# 10edf

 ← 9edf 10edf 11edf →
Prime factorization 2 × 5
Step size 70.1955¢
Octave 17\10edf (1193.32¢)
(semiconvergent)
Semitones (A1:m2) 2:1 (140.4¢ : 70.2¢)
Consistency limit 7
Distinct consistency limit 4

Division of the just perfect fifth into 10 equal parts (10EDF) is related to 17 edo, but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the 7-integer-limit, but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the 4-integer-limit.

Lookalikes: 17edo, 27edt

## Intervals

degree Neptunian notation using 8\10edf Neapolitan notation using 3/10edf
0 C F
1 70.1955 ^C, vDb F^, Gb
2 140.391 C#, Db F#, Gd
3 210.5865 vD G
4 280.782 D G^, Ab
5 350.9775 ^D, vE G#, Ad
6 421.173 E A
7 491.3685 ^E, vF A^, Hb
8 561.564 F A#, Hd
9 631.7595 ^F, vC H
10 701.955 C B
11 772.1505 ^C, vDb B^, Cb
12 842.346 C#, Db B#, Cd
13 912.5415 vD C
14 982.737 D C^, Db
15 1052.9325 ^D, vE C#, Dd
16 1123.128 E D
17 1193.3235 ^E, vF D^, Eb
18 1263.519 F D#, Eb
19 1333.7145 ^F, vC E
20 1403.91 C F

## 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: 68.57143 cents (4\7/10 = 2\35) to 72 cents (3\5/10 = 3\50)

4\7 68.5714
27\47 68.9362
23\40 69.000
42\73 69.0411
19\33 69.0909
53\92 69.1304
34\59 69.1525
49\85 69.1765
15\26 69.2308
56\97 69.2784
41\71 69.2958
67\116 69.3103
26\45 69.3333 Flattone is in this region
63\109 69.3578
37\64 69.3750
48\83 69.3976
11\19 69.4737
51\88 69.5455
40\69 69.5652
69\119 69.5798
29\50 69.6000
66\131 69.6183 Golden meantone (696.2145¢)
47\81 69.6296
65\112 69.6429
18\31 69.6774 Meantone is in this region
61\105 69.7143
43\74 69.7297
68\117 69.7436
25\43 69.7674
57\98 69.7959
32\55 69.8182
39\67 69.8507
7\12 70.000
38\65 70.1539
31\53 70.1887 The fifth closest to a just 3/2 for EDOs less than 200
55\94 70.2128 Garibaldi / Cassandra
24\41 70.2409
65\111 70.2703
41\70 70.2857
58\99 70.3030
17\29 70.3448
61\104 70.3846
44\75 70.4000
71\121 70.4132 Golden neogothic (704.0956¢)
27\46 70.4348 Neogothic is in this region
64\109 70.5487
37\63 70.4762
47\80 70.5000
10\17 70.5882 The generator closest to a just 25/24 for EDOs less than 200
43\73 70.6849
33\56 70.7143
56\95 70.7368
23\39 70.7692
59\100 70.8000
36\61 70.8197
49\83 70.8434
13\22 70.9091 Archy is in this region
42\71 70.9859
29\49 71.0204
45\76 71.0526
16\27 71.1111
35\59 71.1864
19\32 71.2500
22\37 71.3514
3\5 72.000

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.