6edf

 ← 5edf 6edf 7edf →
Prime factorization 2 × 3
Step size 116.993¢
Octave 10\6edf (1169.93¢) (→5\3edf)
Twelfth 16\6edf (1871.88¢) (→8\3edf)
Consistency limit 2
Distinct consistency limit 2
Special properties

6EDF is the equal division of the just perfect fifth into six parts of 116.9925 cents each, corresponding to 10.2571 edo. It is related to the miracle temperament, which tempers out 225/224 and 1029/1024 in the 7-limit.

Intervals

degrees cents ~ cents octave-reduced approximate ratios Neptunian notation
0 0 (perfect unison, 1:1) 1/1 C
1 117 16/15, 15/14 C#
2 234 8/7 Db
3 351 11/9, 27/22 D
4 468 21/16 E
5 585 7/5, 45/32 F
6 702 (just perfect fifth, 3:2) 3/2 C
7 819 8/5, 21/13 C#
8 936 12/7, 55/32 Db
9 1053 11/6 D
10 1170 49/25, 160/81, 2/1 E
11 1287 ~ 87 F
12 1404 ~ 204 (just major whole tone/ninth, 9:4) C
13 1521 ~ 321 C#
14 1638 ~ 438 Db
15 1755 ~ 555 D
16 1872 ~ 672 E
17 1988 ~ 788 F
18 2106 ~ 906 (Pythagorean major sixth, 27:8) C
19 2223 ~ 1023 C#
20 2340 ~ 1140 Db
21 2457 ~ 57 D
22 2574 ~ 174 E
23 2691 ~ 291 F
24 2808 ~ 408 (Pythagorean major third, 81:16) C

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: 114.2857 cents (4\7/6 = 2\21) to 120 cents (3\5/6 = 1\10)

4\7 114.286
27\47 114.894
23\40 115.000
42\73 115.0685
19\33 115.15
53\92 115.217
34\59 115.254
49\85 115.294
15\26 115.385
56\97 115.464
41\71 115.493
67\116 115.517
26\45 115.5 Flattone is in this region
63\109 115.596
37\64 115.625
48\83 115.663
11\19 115.684
51\88 115.90
40\69 115.942
69\119 115.966
29\50 116.000
76\131 116.0305 Golden meantone (696.2145¢)
47\81 116.049
65\112 116.071
18\31 116.129 Meantone is in this region
61\105 116.1905
43\74 116.216
68\117 116.239
25\43 116.279
57\98 116.3265
32\55 116.36
39\67 116.418
7\12 116.6
38\65 116.923
31\53 116.981 The fifth closest to a just 3/2 for EDOs less than 200
55\94 117.021 Garibaldi / Cassandra
24\41 117.073
65\111 117.117
41\70 117.143
58\99 117.17
17\29 117.241
61\104 117.308
44\75 117.3
71\121 117.355 Golden neogothic (704.0956¢)
27\46 117.391 Neogothic is in this region
64\109 117.431
37\63 117.460
47\80 117.500
10\17 117.647
43\73 117.808
33\56 117.857
56\95 117.895
23\39 117.949
59\100 118.000
36\61 118.033
49\83 118.072
13\22 118.18 Archy is in this region
42\71 118.310
29\49 118.367
45\76 118.421
16\27 118.518
35\59 118.644
19\32 118.750
22\37 118.918 The generator closest to a just 15/14 for EDOs less than 1200
3\5 120.000

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.