21edf: Difference between revisions

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'''[[EDF|Division of the just perfect fifth]] into 21 equal parts''' (21EDF) is related to [[36edo|36 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 3.3514 cents stretched and the step size is about 33.4264 cents. Unlike 36edo, it is only consistent up to the [[3-odd-limit|4-integer-limit]], with discrepancy for the 5th harmonic.
{{Infobox ET}}
{{ED intro}}


[[Category:Edf]]
21EDF is related to [[36edo]], but with the 3/2 rather than the 2/1 being just, which stretches the octave by about 3.3514 cents. Unlike 36edo, it is only consistent up to the 4-[[integer-limit]], with discrepancy for the 5th harmonic.
[[Category:Edonoi]]
 
Lookalikes: [[36edo]], [[57edt]], [[93ed6]], [[101ed7]], [[129ed12]]
== Theory ==
21edf acts as a stretched version of 36edo, though under most circumstances the stretch is more than ideal. If used as a fifth-based system, the chord 6:7:8(:9) may act as the fundamental chord of the system. The most important comma tempered out by this system is [[1029/1024]], and the related temperament is a fifth-based version of [[slendric]] with a 1/3-fifth period representing [[8/7]] and a generator of about a sixth-tone. One generator up or down from this period gives [[7/6]] and [[9/8]] respectively. Other edfs supporting this temperament include [[24edf]] and [[45edf]].
== Approximations ==
=== Harmonics ===
{{Harmonics in equal|21|3|2|prec=2|columns=8}}
{{Harmonics in equal|21|3|2|prec=2|columns=8|start=9|title=contd.}}
 
=== 3-limit (Pythagorean) approximations (same as 7edf): ===
2/1 = 1200 cents; 36 degrees of 21edf = 1203.3514... cents.
 
4/3 = 498.045... cents; 15 degrees of 21edf = 501.9634... cents.
 
9/8 = 203.910... cents; 6 degrees of 21edf = 200.5585... cents.
 
16/9 = 996.090... cents; 30 degrees of 21edf = 1002.7928... cents.
 
27/16 = 905.865... cents; 27 degrees of 21edf = 902.5135... cents.
 
32/27 = 294.135... cents; 9 degrees of 21edf = 300.8379... cents.
 
81/64 = 407.820... cents; 12 degrees of 21edf = 401.1171... cents.
 
128/81 = 792.180... cents; 24 degrees of 21edf = 802.2342... cents.
 
=== 7-limit approximations: ===
==== 7 only: ====
7/4 = 968.826... cents; 29 degrees of 21edf = 969.3664... cents.
 
8/7 = 231.174... cents; 7 degrees of 21edf = 233.985... cents.
 
49/32 = 737.652... cents; 22 degrees of 21edf = 733.333... cents.
 
64/49 = 462.348... cents; 14 degrees of 21edf = 467.97... cents.
 
==== 3 and 7: ====
7/6 = 266.871... cents; 8 degrees of 21edf = 267.4114... cents.
 
12/7 = 933.129... cents; 28 degrees of 21edf = 935.94... cents.
 
9/7 = 435.084... cents; 13 degrees of 21edf = 434.5435... cents.
 
14/9 = 764.916... cents; 23 degrees of 21edf = 768.8078... cents.
 
28/27 = 62.961... cents; 2 degrees of 21edf = 66.8528... cents.
 
27/14 = 1137.039... cents; 34 degrees of 21edf = 1136.4985... cents.
 
21/16 = 470.781... cents; 14 degrees of 21edf = 467.97... cents.
 
32/21 = 729.219... cents; 22 degrees of 21edf = 735.3814... cents.
 
49/48 = 35.697... cents; 1 degree of 21edf = 33.4264... cents.
 
96/49 = 1164.303... cents; 35 degrees of 21edf = 1169.925... cents.
 
49/36 = 533.742... cents; 16 degrees of 21edf = 534.8228... cents.
 
72/49 = 666.258... cents; 20 degrees of 21edf = 668.5285... cents.
 
64/63 = 27.264... cents; 1 degree of 21edf = 33.4264... cents.
 
63/32 = 1172.736... cents; 35 degrees of 21edf = 1169.925... cents.
 
== Intervals ==
The following table gives an overview of all degrees of 21edf.
 
{| class="wikitable mw-collapsible" style="text-align: center;"
|+ style="font-size: 105%;" | Intervals of 21edf
|-
! Degree
! Size<br />in [[Cent|cents]]
! Approximate<br />ratios of 2.3.7
! Additional ratios<br />of 2.3.7.13.17
|-
| colspan="2" | 0
| 1/1
|
|-
| 1
| style="text-align: right;" | 33.4264
| 64/63, [[49/48]]
|
|-
| 2
| style="text-align: right;" | 66.8529
| [[28/27]]
|
|-
| 3
| style="text-align: right;" | 100.2793
| 256/243
| [[17/16]], [[18/17]]
|-
| 4
| style="text-align: right;" | 133.7057
| 243/224
| [[14/13]], [[13/12]]
|-
| 5
| style="text-align: right;" | 167.1321
| [[54/49]]
|
|-
| 6
| style="text-align: right;" | 200.5586
| [[9/8]]
|
|-
| 7
| style="text-align: right;" | 233.985
| [[8/7]]
|
|-
| 8
| style="text-align: right;" | 267.4114
| [[7/6]]
|
|-
| 9
| style="text-align: right;" | 300.8379
| [[32/27]]
|
|-
| 10
| style="text-align: right;" | 334.2643
| 98/81
| [[17/14]]
|-
| 11
| style="text-align: right;" | 367.6907
| 243/196
| [[16/13]], [[26/21]], [[21/17]]
|-
| 12
| style="text-align: right;" | 401.1171
| [[81/64]]
|
|-
| 13
| style="text-align: right;" | 434.5436
| [[9/7]]
|
|-
| 14
| style="text-align: right;" | 467.97
| [[64/49]], [[21/16]]
| [[17/13]]
|-
| 15
| style="text-align: right;" | 501.3964
| [[4/3]]
|
|-
| 16
| style="text-align: right;" | 534.8229
| [[49/36]]
|
|-
| 17
| style="text-align: right;" | 568.2493
|
| [[18/13]]
|-
| 18
| style="text-align: right;" | 601.6757
|
|
|-
| 19
| style="text-align: right;" | 635.1021
|
| [[13/9]]
|-
| 20
| style="text-align: right;" | 668.5286
| 72/49
|
|-
| 21
| style="text-align: right;" | 701.955
| [[3/2]]
|
|-
| 22
| style="text-align: right;" | 735.3814
| [[49/32]], [[32/21]]
| [[26/17]]
|-
| 23
| style="text-align: right;" | 768.8079
| [[14/9]]
|
|-
| 24
| style="text-align: right;" | 802.2343
| [[128/81]]
|
|-
| 25
| style="text-align: right;" | 835.6607
| 392/243
| [[13/8]], [[21/13]], [[34/21]]
|-
| 26
| style="text-align: right;" | 869.0871
| 81/49
| [[28/17]]
|-
| 27
| style="text-align: right;" | 902.5136
| [[27/16]]
|
|-
| 28
| style="text-align: right;" | 935.94
| [[12/7]]
|
|-
| 29
| style="text-align: right;" | 969.3664
| [[7/4]]
|
|-
| 30
| style="text-align: right;" | 1002.7929
| [[16/9]]
|
|-
| 31
| style="text-align: right;" | 1036.2193
| 49/27
|
|-
| 32
| style="text-align: right;" | 1069.6457
| 448/243
| [[13/7]], [[24/13]]
|-
| 33
| style="text-align: right;" | 1103.0721
| [[243/128]]
| [[32/17]], [[17/9]]
|-
| 34
| style="text-align: right;" | 1136.4986
| [[27/14]]
|
|-
| 35
| style="text-align: right;" | 1169.925
| 63/32, 96/49
|
|-
| 36
| style="text-align: right;" | 1203.3514
| 2/1
|
|-
| 37
| 1236.7779
| 128/63, 49/24
|
|-
| 38
| 1270.2043
| 56/27
|
|-
| 39
| 1303.6307
| 512/243
| 17/8, 36/17
|-
| 40
| 1337.05715
| 243/112
| 28/13, 13/6
|-
| 41
| 1370.4836
| 108/49
|
|-
| 42
| 1403.91
| 9/4
|
|}
 
== See also ==
* [[36edo]] – relative edo
* [[57edt]] – relative edt
* [[93ed6]] – relative ed6
* [[101ed7]] – relative ed7
* [[129ed12]] – relative ed12, close to the zeta-optimized tuning for 36edo
 
{{todo|expand}}
 
[[Category:36edo]]

Latest revision as of 23:39, 22 October 2025

← 20edf 21edf 22edf →
Prime factorization 3 × 7
Step size 33.4264 ¢ 
Octave 36\21edf (1203.35 ¢) (→ 12\7edf)
Twelfth 57\21edf (1905.31 ¢) (→ 19\7edf)
Consistency limit 4
Distinct consistency limit 4

21 equal divisions of the perfect fifth (abbreviated 21edf or 21ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 21 equal parts of about 33.4 ¢ each. Each step represents a frequency ratio of (3/2)1/21, or the 21st root of 3/2.

21EDF is related to 36edo, but with the 3/2 rather than the 2/1 being just, which stretches the octave by about 3.3514 cents. Unlike 36edo, it is only consistent up to the 4-integer-limit, with discrepancy for the 5th harmonic.

Lookalikes: 36edo, 57edt, 93ed6, 101ed7, 129ed12

Theory

21edf acts as a stretched version of 36edo, though under most circumstances the stretch is more than ideal. If used as a fifth-based system, the chord 6:7:8(:9) may act as the fundamental chord of the system. The most important comma tempered out by this system is 1029/1024, and the related temperament is a fifth-based version of slendric with a 1/3-fifth period representing 8/7 and a generator of about a sixth-tone. One generator up or down from this period gives 7/6 and 9/8 respectively. Other edfs supporting this temperament include 24edf and 45edf.

Approximations

Harmonics

Approximation of harmonics in 21edf
Harmonic 2 3 4 5 6 7 8 9
Error Absolute (¢) +3.35 +3.35 +6.70 -11.92 +6.70 +7.24 +10.05 +6.70
Relative (%) +10.0 +10.0 +20.1 -35.7 +20.1 +21.7 +30.1 +20.1
Steps
(reduced) 		36
(15) 		57
(15) 		72
(9) 		83
(20) 		93
(9) 		101
(17) 		108
(3) 		114
(9)
contd.
Harmonic 10 11 12 13 14 15 16 17
Error Absolute (¢) -8.57 -6.44 +10.05 +5.19 +10.59 -8.57 +13.41 +8.73
Relative (%) -25.6 -19.3 +30.1 +15.5 +31.7 -25.6 +40.1 +26.1
Steps
(reduced) 		119
(14) 		124
(19) 		129
(3) 		133
(7) 		137
(11) 		140
(14) 		144
(18) 		147
(0)

3-limit (Pythagorean) approximations (same as 7edf):

2/1 = 1200 cents; 36 degrees of 21edf = 1203.3514... cents.

4/3 = 498.045... cents; 15 degrees of 21edf = 501.9634... cents.

9/8 = 203.910... cents; 6 degrees of 21edf = 200.5585... cents.

16/9 = 996.090... cents; 30 degrees of 21edf = 1002.7928... cents.

27/16 = 905.865... cents; 27 degrees of 21edf = 902.5135... cents.

32/27 = 294.135... cents; 9 degrees of 21edf = 300.8379... cents.

81/64 = 407.820... cents; 12 degrees of 21edf = 401.1171... cents.

128/81 = 792.180... cents; 24 degrees of 21edf = 802.2342... cents.

7-limit approximations:

7 only:

7/4 = 968.826... cents; 29 degrees of 21edf = 969.3664... cents.

8/7 = 231.174... cents; 7 degrees of 21edf = 233.985... cents.

49/32 = 737.652... cents; 22 degrees of 21edf = 733.333... cents.

64/49 = 462.348... cents; 14 degrees of 21edf = 467.97... cents.

3 and 7:

7/6 = 266.871... cents; 8 degrees of 21edf = 267.4114... cents.

12/7 = 933.129... cents; 28 degrees of 21edf = 935.94... cents.

9/7 = 435.084... cents; 13 degrees of 21edf = 434.5435... cents.

14/9 = 764.916... cents; 23 degrees of 21edf = 768.8078... cents.

28/27 = 62.961... cents; 2 degrees of 21edf = 66.8528... cents.

27/14 = 1137.039... cents; 34 degrees of 21edf = 1136.4985... cents.

21/16 = 470.781... cents; 14 degrees of 21edf = 467.97... cents.

32/21 = 729.219... cents; 22 degrees of 21edf = 735.3814... cents.

49/48 = 35.697... cents; 1 degree of 21edf = 33.4264... cents.

96/49 = 1164.303... cents; 35 degrees of 21edf = 1169.925... cents.

49/36 = 533.742... cents; 16 degrees of 21edf = 534.8228... cents.

72/49 = 666.258... cents; 20 degrees of 21edf = 668.5285... cents.

64/63 = 27.264... cents; 1 degree of 21edf = 33.4264... cents.

63/32 = 1172.736... cents; 35 degrees of 21edf = 1169.925... cents.

Intervals

The following table gives an overview of all degrees of 21edf.

Intervals of 21edf
Degree Size
in cents 		Approximate
ratios of 2.3.7 		Additional ratios
of 2.3.7.13.17
0 1/1
1 33.4264 64/63, 49/48
2 66.8529 28/27
3 100.2793 256/243 17/16, 18/17
4 133.7057 243/224 14/13, 13/12
5 167.1321 54/49
6 200.5586 9/8
7 233.985 8/7
8 267.4114 7/6
9 300.8379 32/27
10 334.2643 98/81 17/14
11 367.6907 243/196 16/13, 26/21, 21/17
12 401.1171 81/64
13 434.5436 9/7
14 467.97 64/49, 21/16 17/13
15 501.3964 4/3
16 534.8229 49/36
17 568.2493 18/13
18 601.6757
19 635.1021 13/9
20 668.5286 72/49
21 701.955 3/2
22 735.3814 49/32, 32/21 26/17
23 768.8079 14/9
24 802.2343 128/81
25 835.6607 392/243 13/8, 21/13, 34/21
26 869.0871 81/49 28/17
27 902.5136 27/16
28 935.94 12/7
29 969.3664 7/4
30 1002.7929 16/9
31 1036.2193 49/27
32 1069.6457 448/243 13/7, 24/13
33 1103.0721 243/128 32/17, 17/9
34 1136.4986 27/14
35 1169.925 63/32, 96/49
36 1203.3514 2/1
37 1236.7779 128/63, 49/24
38 1270.2043 56/27
39 1303.6307 512/243 17/8, 36/17
40 1337.05715 243/112 28/13, 13/6
41 1370.4836 108/49
42 1403.91 9/4

See also

  • 36edo – relative edo
  • 57edt – relative edt
  • 93ed6 – relative ed6
  • 101ed7 – relative ed7
  • 129ed12 – relative ed12, close to the zeta-optimized tuning for 36edo
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