# 18edf

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 ← 17edf 18edf 19edf →
Prime factorization 2 × 32
Step size 38.9975¢
Octave 31\18edf (1208.92¢)
Semitones (A1:m2) 2:3 (78¢ : 117¢)
Consistency limit 4
Distinct consistency limit 4

18EDF is the equal division of the just perfect fifth into 18 parts of 38.9975 cents each, corresponding to 30.7712 edo. It is related to the regular temperament which tempers out 2401/2400 and 8589934592/8544921875 in the 7-limit; 5632/5625, 46656/46585, and 166698/166375 in the 11-limit, which is supported by 31edo, 369edo, 400edo, 431edo, and 462edo.

Lookalikes: 31edo, 49edt

## Intervals

degree cents value corresponding
JI intervals
comments
0 exact 1/1
1 38.9975 45/44
2 77.995
3 116.9925 16/15
4 155.99 128/117
5 194.9875 28/25
6 233.985 8/7
7 272.9825 7/6
8 311.98 6/5
9 350.9775 60/49, 49/40
10 389.975 5/4
11 428.9725 9/7
12 467.97
13 506.9675 75/56
14 545.965
15 584.9625
16 623.96
17 662.9575 22/15
18 701.955 exact 3/2 just perfect fifth
19 740.9525 135/88
20 779.95
21 818.9475 8/5
22 857.945 64/39
23 896.9425 42/25
24 935.94 12/7
25 974.9375 7/4
26 1013.935 9/5
27 1052.9325 90/49, 147/80
28 1091.93 15/8
29 1130.9275 27/14
30 1169.925
31 1208.9225 225/112
32 1247.92
33 1286.9175
34 1325.915
35 1364.9125
36 1403.91 exact 9/4

## Related regular temperaments

The rank-two regular temperament supported by 31edo and 369edo has three equal divisions of the interval which equals an octave minus the step interval of 18EDF as a generator.

### 7-limit 31&369

Commas: 2401/2400, 8589934592/8544921875

POTE generator: ~5/4 = 386.997

Mapping: [<1 19 2 7|, <0 -54 1 -13|]

EDOs: 31, 369, 400, 431, 462

### 11-limit 31&369

Commas: 2401/2400, 5632/5625, 46656/46585

POTE generator: ~5/4 = 386.999

Mapping: [<1 19 2 7 37|, <0 -54 1 -13 -104|]

EDOs: 31, 369, 400, 431, 462

### 13-limit 31&369

Commas: 1001/1000, 1716/1715, 4096/4095, 46656/46585

POTE generator: ~5/4 = 387.003

Mapping: [<1 19 2 7 37 -35|, <0 -54 1 -13 -104 120|]

EDOs: 31, 369, 400, 431, 462

## 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: 28.09524 cents (4\7/18 = 2\63) to 40 cents (3\5/18 = 1\30)

Fifth Cents Comments
4\7 38.0952
27\47 38.2979
23\40 38.3
42\73 38.3562
19\33 38.38
53\92 38.4058
34\59 38.4181
49\85 38.4314
15\26 38.4615
56\97 38.4880
41\71 38.49765
67\116 38.50575
26\45 38.518 Flattone is in this region
63\109 38.5321
37\64 38.5417
48\83 38.5542
11\19 38.5965
51\88 38.63
40\69 38.6473
69\119 38.6555
29\50 38.6
76\131 38.6768 Golden meantone (696.2145¢)
47\81 38.6831
65\112 38.6905
18\31 38.7097 Meantone is in this region
61\105 38.7302
43\74 38.738
68\117 38.7464
25\43 38.7597
57\98 38.7755
32\55 38.78
39\67 38.8060
7\12 38.8
38\65 38.9743
31\53 38.9937 The fifth closest to a just 3/2 for EDOs less than 200
55\94 39.0071 Garibaldi / Cassandra
24\41 39.0244
65\111 39.039
41\70 39.0476
58\99 39.0572
17\29 39.0805
61\104 39.1026
44\75 39.1
71\121 39.1185 Golden neogothic (704.0956¢)
27\46 39.1304 Neogothic is in this region
64\109 39.1437
37\63 39.1534
47\80 39.16
10\17 39.2157
43\73 39.2694
33\56 39.2857
56\95 39.29825
23\39 39.3162
59\100 39.3
36\61 39.3443
49\83 39.3574
13\22 39.39 Archy is in this region
42\71 39.4366
29\49 39.4558
45\76 39.4737
16\27 39.5062
35\59 39.5480
19\32 39.583
22\37 39.639
3\5 40.0000

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.