# 2edf

 ← 1edf 2edf 3edf →
Prime factorization 2 (prime)
Step size 350.978¢
Octave 3\2edf (1052.93¢)
(semiconvergent)
Semitones (A1:m2) 2:-1 (702¢ : -351¢)
Consistency limit 2
Distinct consistency limit 1
Special properties

2edf, if the attempt is made to use it as an actual scale, would divide the just perfect fifth into two equal parts, each of size [[1]] cents, which is to say sqrt (3/2) as a frequency ratio. It corresponds to 3.4190 edo. If we want to consider it to be a temperament, it tempers out 6/5, 9/7, 32/27, and 81/80 in the patent val.

## Factoids about 2edf

60/49 and 49/40 are good rational representations of the square root of 3/2.

2edf's step size is close to the generator of the hemififths temperament, which tempers out 2401/2400 and 5120/5103 in the 7-limit.

# Cents
1 350.98
2 701.96

## 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: 342.8571 cents (4\7/2 = 2\7) to 360 cents (3\5/2 = 3/10)

4\7 342.857
27\47 344.681
23\40 345.000
42\73 345.2055
19\33 345.45
53\92 345.652
34\59 345.763
49\85 345.882
15\26 346.153
56\97 346.392
41\71 346.479
67\116 346.551
26\45 346.6 Flattone is in this region
63\109 346.789
37\64 346.875
48\83 346.988
11\19 347.368 The generator closest to a just 11/9 for EDOs less than 200
51\88 347.72
40\69 347.826
69\119 347.899
29\50 348.000
76\131 348.092 Golden meantone (696.2145¢)
47\81 348.148
65\112 348.214
18\31 348.387 Meantone is in this region
61\105 348.571
43\74 348.648
68\117 348.718
25\43 348.837
57\98 348.980
32\55 349.09
39\67 349.254
7\12 350.000
38\65 350.769
31\53 350.843 The fifth closest to a just 3/2 for EDOs less than 200
55\94 351.064 Garibaldi / Cassandra
24\41 351.2195
65\111 351.351
41\70 351.429
58\99 351.51
17\29 351.724
61\104 351.923
44\75 352.000
71\121 352.066 Golden neogothic (704.0956¢)
27\46 352.174 Neogothic is in this region
64\109 352.294
37\63 352.381
47\80 352.500
10\17 352.941
43\73 353.425
33\56 353.571
56\95 353.684
23\39 353.846
59\100 354.000
36\61 354.098
49\83 354.217
13\22 354.54 Archy is in this region
42\71 354.930
29\49 355.102
45\76 355.263
16\27 355.5
35\59 355.932
19\32 356.250
22\37 356.756
3\5 360.000

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.