# 5edf

 ← 4edf 5edf 6edf →
Prime factorization 5 (prime)
Step size 140.391¢
Octave 9\5edf (1263.52¢)
Twelfth 14\5edf (1965.47¢)
Consistency limit 2
Distinct consistency limit 2

5EDF is the equal division of the just perfect fifth into five parts of 140.391 cents each, corresponding to 8.5476 edo. It is close to the Bleu generator chain and every second step of 17edo. 4 steps of 5edf is a fraction of a cent away to the seventh harmonic (which is 112/81 instead of 7/4 since the equave is 3/2), which is an extremely accurate approximation for the size of this scale.

## Intervals

degree cents value octave-reduced cents value approximate ratios Neptunian notation
0 1/1 perfect unison C
1 140.391 13/12, 49/45 augmented unison, minor second C#, Db
2 280.782 75/64, 20/17, 13/11 major second, minor third D, Eb
3 421.173 14/11, 23/18 major third, diminished fourth E, Fb
4 561.564 11/8, 18/13, 25/18 perfect fourth F
5 701.955 3/2 perfect fifth C
6 842.346 21/13, 13/8, 18/11 augmented fifth, minor sixth C#, Db
7 982.737 7/4, 30/17 major sixth, minor seventh D, Eb
8 1123.128 major seventh, minor octave E, Fb
9 1263.519 63.519 major octave F
10 1403.910 203.910 C
11 1544.301 344.301 C#, Db
12 1684.692 484.692 D, Eb
13 1825.083 625.083 E
14 1965.474 765.474 F
15 2105.865 905.865 C
16 2246.256 1046.256 C#, Db
17 2386.647 1186.647 D

## 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: 137.1429 cents (4\7/5 = 4\35) to 144 cents (3\5/5 = 3\25)

4\7 137.143
27\47 137.872
23\40 138.000
42\73 138.082
19\33 138.18
53\92 138.261
34\59 138.305
49\85 138.353
15\26 138.4615
56\97 138.557
41\71 138.5915 The generator closest to a just 13/12 for EDOs less than 1000
67\116 138.621
26\45 138.6 Flattone is in this region
63\109 138.716
37\64 138.750
48\83 138.795
11\19 138.947
51\88 139.09
40\69 139.130
69\119 139.160
29\50 139.200
76\131 139.237 Golden meantone (696.2145¢)
47\81 139.259
65\112 139.286
18\31 139.355 Meantone is in this region
61\105 139.429
43\74 139.459
68\117 139.487
25\43 139.535
57\98 139.592
32\55 139.63
39\67 139.7015
7\12 140.000
38\65 140.308
31\53 140.377 The fifth closest to a just 3/2 for EDOs less than 200
55\94 140.4255 Garibaldi / Cassandra
24\41 140.488
65\111 140.540
41\70 140.571
58\99 140.60
17\29 140.690
61\104 140.769
44\75 140.800
71\121 140.826 Golden neogothic (704.0956¢)
27\46 140.870 Neogothic is in this region
64\109 140.917
37\63 140.952
47\80 141.000
10\17 141.1765
43\73 141.370
33\56 141.429
56\95 141.474
23\39 141.5385
59\100 141.600
36\61 141.639
49\83 141.687
13\22 141.81 Archy is in this region
42\71 141.972
29\49 142.041
45\76 142.105
16\27 142.2
35\59 142.373
19\32 142.500
22\37 142.702
3\5 144.000

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.