]{{dead link}} by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal]{{dead link}} by Chris Vaisvil
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* 11edo pieces by  [[User:Ayceman|Alexandru Ianu]]:
* 11edo pieces by  [[User:Ayceman|Alexandru Ianu]]:
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<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>
<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>


== 11EDO Zine ==
== 11edo Zine ==
There is an 11EDO Zine! As far as we know, 11EDO is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11EDO Zine]].
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].


[[Category:11edo| ]] <!-- main article -->
[[Category:11edo| ]] <!-- main article -->

Revision as of 02:02, 15 January 2022

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

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

Theory

Template:Primes in equal

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of 22edo, a tuning known as "elevenplus".

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

  • The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
  • The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
  • The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
  • The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
  • The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the 2*11 subgroup 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

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

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.

Degree Size in
cents 		Solfege Approximate Ratios* Sagittal
notation
(22edo subset) 		Up/down notation
with major wider
than minor 		Up/down notation
with major narrower
than minor 		Smitonic

(3rd-gen)

notation

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

11edo in Sagittal notation:

Sagittal11EDO.jpg

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

Pentatonic 5th-generated: D * * E G * * A C * * D (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 (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...

Commas

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

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

JI Intervals

Harmonic 8 9 11 14 16
JI interval from 1/1 1/1 = 0 cents 9/8 = 204 11/8 = 551 7/4 = 969 2/1 = 1200
nearest 11edo interval 0\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

MOS Scales

Although 11edo has one fewer interval in the octave than 12edo, in terms of moment-of-symmetry scales, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

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

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

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

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

See 11edo Modes

Pathological Modes

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

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

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

Instruments

11edo ukulele:

11-edo-ukulele.JPG

In February 2011, Oddmusic U-C, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

Music

See also: Category:11edo tracks

Videos

The Stuffed Ones: Goopy, Ziggy, Ellie, Towelbear by 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. See 11edo Zine.

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