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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, 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.
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. Akin to [[16edt]] with [[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}}  


== Intervals ==
== 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.


{| class="wikitable"
{| class="wikitable"
|-
|-
! | Degree
! | Degree
! | Note (BPS-Lambda notation)
! | Note ([[4L 5s (3/1-equivalent)#Notation|BPS-Lambda notation]])
! | Note (Macrodiatonic notation)
! | Note (Macrodiatonic notation)
! | Approximate 3.7.11 subgroup interval
! | Approximate 3.7.11 subgroup interval
Line 20: Line 22:
|-
|-
| | 0
| | 0
| | E
| | J
| | E
| | E
| | 1/1
| | 1/1
Line 27: Line 29:
|-
|-
| | 1
| | 1
| | E# = Fb
| | J# = Kb
| | F
| | F
| | 81/77, 363/343
| | 81/77, 363/343
Line 34: Line 36:
|-
|-
| | 2
| | 2
| | F
| | K
| | Gb = Dx
| | Gb = Dx
| | 2673/2401, 6561/5929
| | 2673/2401, 6561/5929
Line 41: Line 43:
|-
|-
| | 3
| | 3
| | F#
| | K#
| | E# = Abb
| | E# = Abb
| | 343/297, 847/729
| | 343/297, 847/729
Line 48: Line 50:
|-
|-
| | 4
| | 4
| | Gb
| | Lb
| | F#
| | F#
| | 11/9, 147/121
| | 11/9, 147/121
Line 55: Line 57:
|-
|-
| | 5
| | 5
| | G
| | L
| | G
| | G
| | 9/7
| | 9/7
Line 62: Line 64:
|-
|-
| | 6
| | 6
| | G# = Hb
| | L# = Mb
| | Ab = Ex
| | Ab = Ex
| | 729/539
| | 729/539
Line 69: Line 71:
|-
|-
| | 7
| | 7
| | H
| | M
| | Fx = Bbb
| | Fx = Bbb
| | 343/243
| | 343/243
Line 76: Line 78:
|-
|-
| | 8
| | 8
| | H#
| | M#
| | G#
| | G#
| | 49/33, 121/81
| | 49/33, 121/81
Line 83: Line 85:
|-
|-
| | 9
| | 9
| | Jb
| | Nb
| | A
| | A
| | 11/7
| | 11/7
Line 90: Line 92:
|-
|-
| | 10
| | 10
| | J
| | N
| | Bb
| | Bb
| | 81/49
| | 81/49
Line 97: Line 99:
|-
|-
| | 11
| | 11
| | J# = Ab
| | N# = Ob
| | Cb = Gx
| | Cb = Gx
| | 3773/2187, 6561/3773
| | 3773/2187, 6561/3773
Line 104: Line 106:
|-
|-
| | 12
| | 12
| | A
| | O
| | A# = Dbb
| | A# = Dbb
| | 49/27
| | 49/27
Line 111: Line 113:
|-
|-
| | 13
| | 13
| | A#
| | O#
| | B
| | B
| | 21/11
| | 21/11
Line 118: Line 120:
|-
|-
| | 14
| | 14
| | Bb
| | Pb
| | C
| | C
| | 99/49, 243/121
| | 99/49, 243/121
Line 125: Line 127:
|-
|-
| | 15
| | 15
| | B
| | P
| | Db = Ax
| | Db = Ax
| | 729/343
| | 729/343
Line 132: Line 134:
|-
|-
| | 16
| | 16
| | B# = Cb
| | P# = Qb
| | B# = Ebb
| | B# = Ebb
| | 539/243
| | 539/243
Line 139: Line 141:
|-
|-
| | 17
| | 17
| | C
| | Q
| | C#
| | C#
| | 7/3
| | 7/3
Line 146: Line 148:
|-
|-
| | 18
| | 18
| | C#
| | Q#
| | D
| | D
| | 27/11, 121/49
| | 27/11, 121/49
Line 153: Line 155:
|-
|-
| | 19
| | 19
| | Db
| | Rb
| | Eb
| | Eb
| | 891/343, 2187/847
| | 891/343, 2187/847
Line 160: Line 162:
|-
|-
| | 20
| | 20
| | D
| | R
| | Fb = Cx
| | Fb = Cx
| | 2401/891, 5929/2187
| | 2401/891, 5929/2187
Line 167: Line 169:
|-
|-
| | 21
| | 21
| | D# = Eb
| | R# = Jb
| | D# = Gbb
| | D# = Gbb
| | 77/27, 343/121
| | 77/27, 343/121
Line 174: Line 176:
|-
|-
| | 22
| | 22
| | E
| | J
| | E
| | E
| | 3/1
| | 3/1
Line 181: Line 183:
|}
|}


== 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]]

Latest revision as of 15:31, 31 July 2025

← 21edt 22edt 23edt →
Prime factorization 2 × 11
Step size 86.4525 ¢ 
Octave 14\22edt (1210.34 ¢) (→ 7\11edt)
Consistency limit 7
Distinct consistency limit 5

22 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 22edt or 22ed3), is a nonoctave tuning system that divides the interval of 3/1 into 22 equal parts of about 86.5 ¢ each. Each step represents a frequency ratio of 31/22, or the 22nd root of 3. It supports mintaka temperament.

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.

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 (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 (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.


Approximation of prime harmonics in 22edt
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Error Absolute (¢) +10.3 +0.0 -19.8 +2.8 -1.6 -31.5 +22.8 +3.2 +18.2 -37.3 +20.2 -26.8 -31.6 -27.6 -8.7
Relative (%) +12.0 +0.0 -22.9 +3.3 -1.8 -36.4 +26.4 +3.7 +21.1 -43.1 +23.4 -31.0 -36.5 -31.9 -10.0
Steps
(reduced) 		14
(14) 		22
(0) 		32
(10) 		39
(17) 		48
(4) 		51
(7) 		57
(13) 		59
(15) 		63
(19) 		67
(1) 		69
(3) 		72
(6) 		74
(8) 		75
(9) 		77
(11)

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.

Degree Note (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.

Audio examples

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

Music

Peter Kosmorsky
Ray Perlner
Chris Vaisvil
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