# 19edf

 ← 18edf 19edf 20edf →
Prime factorization 19 (prime)
Step size 36.945¢
Octave 32\19edf (1182.24¢)
Semitones (A1:m2) 5:1 (184.7¢ : 36.95¢)
Consistency limit 2
Distinct consistency limit 2

19EDF is the equal division of the just perfect fifth into 19 parts of 36.945 cents each, corresponding to 32.4807 edo (similar to every second step of 65edo). It tempers out the same commas as 65edo with the addition of |-103/19 65/19> (1.425 cents) resulting from its inexact 4/1.

## Intervals

degree cents value corresponding
JI intervals
0 exact 1/1
1 36.945
2 73.89 24/23
3 110.835 16/15
4 147.78 12/11
5 184.725 10/9
6 221.67 25/22
7 258.615 36/31
8 295.56 19/16
9 332.505 63/52, 40/33
10 369.45 26/21
11 406.395 24/19, 19/15
12 443.34 31/24
13 480.285 33/25
14 517.23 27/20
15 554.175 11/8
16 591.12 45/32
17 628.065 23/16
18 665.01 22/15
19 701.955 exact 3/2 just perfect fifth
20 738.9
21 775.845
22 812.79 8/5
23 849.735 18/11
24 886.68 5/3
25 923.625
26 960.57
27 997.515 16/9
28 1034.46 20/11
29 1071.405 13/7
30 1108.35 36/19
31 1145.295 31/16
32 1182.24
33 1219.185
34 1256.13
35 1293.075
36 1330.02
37 1366.965
38 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: 36.09023 cents (4\7/19 = 4\133) to 37.89474 cents (3\5/19 = 3\95)

4\7 36.0902
27\47 36.2822
23\40 36.3158
42\73 36.3374
19\33 36.36
53\92 36.3844
34\59 36.3961
49\85 36.4087
15\26 36.43725
56\97 36.4633
41\71 36.4175
67\116 36.4791
26\45 36.4912 Flattone is in this region
63\109 36.5041
37\64 36.5132
48\83 36.52505
11\19 36.5651
51\88 36.6029
40\69 36.6133
69\119 36.6210
29\50 36.6316
76\131 36.6412 Golden meantone (696.2145¢)
47\81 36.6472
65\112 36.6541
18\31 36.6723 Meantone is in this region
61\105 36.6917
43\74 36.6999
68\117 36.70175
25\43 36.7197
57\98 36.7347
32\55 36.7464
39\67 36.76355
7\12 36.8421
38\65 36.9231
31\53 36.9414 The fifth closest to a just 3/2 for EDOs less than 200
55\94 36.9541 Garibaldi / Cassandra
24\41 36.9705
65\111 36.98435
41\70 36.9925
58\99 37.0016
17\29 37.0236
61\104 37.0445
44\75 37.0526
71\121 37.0596 Golden neogothic (704.0956¢)
27\46 37.0709 Neogothic is in this region
64\109 37.0835
37\63 37.0927
47\80 37.1053
10\17 37.1517
43\73 37.2026
33\56 37.21805
56\95 37.2299
23\39 37.2470
59\100 37.2632
36\61 37.2735
49\83 37.2388
13\22 37.3206 Archy is in this region
42\71 37.3610
29\49 37.3792
45\76 37.3961
16\27 37.4269
35\59 37.46655
19\32 37.5000
22\37 37.5533
3\5 37.8947

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.