22edt: Difference between revisions

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'''22edt''' is the '''equal division of the third harmonic''' ([[edt]]) into '''22 tones''', each 86.4525 [[cent]]s in size.
'''22edt''' is the '''equal division of the third harmonic''' ([[edt]]) into '''22 tones''', each 86.4525 [[cent]]s in size.


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 enneatonic) 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.
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.
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.
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== Intervals ==
== Intervals ==


{| class="wikitable" style="text-align: right"
{| class="wikitable"
! Steps !! [[Cent]]s
!hekts
|-
|-
| 1 || 86.453
! | Degree
|59.091
! | Note (BPS-Lambda notation)
! | Note (Macrodiatonic notation)
! | Approximate 3.7.11 subgroup interval
! | cents value
! | hekts
|-
|-
| 2 || 172.905
| | 0
|118.182
| | E
| | E
| | 1/1
| | 0
| | 0
|-
|-
| 3 || 259.358
| | 1
|177.273
| | E# = Fb
| | F
| | 81/77, 363/343
| | 86.453
| | 59.091
|-
|-
| 4 || 345.81
| | 2
|236.364
| | F
| | Gb = Dx
| | 2673/2401, 6561/5929
| | 172.905
| | 118.182
|-
|-
| 5 || 432.263
| | 3
|295.4545
| | F#
| | E# = Abb
| | 343/297, 847/729
| | 259.358
| | 177.273
|-
|-
| 6 || 518.715
| | 4
|354.5455
| | Gb
| | F#
| | 11/9, 147/121
| | 345.810
| | 236.364
|-
|-
| 7 || 605.168
| | 5
|413.636
| | G
| | G
| | 9/7
| | 432.263
| | 295.455
|-
|-
| 8 || 691.620
| | 6
|472.727
| | G# = Hb
| | Ab = Ex
| | 729/539
| | 518.715
| | 354.545
|-
|-
| 9 || 778.073
| | 7
|531.818
| | H
| | Fx = Bbb
| | 343/243
| | 605.168
| | 413.636
|-
|-
| 10 || 864.525
| | 8
|590.909
| | H#
| | G#
| | 49/33, 121/81
| | 691.620
| | 472.727
|-
|-
| 11 || 950.978
| | 9
|650
| | Jb
| | A
| | 11/7
| | 778.073
| | 531.818
|-
|-
| 12 || 1037.430
| | 10
|709.091
| | J
| | Bb
| | 81/49
| | 864.525
| | 590.909
|-
|-
| 13 || 1123.883
| | 11
|768.182
| | J# = Ab
| | Cb = Gx
| | 3773/2187, 6561/3773
| | 950.978
| | 650.
|-
|-
| 14 || 1210.335
| | 12
|827.273
| | A
| | A# = Dbb
| | 49/27
| | 1037.430
| | 709.091
|-
|-
| 15 || 1296.788
| | 13
|886.364
| | A#
| | B
| | 21/11
| | 1123.883
| | 768.182
|-
|-
| 16 || 1383.24
| | 14
|945.4545
| | Bb
| | C
| | 99/49, 243/121
| | 1210.335
| | 827.273
|-
|-
| 17 || 1469.693
| | 15
|1004.5455
| | B
| | Db = Ax
| | 729/343
| | 1296.788
| | 886.364
|-
|-
| 18 || 1556.145
| | 16
|1063.636
| | B# = Cb
| | B# = Ebb
| | 539/243
| | 1383.240
| | 945.455
|-
|-
| 19 || 1642.598
| | 17
|1122.727
| | C
| | C#
| | 7/3
| | 1469.693
| | 1004.545
|-
|-
| 20 || 1729.05
| | 18
|1181.818
| | C#
| | D
| | 27/11, 121/49
| | 1556.145
| | 1063.636
|-
|-
| 21 || 1815.503
| | 19
|1240.909
| | Db
| | Eb
| | 891/343, 2187/847
| | 1642.598
| | 1122.727
|-
|-
| 22 || 1901.955
| | 20
|1300
| | D
| | Fb = Cx
| | 2401/891, 5929/2187
| | 1729.050
| | 1181.818
|-
| | 21
| | D# = Eb
| | D# = Gbb
| | 77/27, 343/121
| | 1815.503
| | 1240.909
|-
| | 22
| | E
| | E
| | 3/1
| | 1901.955
| | 1300.
|}
|}


Revision as of 08:06, 21 August 2024

← 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

22edt is the equal division of the third harmonic (edt) into 22 tones, each 86.4525 cents in size.

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.

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.


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

Degree Note (BPS-Lambda notation) Note (Macrodiatonic notation) Approximate 3.7.11 subgroup interval cents value hekts
0 E E 1/1 0 0
1 E# = Fb F 81/77, 363/343 86.453 59.091
2 F Gb = Dx 2673/2401, 6561/5929 172.905 118.182
3 F# E# = Abb 343/297, 847/729 259.358 177.273
4 Gb F# 11/9, 147/121 345.810 236.364
5 G G 9/7 432.263 295.455
6 G# = Hb Ab = Ex 729/539 518.715 354.545
7 H Fx = Bbb 343/243 605.168 413.636
8 H# G# 49/33, 121/81 691.620 472.727
9 Jb A 11/7 778.073 531.818
10 J Bb 81/49 864.525 590.909
11 J# = Ab Cb = Gx 3773/2187, 6561/3773 950.978 650.
12 A A# = Dbb 49/27 1037.430 709.091
13 A# B 21/11 1123.883 768.182
14 Bb C 99/49, 243/121 1210.335 827.273
15 B Db = Ax 729/343 1296.788 886.364
16 B# = Cb B# = Ebb 539/243 1383.240 945.455
17 C C# 7/3 1469.693 1004.545
18 C# D 27/11, 121/49 1556.145 1063.636
19 Db Eb 891/343, 2187/847 1642.598 1122.727
20 D Fb = Cx 2401/891, 5929/2187 1729.050 1181.818
21 D# = Eb D# = Gbb 77/27, 343/121 1815.503 1240.909
22 E E 3/1 1901.955 1300.

Compositions

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