# 15edf

 ← 14edf 15edf 16edf →
Prime factorization 3 × 5
Step size 46.797¢
Octave 26\15edf (1216.72¢)
Twelfth 41\15edf (1918.68¢)
Consistency limit 2
Distinct consistency limit 2

15EDF is the equal division of the just perfect fifth into 15 parts of 46.797 cents each, corresponding to 25.6427 edo (similar to every third step of 77edo). The 3edf~5edo correspondence has completely collapsed by this point, with this EDF being closer to 26edo than 25edo.

Lookalikes: 26edo, 41edt

## Intervals

degree cents value corresponding
JI intervals
0 exact 1/1
1 46.797
2 93.594 19/18
3 140.391 13/12
4 187.188 10/9
5 233.985 8/7
6 280.782 20/17
7 327.579
8 374.376
9 421.173 51/40
10 467.97 21/16
11 514.767 27/20
12 561.564 18/13
13 608.361 27/19
14 655.158
15 701.955 exact 3/2 just perfect fifth
16 748.752
17 795.549 19/12
18 842.346 13/8
19 889.143 5/3
20 935.94 12/7
21 982.737 30/17
22 1029.534
23 1076.331
24 1123.128 153/80
25 1168.925 63/32
26 1216.722 81/40
27 1263.519 27/13
28 1310.316 81/38
29 1357.113
30 1403.91 exact 9/4

## 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: 45.71429 cents (4\7/15 = 4\105) to 48 cents (3\5/15 = 1\25)

4\7 45.7143
27\47 45.95745
23\40 46.0000
42\73 46.0274
19\33 46.06
53\92 46.0870
34\59 46.1017
49\85 46.11765
15\26 46.15385
56\97 46.1856
41\71 46.1972
67\116 46.2069
26\45 46.2 Flattone is in this region
63\109 46.2385
37\64 46.2500
48\83 46.2651
11\19 46.3158
51\88 46.36
40\69 46.3768
69\119 46.38655
29\50 46.4000
76\131 46.4122 Golden meantone (696.2145¢)
47\81 46.41975
65\112 46.4286
18\31 46.4516 Meantone is in this region
61\105 46.4762
43\74 46.486
68\117 46.4957
25\43 46.5116
57\98 46.5306
32\55 46.54
39\67 46.5672
7\12 46.6
38\65 46.7692
31\53 46.79245 The fifth closest to a just 3/2 for EDOs less than 200
55\94 46.8085 Garibaldi / Cassandra
24\41 46.8293
65\111 46.843
41\70 46.8571
58\99 46.86
17\29 46.89655
61\104 46.9231
44\75 46.93
71\121 46.94215 Golden neogothic (704.0956¢)
27\46 46.9565 Neogothic is in this region
64\109 46.9725
37\63 46.9841
47\80 47.0000
10\17 47.0588 The generator closest to a just 21/16 for EDOs less than 200
43\73 47.1233
33\56 47.1429
56\95 47.1579
23\39 47.1795
59\100 47.2000
36\61 47.2131
49\83 47.2289
13\22 47.27 Archy is in this region
42\71 47.3239
29\49 47.3469
45\76 47.3864
16\27 47.407
35\59 47.4576
19\32 47.5000
22\37 47.567
3\5 48.0000

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.