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| ja = 11平均律
| ja = 11平均律
}}__FORCETOC__
}}__FORCETOC__
=11 tone equal temperament=
==11 tone equal temperament==


11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].
11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].
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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".
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".


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


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


=Subgroup=
==Subgroup==
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.
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=
==Notation==
 
{| class="wikitable"
|-
! | Harmonic
! | 8
! |
! | 9
! |
! | 11
! |
! | 14
! |
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| |
| | 9/8 = 204
| |
| | 11/8 = 551
| |
| | 7/4 = 969
| |
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| |
| | 2\11 = 218¢
| |
| | 5\11 = 545
| |
| | 9\11 = 982
| |
| | 11\11 = 1200
|-
! | difference
| | 0
| |
| | +14¢
| |
| | -6¢
| |
| | +13¢
| |
| | 0¢
|-
! | JI interval between
| |
| | 9:8 = 204¢
| |
| | 11:9 = 347
| |
| | 14:11 = 418
| |
| | 8:7 = 231
| |
|-
! | nearest 11edo interval
| |
| | 2\11 = 218¢
| |
| | 3\11 = 327
| |
| | 4\11 = 436
| |
| | 2\11 = 218
| |
|-
! | difference
| |
| | +14¢
| |
| | -20¢
| |
| | +18¢
| |
| | -13¢
| |
|}
 
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:
 
[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]
 
==11edo solfege==
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'''.
 
=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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| style="text-align:center;" | AII\ or B!!/
| style="text-align:center;" | AII\ or B!!/
| style="text-align:center;" | ^1, m2
| style="text-align:center;" | ^1, m2
| style="text-align:center;" | A^, B
| style="text-align:center;" | ^A, B
| style="text-align:center;" | ^1, M2
| style="text-align:center;" | ^1, M2
| style="text-align:center;" | A^, B
| style="text-align:center;" | ^A, B
| style="text-align:center;" | Q#\Rb
| style="text-align:center;" | Q#\Rb
|-
|-
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| style="text-align:center;" | B
| style="text-align:center;" | B
| style="text-align:center;" | ~2, m3
| style="text-align:center;" | ~2, m3
| style="text-align:center;" | B^, Cb
| style="text-align:center;" | ^B, Cb
| style="text-align:center;" | ~2, M3
| style="text-align:center;" | ~2, M3
| style="text-align:center;" | B^, C#
| style="text-align:center;" | ^B, C#
| style="text-align:center;" | R
| style="text-align:center;" | R
|-
|-
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| style="text-align:center;" | C/I or BII\ or D\!!/
| style="text-align:center;" | C/I or BII\ or D\!!/
| style="text-align:center;" | M2, ~3
| style="text-align:center;" | M2, ~3
| style="text-align:center;" | B#, Cv
| style="text-align:center;" | B#, vC
| style="text-align:center;" | m2, ~3
| style="text-align:center;" | m2, ~3
| style="text-align:center;" | Bb, Cv
| style="text-align:center;" | Bb, vC
| style="text-align:center;" | R#\Sb
| style="text-align:center;" | R#\Sb
|-
|-
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| style="text-align:center;" | D\! or C/II\
| style="text-align:center;" | D\! or C/II\
| style="text-align:center;" | M3, v4
| style="text-align:center;" | M3, v4
| style="text-align:center;" | C, Dv
| style="text-align:center;" | C, vD
| style="text-align:center;" | m3, v4
| style="text-align:center;" | m3, v4
| style="text-align:center;" | C, Dv
| style="text-align:center;" | C, vD
| style="text-align:center;" | S
| style="text-align:center;" | S
|-
|-
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| style="text-align:center;" | D/I or E\!!/
| style="text-align:center;" | D/I or E\!!/
| style="text-align:center;" | P4, v5
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, Ev
| style="text-align:center;" | D, vE
| style="text-align:center;" | P4, v5
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, Ev
| style="text-align:center;" | D, vE
| style="text-align:center;" | S#\Tb
| style="text-align:center;" | S#\Tb
|-
|-
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| style="text-align:center;" | E\! or D/II\
| style="text-align:center;" | E\! or D/II\
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | D^, E
| style="text-align:center;" | ^D, E
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | D^, E
| style="text-align:center;" | ^D, E
| style="text-align:center;" | T
| style="text-align:center;" | T
|-
|-
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| style="text-align:center;" | F
| style="text-align:center;" | F
| style="text-align:center;" | ^5, m6
| style="text-align:center;" | ^5, m6
| style="text-align:center;" | E^, Fb
| style="text-align:center;" | ^E, Fb
| style="text-align:center;" | ^5, M6
| style="text-align:center;" | ^5, M6
| style="text-align:center;" | E^, F#
| style="text-align:center;" | ^E, F#
| style="text-align:center;" | T#\Ub
| style="text-align:center;" | T#\Ub
|-
|-
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| style="text-align:center;" | FII\ or G!!/
| style="text-align:center;" | FII\ or G!!/
| style="text-align:center;" | ~6, m7
| style="text-align:center;" | ~6, m7
| style="text-align:center;" | Fv, Gb
| style="text-align:center;" | vF, Gb
| style="text-align:center;" | ~6, M7
| style="text-align:center;" | ~6, M7
| style="text-align:center;" | Fv, G#
| style="text-align:center;" | vF, G#
| style="text-align:center;" | U
| style="text-align:center;" | U
|-
|-
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| style="text-align:center;" | G
| style="text-align:center;" | G
| style="text-align:center;" | M6, ~7
| style="text-align:center;" | M6, ~7
| style="text-align:center;" | F, Gv
| style="text-align:center;" | F, vG
| style="text-align:center;" | m6, ~7
| style="text-align:center;" | m6, ~7
| style="text-align:center;" | F, Gv
| style="text-align:center;" | F, vG
| style="text-align:center;" | U#\Pb
| style="text-align:center;" | U#\Pb
|-
|-
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| style="text-align:center;" | GII\ or A!!/
| style="text-align:center;" | GII\ or A!!/
| style="text-align:center;" | M7, v8
| style="text-align:center;" | M7, v8
| style="text-align:center;" | G, Av
| style="text-align:center;" | G, vAv
| style="text-align:center;" | m7, v8
| style="text-align:center;" | m7, v8
| style="text-align:center;" | G, Av
| style="text-align:center;" | G, vAv
| style="text-align:center;" | P\Qb
| style="text-align:center;" | P\Qb
|-
|-
Line 284: Line 193:


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).
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==
==Commas==
Line 311: Line 207:
|-
|-
| style="text-align:center;" | [[135/128|135/128]]
| style="text-align:center;" | [[135/128|135/128]]
| | | -7 3 1 &gt;
| |<nowiki> | -7 3 1 </nowiki>&gt;
| style="text-align:right;" | 92.18
| style="text-align:right;" | 92.18
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Chroma
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|-
|-
| style="text-align:center;" | 9931568/9752117
| style="text-align:center;" | 9931568/9752117
| | | -25 7 6 &gt;
| |<nowiki> | -25 7 6 </nowiki>&gt;
| style="text-align:right;" | 31.57
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | Ampersand's Comma
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|-
|-
| style="text-align:center;" | 1776337/1773750
| style="text-align:center;" | 1776337/1773750
| | | -68 18 17 &gt;
| |<nowiki> | -68 18 17 </nowiki>&gt;
| style="text-align:right;" | 2.52
| style="text-align:right;" | 2.52
| style="text-align:center;" | Vavoom
| style="text-align:center;" | Vavoom
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|-
|-
| style="text-align:center;" | 9859966/9733137
| style="text-align:center;" | 9859966/9733137
| | | -10 7 8 -7 &gt;
| |<nowiki> | -10 7 8 -7 </nowiki>&gt;
| style="text-align:right;" | 22.41
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | Blackjackisma
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|-
|-
| style="text-align:center;" | 1029/1024
| style="text-align:center;" | 1029/1024
| | | -10 1 0 3 &gt;
| |<nowiki> | -10 1 0 3 </nowiki>&gt;
| style="text-align:right;" | 8.43
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | Gamelisma
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|-
|-
| style="text-align:center;" | [[225/224|225/224]]
| style="text-align:center;" | [[225/224|225/224]]
| | | -5 2 2 -1 &gt;
| |<nowiki> | -5 2 2 -1 </nowiki>&gt;
| style="text-align:right;" | 7.71
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Septimal Kleisma
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|-
|-
| style="text-align:center;" | 16875/16807
| style="text-align:center;" | 16875/16807
| | | 0 3 4 -5 &gt;
| |<nowiki> | 0 3 4 -5 </nowiki>&gt;
| style="text-align:right;" | 6.99
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | Mirkwai
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|-
|-
| style="text-align:center;" | 2401/2400
| style="text-align:center;" | 2401/2400
| | | -5 -1 -2 4 &gt;
| |<nowiki> | -5 -1 -2 4 </nowiki>&gt;
| style="text-align:right;" | 0.72
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" | Breedsma
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|-
|-
| style="text-align:center;" | 121/120
| style="text-align:center;" | 121/120
| | | -3 -1 -1 0 2 &gt;
| |<nowiki> | -3 -1 -1 0 2 </nowiki>&gt;
| style="text-align:right;" | 14.37
| style="text-align:right;" | 14.37
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | Biyatisma
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|-
|-
| style="text-align:center;" | 65536/65219
| style="text-align:center;" | 65536/65219
| | | 16 0 0 -2 -3 &gt;
| |<nowiki> | 16 0 0 -2 -3 </nowiki>&gt;
| style="text-align:right;" | 8.39
| style="text-align:right;" | 8.39
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | Orgonisma
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|}
|}


=11edo Instant Ensemble=
==Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! |
! | 9
! |
! | 11
! |
! | 14
! |
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| |
| | 9/8 = 204
| |
| | 11/8 = 551
| |
| | 7/4 = 969
| |
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| |
| | 2\11 = 218¢
| |
| | 5\11 = 545
| |
| | 9\11 = 982
| |
| | 11\11 = 1200
|-
! | difference
| | 0
| |
| | +14¢
| |
| | -6¢
| |
| | +13¢
| |
| | 0¢
|-
! | JI interval between
| |
| | 9:8 = 204¢
| |
| | 11:9 = 347
| |
| | 14:11 = 418
| |
| | 8:7 = 231
| |
|-
! | nearest 11edo interval
| |
| | 2\11 = 218¢
| |
| | 3\11 = 327
| |
| | 4\11 = 436
| |
| | 2\11 = 218
| |
|-
! | difference
| |
| | +14¢
| |
| | -20¢
| |
| | +18¢
| |
| | -13¢
| |
|}
 
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:
 
[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]
 
==11edo solfege==
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'''.
 
==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]]
 
==Instruments==
[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]
 
==11edo Instant Ensemble==
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.


Line 387: Line 389:
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|11edo Zine]].


=Compositions=
==Compositions==
[[File:11EDO-improv.mp3]]
[[File:11EDO-improv.mp3]]
<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://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.
Line 429: Line 431:
''[http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]'' by [[Aaron_Krister_Johnson|Aaron Krister Johnson]]
''[http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]'' by [[Aaron_Krister_Johnson|Aaron Krister Johnson]]


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


''<span style="">[http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by [[Chris_Vaisvil|Chris Vaisvil]]</span>''
''<span style="">[http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by [[Chris_Vaisvil|Chris Vaisvil]]</span>''
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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>
==Instruments==
[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


[[Category:11-tone]]
[[Category:11-tone]]
Retrieved from "https://en.xen.wiki/w/11edo"