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{{Infobox ET}}
{{Infobox ET}}
'''22edt''' is the '''equal division of the third harmonic''' ([[edt]]) into '''22 tones''', each 86.4525 [[cent]]s in size.
{{ED intro}} It supports [[mintaka]] temperament.


22edt has good approximations of the 7th, 11th, 19th and 20th harmonics. It also has the 4L+5s MOS with L=3 and s=2 approximating 5/3 somewhat fuzzily.
Like [[11edt]], both the [[octave]] and [[small whole tone]] ([[10/9]]) are about 10c off (sharp and flat respectively) dissonant but recognizable. Akin to [[16edt]] with [[Blackwood]], admitting the octave induces an interpretation into a tritave-based version of [[Whitewood]] temperament, therefore allowing the system to function as an octave stretch of [[14edo]]. However, it can just as well be treated as a pure no-twos system, which is the main interpretation used in the below article.


Like [[11edt]], both the [[octave]] and [[small whole tone]] ([[10/9]]) are about 10c off (sharp and flat respectively) dissonant but recognizable. Like [[16edt]] and Blackwood, admitting the octave induces an interpretation into a tritave-based version of Whitewood temperament.
22edt has good approximations of the 7th, 11th, 19th and 20th harmonics, being better for its size in the 3.7.11 subgroup than even [[13edt]] is in 3.5.7. In this subgroup, it tempers out the commas [[1331/1323]] and [[387420489/386683451]], with the former comma allowing a hard [[5L 2s (3/1-equivalent)|5L 2s]] (macrodiatonic) scale generated by [[11/7]], two of which are equated to [[27/11]] and three of which are equated to [[9/7]] up a tritave. This [[9/7]] can also serve as the generator for a [[4L 5s (3/1-equivalent)|4L 5s]] (BPS Lambda) scale, supporting [[Bohlen-Pierce-Stearns]] harmony by tempering out [[245/243]], although its representation of the 3.5.7 subgroup is less accurate than that of 13edt, and tempered in the wrong direction relative to 13edt for ideal BPS.


{{Harmonics in equal|22|3|1|intervals=prime|columns=15}}  
{{Harmonics in equal|22|3|1|intervals=prime|columns=15}}  
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== Intervals ==
== Intervals ==


{| class="wikitable" style="text-align: right"
The notation schemes below are based on the BPS-Lambda enneatonic scale presented in the symmetric (sLsLsLsLs, Cassiopeian) mode in J, and the Mintaka macrodiatonic scale presented in the macro-Phrygian (sLLLsLL) mode in E.
! Steps !! [[Cent]]s
 
!hekts
{| class="wikitable"
|-
! | Degree
! | Note ([[4L 5s (3/1-equivalent)#Notation|BPS-Lambda notation]])
! | Note (Macrodiatonic notation)
! | Approximate 3.7.11 subgroup interval
! | cents value
! | hekts
|-
| | 0
| | J
| | E
| | 1/1
| | 0
| | 0
|-
|-
| 1 || 86.453
| | 1
|59.091
| | J# = Kb
| | F
| | 81/77, 363/343
| | 86.453
| | 59.091
|-
|-
| 2 || 172.905
| | 2
|118.182
| | K
| | Gb = Dx
| | 2673/2401, 6561/5929
| | 172.905
| | 118.182
|-
|-
| 3 || 259.358
| | 3
|177.273
| | K#
| | E# = Abb
| | 343/297, 847/729
| | 259.358
| | 177.273
|-
|-
| 4 || 345.81
| | 4
|236.364
| | Lb
| | F#
| | 11/9, 147/121
| | 345.810
| | 236.364
|-
|-
| 5 || 432.263
| | 5
|295.4545
| | L
| | G
| | 9/7
| | 432.263
| | 295.455
|-
|-
| 6 || 518.715
| | 6
|354.5455
| | L# = Mb
| | Ab = Ex
| | 729/539
| | 518.715
| | 354.545
|-
|-
| 7 || 605.168
| | 7
|413.636
| | M
| | Fx = Bbb
| | 343/243
| | 605.168
| | 413.636
|-
|-
| 8 || 691.620
| | 8
|472.727
| | M#
| | G#
| | 49/33, 121/81
| | 691.620
| | 472.727
|-
|-
| 9 || 778.073
| | 9
|531.818
| | Nb
| | A
| | 11/7
| | 778.073
| | 531.818
|-
|-
| 10 || 864.525
| | 10
|590.909
| | N
| | Bb
| | 81/49
| | 864.525
| | 590.909
|-
|-
| 11 || 950.978
| | 11
|650
| | N# = Ob
| | Cb = Gx
| | 3773/2187, 6561/3773
| | 950.978
| | 650.
|-
|-
| 12 || 1037.430
| | 12
|709.091
| | O
| | A# = Dbb
| | 49/27
| | 1037.430
| | 709.091
|-
|-
| 13 || 1123.883
| | 13
|768.182
| | O#
| | B
| | 21/11
| | 1123.883
| | 768.182
|-
|-
| 14 || 1210.335
| | 14
|827.273
| | Pb
| | C
| | 99/49, 243/121
| | 1210.335
| | 827.273
|-
|-
| 15 || 1296.788
| | 15
|886.364
| | P
| | Db = Ax
| | 729/343
| | 1296.788
| | 886.364
|-
|-
| 16 || 1383.24
| | 16
|945.4545
| | P# = Qb
| | B# = Ebb
| | 539/243
| | 1383.240
| | 945.455
|-
|-
| 17 || 1469.693
| | 17
|1004.5455
| | Q
| | C#
| | 7/3
| | 1469.693
| | 1004.545
|-
|-
| 18 || 1556.145
| | 18
|1063.636
| | Q#
| | D
| | 27/11, 121/49
| | 1556.145
| | 1063.636
|-
|-
| 19 || 1642.598
| | 19
|1122.727
| | Rb
| | Eb
| | 891/343, 2187/847
| | 1642.598
| | 1122.727
|-
|-
| 20 || 1729.05
| | 20
|1181.818
| | R
| | Fb = Cx
| | 2401/891, 5929/2187
| | 1729.050
| | 1181.818
|-
|-
| 21 || 1815.503
| | 21
|1240.909
| | R# = Jb
| | D# = Gbb
| | 77/27, 343/121
| | 1815.503
| | 1240.909
|-
|-
| 22 || 1901.955
| | 22
|1300
| | J
| | E
| | 3/1
| | 1901.955
| | 1300.
|}
|}


== Compositions ==
== Audio examples ==
* http://www.archive.org/details/TuneIn22Edt by [[Peter Kosmorsky]]
[[File:22ed3-1.mp3]]
* [http://micro.soonlabel.com/22-edt/daily20111206-22edt-improv.mp3 22 edt piano improvisation] by [[Chris Vaisvil]]
 
A short composition by [[Wensik]], based on the 7:9:11 chord and its inversion, 63:77:99.
 
== Music ==
; [[Peter Kosmorsky]]
* [http://www.archive.org/details/TuneIn22Edt Tune in 22edt] (2011)
 
; [[Ray Perlner]]
* [https://www.youtube.com/watch?v=EWy0y_WsVNk ''Fugue in 22EDT Mintaka[7] sLLLsLL "Macro-Phrygian"''] (2025)
 
; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/22-edt/daily20111206-22edt-improv.mp3 22 edt piano improvisation] {{dead link}}


[[Category:Edt]]
[[Category:Nonoctave]]
[[Category:Nonoctave]]
[[Category:Listen]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/22edt"