# 13edf

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 ← 12edf 13edf 14edf →
Prime factorization 13 (prime)
Step size 53.9965¢
Octave 22\13edf (1187.92¢)
Semitones (A1:m2) 3:1 (162¢ : 54¢)
Consistency limit 4
Distinct consistency limit 4

13EDF is the equal division of the just perfect fifth into 13 parts of 53.9965 cents each, corresponding to 22.2236 edo. It is nearly identical to every ninth step of 200edo.

## Intervals

degree cents value corresponding
JI intervals
comments
0 exact 1/1
1 53.9965 33/32 pseudo-25/24
2 107.9931 17/16, 117/110, 16/15
3 161.9896 11/10
4 215.9862 17/15
5 269.9827 7/6
6 323.9792 77/64 pseudo-6/5
7 377.9758 56/45 pseudo-5/4
8 431.9723 9/7
9 485.9688 45/34 pseudo-4/3
10 539.9654 15/11
11 593.9619 55/39, 24/17
12 647.9585 16/11
13 701.9550 exact 3/2 just perfect fifth
14 755.9515 99/64
15 809.9481 51/32, 8/5
16 863.9446 33/20
17 917.9412 17/10
18 971.9377 7/4
19 1025.9342 29/16 pseudo-9/5
20 1079.9308 28/15 pseudo-15/8
21 1133.9273 52/27, 27/14
22 1187.9238 135/68 pseudo-octave
23 1241.9204 45/22
24 1295.9169 19/9, 36/17
25 1349.9135 24/11
26 1403.9100 exact 9/4 pythagorean major ninth

## 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: 52.74725 cents (4\7/13 = 4\91) to 55.384615 cents (3\5/13 = 3\65)

Fifth Cents Comments
4\7 52.74725
27\47 53.0278
23\40 53.0769
42\73 53.1085
19\33 53.14685
53\92 53.1773
34\59 53.1943
49\85 53.2123
15\26 53.2544
56\97 53.2910
41\71 53.3044
67\116 53.31565
26\45 53.3 Flattone is in this region
63\109 53.35215
37\64 53.3654
48\83 53.3828
11\19 53.4413
51\88 53.4965
40\69 53.5117
69\119 53.52295
29\50 53.5385
76\131 53.55255 Golden meantone (696.2145¢)
47\81 53.56125
65\112 53.5714
18\31 53.5980 Meantone is in this region
61\105 53.6264
43\74 53.63825
68\117 53.6489
25\43 53.6673
57\98 53.6892
32\55 53.7063
39\67 53.73135
7\12 53.84615
38\65 53.9645
31\53 53.9913 The fifth closest to a just 3/2 for EDOs less than 200
55\94 54.0098 Garibaldi / Cassandra
24\41 54.0338
65\111 54.054
41\70 54.0659
58\99 54.07925
17\29 54.1114
61\104 54.1420
44\75 54.15385
71\121 54.1640 Golden neogothic (704.0956¢)
27\46 54.1806 Neogothic is in this region
64\109 54.1990
37\63 54.21245
47\80 54.2308
10\17 54.2986
43\73 54.3730
33\56 54.3956
56\95 54.4130
23\39 54.4734
59\100 54.4615
36\61 54.4767
49\83 54.4949
13\22 54.54 Archy is in this region
42\71 54.60455
29\49 54.6311
45\76 54.6558
16\27 54.70085
35\59 54.7588
19\32 54.8077
22\37 54.88565
3\5 55.3846

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.