# 21edf

 ← 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

Division of the just perfect fifth into 21 equal parts (21EDF) is related to 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 4-integer-limit, with discrepancy for the 5th harmonic.

Lookalikes: 36edo, 57edt

# Approximations

## Harmonics

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

## 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.

The following table gives an overview of all degrees of 36edo.

Degree Size

in cents

Approximate

ratios of 2.3.7

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