# 4edf

 ← 3edf 4edf 5edf →
Prime factorization 22
Step size 175.489¢
Octave 7\4edf (1228.42¢)
(semiconvergent)
Semitones (A1:m2) 0:1 (0¢ : 175.5¢)
Consistency limit 6
Distinct consistency limit 2
Special properties

4EDF is the equal division of the just perfect fifth into four parts of 175.489 cents each, corresponding to 6.8380 edo. It is related to the tetracot temperament, which tempers out 20000/19683 in the 5-limit.

This division is said to be used in traditional scales of Georgia. See the discussion on the Yahoo tuning list.

## Intervals

degree cents value octave-reduced cents value Equalized Neptunian notation
0 C
1 175.489 D
2 350.978 E
3 526.466 F
4 701.955 C
5 877.444 D
6 1052.933 E
second octave
7 1228.421 28.421 F
8 1403.910 203.910 C
nonet
9 1579.399 379.399 D
10 1754.888 554.888 E
11 1930.376 730.376 F
12 2105.865 905.865 C
13 2281.354 1081.354 D
third octave
14 2456.843 56.843 E

## 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: 171.4286 cents (4\7/4 = 1\7) to 180 cents (3\5/4 = 3\20)

4\7 171.429
27\47 172.340
23\40 172.500
42\73 172.603
19\33 172.72
53\92 172.826
34\59 172.881
49\85 172.941
15\26 173.076
56\97 173.196
41\71 173.239
67\116 173.276
26\45 173.3 Flattone is in this region
63\109 173.3945
37\64 173.4375
48\83 173.494
11\19 173.684
51\88 173.863
40\69 173.913
69\119 173.950
29\50 174.000
76\131 174.046 Golden meantone (696.2145¢)
47\81 174.074
65\112 174.107
18\31 174.193 Meantone is in this region
61\105 174.286
43\74 174.324
68\117 174.359
25\43 174.419
57\98 174.490
32\55 174.54
39\67 174.627
7\12 175.000
38\65 175.385
31\53 175.472 The fifth closest to a just 3/2 for EDOs less than 200
55\94 175.532 Garibaldi / Cassandra
24\41 175.610
65\111 175.675
41\70 175.714
58\99 175.75
17\29 175.862
61\104 175.9615
44\75 176.000
71\121 176.033 Golden neogothic (704.0956¢)
27\46 176.087 Neogothic is in this region
64\109 176.147
37\63 176.1905
47\80 176.250
10\17 176.471
43\73 176.712
33\56 176.786
56\95 176.842
23\39 176.923
59\100 177.000
36\61 177.049
49\83 177.108
13\22 177.27 Archy is in this region
42\71 177.648
29\49 177.551
45\76 177.532
16\27 177.7
35\59 177.966
19\32 178.125
22\37 178.378
3\5 180.000

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.