# 21edf

 ← 20edf 21edf 22edf →
Prime factorization 3 × 7
Step size 33.4264¢
Octave 36\21edf (1203.35¢) (→12\7edf)
Semitones (A1:m2) 3:3 (100.3¢ : 100.3¢)
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

## 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

## Scale tree

If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking "freshman sums" of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.

If we carry this freshman-summing out a little further, new, larger EDOs pop up in our continuum.

Generator range: 32.65306 cents (4\7/21 = 4\147) to 34.28571 cents (3\5/21 = 1\35)

4\7 32.6531
27\47 32.82675
23\40 32.8571
42\73 32.8767
19\33 32.9004
53\92 32.91925
34\59 32.9298
49\85 32.9412
15\26 33.9670
56\97 32.9897
41\71 32.9980
67\116 33.0049
26\45 33.0159 Flattone is in this region
63\109 33.0275
37\64 33.0357
48\83 33.0465
11\19 33.0827
51\88 33.1167
40\69 33.1263
69\119 33.13325
29\50 33.1429
76\131 33.1516 Golden meantone (696.2145¢)
47\81 33.1570
65\112 33.1633
18\31 33.1797 Meantone is in this region
61\105 33.1973
43\74 33.2046
68\117 33.2112
25\43 33.2226
57\98 33.23615
32\55 33.24675
39\67 33.2623
7\12 33.3
38\65 33.4066
31\53 33.4232 The fifth closest to a just 3/2 for EDOs less than 200
55\94 33.43465 Garibaldi / Cassandra
24\41 33.4495
65\111 33.4620
41\70 33.4694
58\99 33.4776
17\29 33.4975
61\104 33.5165
44\75 33.5328
71\121 33.5301 Golden neogothic (704.0956¢)
27\46 33.5404 Neogothic is in this region
64\109 33.5518
37\63 33.5601
47\80 33.5714
10\17 33.61345 The generator closest to a just 17/14 for EDOs less than 4200
43\73 33.6595
33\56 33.6735
56\95 33.6842
23\39 33.6996
59\100 33.7143
36\61 33.72365
49\83 33.7349
13\22 33.7662 Archy is in this region
42\71 33.8028
29\49 33.8192
45\76 33.8346
16\27 33.8624
35\59 33.8983
19\32 33.9286
22\37 33.9768
3\5 34.2857

Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.

Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.