# 3edf

 ← 2edf 3edf 4edf →
Prime factorization 3 (prime)
Step size 233.985¢
Octave 5\3edf (1169.93¢)
(convergent)
Semitones (A1:m2) 1:0 (234¢ : 0¢)
Consistency limit 10
Distinct consistency limit 2

3edf, if the attempt is made to use it as an actual scale, would divide the just perfect fifth into three equal parts, each of size 233.985 cents, which is to say (3/2)1/3 as a frequency ratio. It corresponds to 5.1285 edo. If we want to consider it to be a temperament, it tempers out 16/15, 21/20, 28/27, 81/80, and 256/243 as well as 5edo.

## Factoids about 3edf

3edf's step size is close to the slendric temperament, which tempers out 1029/1024 in the 2.3.7 subgroup.

# Cents
1 233.99
2 467.97
3 701.96

## Scale tree

EDF scales can be approximated in EDOs by subdividing diatonic fifths. 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: 228.5714 cents (4\7/3 = 4\21) to 240 cents (3\5/3 = 1\5)

4\7 228.571
27\47 229.787
23\40 230.000
42\73 230.137
19\33 230.30
53\92 230.435
34\59 230.5085
49\85 230.588
15\26 230.769
56\97 230.928
41\71 230.986
67\116 231.0345
26\45 231.1 Flattone is in this region
63\109 231.193 The generator closest to a just 8/7 for EDOs less than 600
37\64 231.25
48\83 231.325
11\19 231.579
51\88 231.81
40\69 231.884
69\119 231.933
29\50 232.000
76\131 232.061 Golden meantone (696.2145¢)
47\81 232.099
65\112 232.143
18\31 232.258 Meantone is in this region
61\105 232.381
43\74 232.432
68\117 232.479
25\43 232.558
57\98 232.653
32\55 232.72
39\67 232.836
7\12 233.3
38\65 233.846
31\53 233.962 The fifth closest to a just 3/2 for EDOs less than 200
55\94 234.043 Garibaldi / Cassandra
24\41 234.146
65\111 234.234
41\70 234.286
58\99 234.34
17\29 234.483
61\104 234.615
44\75 234.6
71\121 234.711 Golden neogothic (704.0956¢)
27\46 234.783 Neogothic is in this region
64\109 234.862
37\63 234.921
47\80 235.000
10\17 235.294
43\73 235.616
33\56 235.714
56\95 235.7895
23\39 235.897
59\100 236.000
36\61 236.066
49\83 236.145
13\22 236.36 Archy is in this region
42\71 236.620
29\49 236.735
45\76 236.842
16\27 237.037
35\59 237.288
19\32 237.500
22\37 237.837
3\5 240.000

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.