# 22edf

 ← 21edf 22edf 23edf →
Prime factorization 2 × 11
Step size 31.907¢
Octave 38\22edf (1212.47¢) (→19\11edf)
Twelfth 60\22edf (1914.42¢) (→30\11edf)
Consistency limit 2
Distinct consistency limit 2

22EDF is the equal division of the just perfect fifth into 22 parts of 31.907 cents each, corresponding to 37.6092 edo (similar to every fifth step of 188edo).

## Intervals

degree cents value corresponding
JI intervals
0 exact 1/1
1 31.907 55/54
2 63.8141 (28/27), (27/26)
3 95.7211
4 127.6282 14/13
5 159.5352 57/52
6 191.4423
7 223.3493 8/7
8 255.2564
9 287.1634 13/11
10 319.0705 6/5
11 350.9775 60/49, 49/40
12 382.8845 5/4
13 414.7916 14/11
14 446.6986
15 478.6057
16 510.5127
17 542.4198 26/19
18 574.3268 39/28
19 606.2339 64/45
20 638.1409 (13/9)
21 670.048 81/55
22 701.955 exact 3/2 just perfect fifth
23 733.862 55/36
24 765.7691 14/9, 81/52
25 797.6761
26 828.5732 21/13
27 861.4902 171/104
28 893.3973
29 925.3043 12/7
30 956.2114
31 988.1184 39/22
32 1020.0255 9/5
33 1052.9235 90/49, 147/80
34 1084.8395 15/8
35 1116.7466 21/11
36 1148.6536
37 1180.5607
38 1211.4677
39 1244.3748 39/19
40 1276.2816 117/56
41 1308.1889 32/15
42 1340.0959 13/6
43 1372.003 243/110
44 1403.91 exact 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: 31.16883 cents (4\7/22 = 2\77) to 32.72 cents (3\5/22 = 3\110)

4\7 31.1688
27\47 31.3346
23\40 31.36
42\73 31.3823
19\33 31.4045
53\92 31.4229
34\59 31.4330
49\85 31.44385
15\26 31.4685
56\97 31.4902
41\71 31.4981
67\116 31.5047
26\45 31.51 Flattone is in this region
63\109 31.5263
37\64 31.53409
48\83 31.5444
11\19 31.57895
51\88 31.6116
40\69 31.62055
69\119 31.6272
29\50 31.63
76\131 31.6447 Golden meantone (696.2145¢)
47\81 31.6498
65\112 31.6558
18\31 31.67155 Meantone is in this region
61\105 31.6883
43\74 31.6953
68\117 31.7016
25\43 31.7125
57\98 31.7254
32\55 31.7355
39\67 31.7503
7\12 31.81
38\65 31.8881
31\53 31.90395 The fifth closest to a just 3/2 for EDOs less than 200
55\94 31.9149 Garibaldi / Cassandra
24\41 31.92905
65\111 31.9410
41\70 31.94805
58\99 31.9559
17\29 31.9749
61\104 31.9930
44\75 32.0000
71\121 32.0060 Golden neogothic (704.0956¢)
27\46 32.0158 Neogothic is in this region
64\109 32.0267
37\63 32.0346
47\80 32.054
10\17 32.0856
43\73 32.1295
33\56 32.1429
56\95 32.1531
23\39 32.1768
59\100 32.18
36\61 32.1908
49\83 32.2015
13\22 32.2314 Archy is in this region
42\71 32.2663
29\49 32.2820
45\76 32.29665
16\27 32.32
35\59 32.3575
19\32 32.3863
22\37 32.432
3\5 32.72

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.