11edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|11}}
{{ED intro}}
== Theory ==
== Theory ==
{{Harmonics in equal|11|intervals=odd}}
Compared to 12edo, the intervals of 11edo are stretched:


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".
* 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}}


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


* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
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.
* 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 [[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 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.


11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].
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 has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 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.]]


== Intervals and notation ==
== 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.
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.


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! Solfege
! Solfege
! Approximate Ratios*
! Approximate Ratios*
! 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
! [[Smitonic]]<br>(3rd-gen)<br>notation  
! [[Smitonic]]<br>(3rd-gen)<br>notation  
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| '''do'''
| '''do'''
| 1/1
| 1/1
| A
| P1
| P1
| A
| A
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| '''ra'''
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^1, m2
| ^A, B
| ^A, B
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| '''re'''
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ~2, m3
| ^B, Cb
| ^B, Cb
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| '''me'''
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| M2, ~3
| B#, vC
| B#, vC
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| '''mo'''
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| M3, v4
| C, vD
| C, vD
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| '''fu'''
| '''fu'''
| [[11/8]], [[15/11]]
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| P4, v5
| D, vE
| D, vE
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| '''su'''
| '''su'''
| [[16/11]], [[22/15]]
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^4, P5
| ^D, E
| ^D, E
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| '''lo'''
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^5, m6
| ^E, Fb
| ^E, Fb
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| '''la'''
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| ~6, m7
| vF, Gb
| vF, Gb
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| '''ta'''
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| M6, ~7
| F, vG
| F, vG
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| '''ti'''
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| M7, v8
| G, vAv
| G, vAv
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| '''do'''
| '''do'''
| 2/1
| 2/1
| A
| P8
| P8
| A
| A
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|[[File:0-1200 octave.mp3|frameless]]
|[[File:0-1200 octave.mp3|frameless]]
|}
|}
*in 2.7.9.11.15.17 subgroup
<nowiki>* in 2.7.9.11.15.17 subgroup</nowiki>
 
11edo in [[Sagittal notation]]:
 
[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|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.
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)
'''<u>Pentatonic 5th-generated:</u>''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)
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== Regular temperament properties ==
== Regular temperament properties ==
=== Uniform maps ===
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}
{{Uniform map|edo=11}}


=== Commas ===
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)
11et [[tempering out|tempers out]] the following [[comma]]s. 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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| Layobi
| Layobi
| Major chroma
| Major chroma
|-
| 5
| [[144/125]]
| {{Monzo| 4 2 -3 }}
| 244.97
| Trigu
| University comma
|-
|-
| 5
| 5
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| 31.57
| 31.57
| Lala-tribiyo
| Lala-tribiyo
| [[Ampersand]]'s comma
| [[Ampersand comma]]
|-
|-
| 5
| 5
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| 2.52
| 2.52
| Quinla-seyo
| Quinla-seyo
| [[Vavoom]]
| [[Vavoom comma]]
|-
|-
| 7
| 7
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| 6.99
| 6.99
| Quinru-aquadyo
| Quinru-aquadyo
| Mirkwai
| Mirkwai comma
|-
|-
| 7
| 7
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[[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]]
== Octave stretch or compression ==
11edo has about equally bad sharp and flat mappings of [[prime]]s 3 and 5. The 7 and 13 are quite sharp, but the 11 is a little flat. To use it as a 2.7.11.13 tuning, slight [[octave shrinking]] is advisable. Examples of slightly compressed versions of 11edo include (least to most compressed) [[ed6|28ed6]], [[ed12|39ed12]], [[zpi|30zpi]], [[equal tuning|35ed9]] and [[ed7|31ed7]].
To use its primes 3 or 5, extreme octave shrinking can be used, at the cost of making the octaves sound significantly weaker. [[equal tuning|37ed10]] is a very compressed version of 11edo.


== Scales ==
== Scales ==
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{{Main|List of 11edo 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.
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.
=== 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 ==
== Instruments ==
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[[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]]


'''Ensembles'''
'''Ensembles'''


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


'''Lumatone'''
'''Lumatone'''


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


== Introductory Materials ==
== Introductory Materials ==
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== Music ==
== Music ==
{{Catrel|11edo tracks}}
{{Catrel|11edo tracks}}
=== Strict 11edo ===
=== 11 equal divisions of the octave (11edo proper) ===
==== Modern renderings ====
==== Modern renderings ====
; {{W|Arthur Schutt}}
; {{W|Arthur Schutt}}
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; [[Francium]]
; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* [https://www.youtube.com/watch?v=hM0BAC_YZnQ ''Sleep Slope''] (2024)
* "Sleep Slope" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/2GnUioPrMlJmMaacE6DK7i Spotify] | [https://francium223.bandcamp.com/track/sleep-slope Bandcamp] | [https://www.youtube.com/watch?v=hM0BAC_YZnQ YouTube]


; [[David Hamill]]
; [[David Hamill]]
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* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning
* "Search Party" from ''No Fun House'' (2025) – [https://open.spotify.com/track/7CMiwDuuRuFpB0skfwT2Ap Spotify] | [https://mundoworld.bandcamp.com/track/search-party Bandcamp] | [https://www.youtube.com/watch?v=hLs6MjuousI YouTube]


; [[User:GlitchyDarkness|No Clue Music]]
; [[User:GlitchyDarkness|No Clue Music]]
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=== Unequal Derivatives of 11edo ===
=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=z0lWcguNsNs ''11 Tone March''] (2024)
* ''11 Tone March'' (2023/2024)
** [https://www.youtube.com/shorts/K2QVvaRUXIQ <nowiki>[short clip]</nowiki>] (2023, with Lumatone view)
** [https://www.youtube.com/watch?v=z0lWcguNsNs <nowiki>[full version]</nowiki>] (2024, with tuning specification in video description)


== Videos ==
== Videos ==
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* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
* [https://www.youtube.com/watch?v=4WlTPfRDPCY untitled1], computer


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


== Notes ==
== Notes ==
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[[Category:Listen]]
[[Category:Listen]]
[[Category:Macrotonal]]
{{Todo|add rank 2 temperaments table}}
[[Category:Todo:add rank 2 temperaments table]]

Latest revision as of 03:19, 8 May 2026

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

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

Theory

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

  • The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
  • The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
  • The "minor third" at 327.27 cents, is rather sharp and encroaching upon "neutral third".
  • The "major third" at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
  • The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.
Approximation of odd harmonics in 11edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -47.4 +50.0 +13.0 +14.3 -5.9 +32.2 +2.6 +4.1 +29.8 -34.4 +26.3
Relative (%) -43.5 +45.9 +11.9 +13.1 -5.4 +29.5 +2.4 +3.8 +27.3 -31.5 +24.1
Steps
(reduced) 		17
(6) 		26
(4) 		31
(9) 		35
(2) 		38
(5) 		41
(8) 		43
(10) 		45
(1) 		47
(3) 		48
(4) 		50
(6)

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

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

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

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

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

Intervals and Notation

Ups and downs notation

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the melodic meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the harmonic meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

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

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

* in 2.7.9.11.15.17 subgroup

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Sagittal notation

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

Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation22-EDO notation

Revo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation22-EDO notation

Regular temperament properties

Uniform maps

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

Commas

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

Prime
limit 		Ratio[note 1] Monzo Cents Color name Name(s)
3 177147/131072 [-17 11⟩ 521.50 sasawa 3rd Pythagorean augmented third
5 135/128 [-7 3 1⟩ 92.18 Layobi Major chroma
5 144/125 [4 2 -3⟩ 244.97 Trigu University comma
5 (16 digits) [-25 7 6⟩ 31.57 Lala-tribiyo Ampersand comma
5 (42 digits) [-68 18 17⟩ 2.52 Quinla-seyo Vavoom comma
7 (18 digits) [-10 7 8 -7⟩ 22.41 Lasepru-aquadbiyo Blackjackisma
7 1029/1024 [-10 1 0 3⟩ 8.43 Latrizo Gamelisma
7 225/224 [-5 2 2 -1⟩ 7.71 Ruyoyo Marvel comma
7 16875/16807 [0 3 4 -5⟩ 6.99 Quinru-aquadyo Mirkwai comma
7 2401/2400 [-5 -1 -2 4⟩ 0.72 Bizozogu Breedsma
11 121/120 [-3 -1 -1 0 2⟩ 14.37 Lologu Biyatisma
11 65536/65219 [16 0 0 -2 -3⟩ 8.39 Satrilu-aruru Orgonisma

Approximation to JI

Harmonic 8 9 11 14 16
JI interval from 1/1 1/1 = 0 cents 9/8 = 204 11/8 = 551 7/4 = 969 2/1 = 1200
Nearest 11edo interval 0\11 = 0¢ 2\11 = 218¢ 5\11 = 545 9\11 = 982 11\11 = 1200
Difference 0 +14¢ -6¢ +13¢ 0¢
JI interval between 9:8 = 204¢ 11:9 = 347 14:11 = 418 8:7 = 231
Nearest 11edo interval 2\11 = 218¢ 3\11 = 327 4\11 = 436 2\11 = 218
Difference +14¢ -20¢ +18¢ -13¢

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

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

Octave stretch or compression

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

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

Scales

Main article: 11edo modes

MOS scales

Main article: List of 11edo MOS scales

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

Instruments

11edo ukulele

11-edo-ukulele.JPG

Ensembles

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

Lumatone

Lumatone mappings for 11edo are available.

Introductory Materials

Music

See also: Category:11edo tracks

11 equal divisions of the octave (11edo proper)

Modern renderings

Arthur Schutt

20th century

George Secor
Bill Sethares

21st century

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

Unequal Derivatives of 11edo

Bryan Deister
  • 11 Tone March (2023/2024)

Videos

See also

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

Notes

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