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'''22edt''' is the [[edt|equal division of the third harmonic]] into 22 tones, each 86.453 [[cent|cent]]s in size.
{{Infobox ET}}
{{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|11edt]], both the [[Octave|octave]] and [[minor_whole_tone|minor whole tone]] [[10/9|10:9]] are about 10c off (sharp and flat respectively) dissonant but recognizable. Like [[16edt|16edt]] and Blackwood, admitting the octave induces an interpretation into a tritave-based version of Whitewood temperament.
== Harmonics ==
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.


='''Intervals'''=
{{Harmonics in equal|22|3|1|intervals=prime|columns=15}}
86.4525


172.905
== Intervals ==
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.


259.3575
{| 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
| | J# = Kb
| | F
| | 81/77, 363/343
| | 86.453
| | 59.091
|-
| | 2
| | K
| | Gb = Dx
| | 2673/2401, 6561/5929
| | 172.905
| | 118.182
|-
| | 3
| | K#
| | E# = Abb
| | 343/297, 847/729
| | 259.358
| | 177.273
|-
| | 4
| | Lb
| | F#
| | 11/9, 147/121
| | 345.810
| | 236.364
|-
| | 5
| | L
| | G
| | 9/7
| | 432.263
| | 295.455
|-
| | 6
| | L# = Mb
| | Ab = Ex
| | 729/539
| | 518.715
| | 354.545
|-
| | 7
| | M
| | Fx = Bbb
| | 343/243
| | 605.168
| | 413.636
|-
| | 8
| | M#
| | G#
| | 49/33, 121/81
| | 691.620
| | 472.727
|-
| | 9
| | Nb
| | A
| | 11/7
| | 778.073
| | 531.818
|-
| | 10
| | N
| | Bb
| | 81/49
| | 864.525
| | 590.909
|-
| | 11
| | N# = Ob
| | Cb = Gx
| | 3773/2187, 6561/3773
| | 950.978
| | 650.
|-
| | 12
| | O
| | A# = Dbb
| | 49/27
| | 1037.430
| | 709.091
|-
| | 13
| | O#
| | B
| | 21/11
| | 1123.883
| | 768.182
|-
| | 14
| | Pb
| | C
| | 99/49, 243/121
| | 1210.335
| | 827.273
|-
| | 15
| | P
| | Db = Ax
| | 729/343
| | 1296.788
| | 886.364
|-
| | 16
| | P# = Qb
| | B# = Ebb
| | 539/243
| | 1383.240
| | 945.455
|-
| | 17
| | Q
| | C#
| | 7/3
| | 1469.693
| | 1004.545
|-
| | 18
| | Q#
| | D
| | 27/11, 121/49
| | 1556.145
| | 1063.636
|-
| | 19
| | Rb
| | Eb
| | 891/343, 2187/847
| | 1642.598
| | 1122.727
|-
| | 20
| | R
| | Fb = Cx
| | 2401/891, 5929/2187
| | 1729.050
| | 1181.818
|-
| | 21
| | R# = Jb
| | D# = Gbb
| | 77/27, 343/121
| | 1815.503
| | 1240.909
|-
| | 22
| | J
| | E
| | 3/1
| | 1901.955
| | 1300.
|}


345.81
== Audio examples ==
[[File:22ed3-1.mp3]]


432.2625
A short composition by [[Wensik]], based on the 7:9:11 chord and its inversion, 63:77:99.


518.715
== Music ==
; [[Peter Kosmorsky]]
* [http://www.archive.org/details/TuneIn22Edt ''Tune in 22edt''] (2011)


605.1675
; [[Merct]]
* [https://soundcloud.com/merct/dysfunctional?in=merct/sets/breathe ''dysfunctional'' from the album ''breathe''] (2025)


691.62
; [[Ray Perlner]]
* [https://www.youtube.com/watch?v=EWy0y_WsVNk ''<nowiki>Fugue in 22EDT Mintaka[7] sLLLsLL "Macro-Phrygian"</nowiki>''] (2025)
* [https://www.youtube.com/watch?v=edyA0QN-Zmc ''<nowiki>7/8 Fugue in 22EDT Mintaka[7] LLsLLLs "Macro-Ionian"</nowiki>''] (2026)


778.0725
; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/22-edt/daily20111206-22edt-improv.mp3 ''22 edt piano improvisation''] {{dead link}}


864.525
[[Category:Nonoctave]]
 
[[Category:Listen]]
950.9775
 
1037.43
 
1123.8825
 
1210.335
 
1296.7875
 
1383.24
 
1469.6925
 
1556.145
 
1642.5975
 
1729.05
 
1815.5025
 
3/1
 
=Compositions=
[http://www.archive.org/details/TuneIn22Edt http://www.archive.org/details/TuneIn22Edt] - Kosmorsky
 
<span style="">[http://micro.soonlabel.com/22-edt/daily20111206-22edt-improv.mp3 22 edt piano improvisation]</span> by [[Chris_Vaisvil|Chris Vaisvil]]
 
[[Category:22edt]]
[[Category:edonoi]]
[[Category:edt]]
[[Category:equal]]
[[Category:nonoctave]]
Retrieved from "https://en.xen.wiki/w/22edt"