# 66edo

 ← 65edo 66edo 67edo →
Prime factorization 2 × 3 × 11
Step size 18.1818¢
Fifth 39\66 (709.091¢) (→13\22)
Semitones (A1:m2) 9:3 (163.6¢ : 54.55¢)
Dual sharp fifth 39\66 (709.091¢) (→13\22)
Dual flat fifth 38\66 (690.909¢) (→19\33)
Dual major 2nd 11\66 (200¢) (→1\6)
Consistency limit 3
Distinct consistency limit 3

66 equal divisions of the octave (abbreviated 66edo or 66ed2), also called 66-tone equal temperament (66tet) or 66 equal temperament (66et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 66 equal parts of about 18.2 ¢ each. Each step represents a frequency ratio of 21/66, or the 66th root of 2.

## Theory

The patent val of 66edo is contorted in the 5-limit, tempering out the same commas (250/243, 2048/2025, 3125/3072, etc.) as 22edo. In the 7-limit it tempers out 686/675 and 1029/1024, in the 11-limit 55/54, 100/99 and 121/120, in the 13-limit 91/90, 169/168, 196/195 and in the 17-limit 136/135 and 256/255. It provides the optimal patent val for the 11- and 13-limit ammonite temperament.

The 66b val tempers out 16875/16384 in the 5-limit, 126/125, 1728/1715 and 2401/2400 in the 7-limit, 99/98 and 385/384 in the 11-limit, and 105/104, 144/143 and 847/845 in the 13-limit.

109 steps of 66edo is extremely close to the acoustic pi with only +0.023 cents of error.

### Odd harmonics

Approximation of odd harmonics in 66edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +7.14 -4.50 -5.19 -3.91 -5.86 -4.16 +2.64 +4.14 -6.60 +1.95 +8.09
Relative (%) +39.2 -24.7 -28.5 -21.5 -32.2 -22.9 +14.5 +22.7 -36.3 +10.7 +44.5
Steps
(reduced)
105
(39)
153
(21)
185
(53)
209
(11)
228
(30)
244
(46)
258
(60)
270
(6)
280
(16)
290
(26)
299
(35)

### Subsets and supersets

Since 66 factors into 2 × 3 × 11, 66edo has subset edos 2, 3, 6, 11, 22, and 33. 198edo, which triples it, corrects its approximation to many of the lower harmonics.

## Interval table

Steps Cents Approximate Ratios Ups and Downs Notation
(Dual Flat Fifth 38\66)
Ups and Downs Notation
(Dual Sharp Fifth 39\66)
0 0 1/1 D D
1 18.182 ^D, vE♭♭♭♭ ^D, vvE♭
2 36.364 50/49, 56/55 D♯, E♭♭♭♭ ^^D, vE♭
3 54.545 33/32 ^D♯, vE♭♭♭ ^3D, E♭
4 72.727 26/25, 80/77 D𝄪, E♭♭♭ ^4D, v8E
5 90.909 21/20, 55/52 ^D𝄪, vE♭♭ ^5D, v7E
6 109.091 16/15, 52/49 D♯𝄪, E♭♭ ^6D, v6E
7 127.273 14/13 ^D♯𝄪, vE♭ ^7D, v5E
8 145.455 D𝄪𝄪, E♭ ^8D, v4E
9 163.636 11/10 ^D𝄪𝄪, vE D♯, v3E
10 181.818 49/44 E ^D♯, vvE
11 200 28/25, 55/49 ^E, vF♭♭♭ ^^D♯, vE
12 218.182 25/22 E♯, F♭♭♭ E
13 236.364 8/7 ^E♯, vF♭♭ ^E, vvF
14 254.545 15/13, 65/56 E𝄪, F♭♭ ^^E, vF
15 272.727 75/64 ^E𝄪, vF♭ F
16 290.909 13/11, 33/28, 77/65 E♯𝄪, F♭ ^F, vvG♭
17 309.091 ^E♯𝄪, vF ^^F, vG♭
18 327.273 40/33 F ^3F, G♭
19 345.455 39/32, 49/40 ^F, vG♭♭♭♭ ^4F, v8G
20 363.636 16/13, 26/21 F♯, G♭♭♭♭ ^5F, v7G
21 381.818 5/4 ^F♯, vG♭♭♭ ^6F, v6G
22 400 44/35 F𝄪, G♭♭♭ ^7F, v5G
23 418.182 14/11, 33/26 ^F𝄪, vG♭♭ ^8F, v4G
24 436.364 F♯𝄪, G♭♭ F♯, v3G
25 454.545 13/10 ^F♯𝄪, vG♭ ^F♯, vvG
26 472.727 21/16 F𝄪𝄪, G♭ ^^F♯, vG
27 490.909 4/3, 65/49 ^F𝄪𝄪, vG G
28 509.091 35/26, 75/56 G ^G, vvA♭
29 527.273 ^G, vA♭♭♭♭ ^^G, vA♭
30 545.455 11/8 G♯, A♭♭♭♭ ^3G, A♭
31 563.636 ^G♯, vA♭♭♭ ^4G, v8A
32 581.818 7/5 G𝄪, A♭♭♭ ^5G, v7A
33 600 ^G𝄪, vA♭♭ ^6G, v6A
34 618.182 10/7 G♯𝄪, A♭♭ ^7G, v5A
35 636.364 75/52 ^G♯𝄪, vA♭ ^8G, v4A
36 654.545 16/11 G𝄪𝄪, A♭ G♯, v3A
37 672.727 65/44, 77/52 ^G𝄪𝄪, vA ^G♯, vvA
38 690.909 52/35 A ^^G♯, vA
39 709.091 3/2 ^A, vB♭♭♭♭ A
40 727.273 32/21 A♯, B♭♭♭♭ ^A, vvB♭
41 745.455 20/13, 77/50 ^A♯, vB♭♭♭ ^^A, vB♭
42 763.636 A𝄪, B♭♭♭ ^3A, B♭
43 781.818 11/7, 52/33 ^A𝄪, vB♭♭ ^4A, v8B
44 800 35/22 A♯𝄪, B♭♭ ^5A, v7B
45 818.182 8/5 ^A♯𝄪, vB♭ ^6A, v6B
46 836.364 13/8, 21/13 A𝄪𝄪, B♭ ^7A, v5B
47 854.545 64/39, 80/49 ^A𝄪𝄪, vB ^8A, v4B
48 872.727 33/20 B A♯, v3B
49 890.909 ^B, vC♭♭♭ ^A♯, vvB
50 909.091 22/13, 56/33 B♯, C♭♭♭ ^^A♯, vB
51 927.273 75/44 ^B♯, vC♭♭ B
52 945.455 26/15 B𝄪, C♭♭ ^B, vvC
53 963.636 7/4 ^B𝄪, vC♭ ^^B, vC
54 981.818 44/25 B♯𝄪, C♭ C
55 1000 25/14 ^B♯𝄪, vC ^C, vvD♭
56 1018.182 C ^^C, vD♭
57 1036.364 20/11 ^C, vD♭♭♭♭ ^3C, D♭
58 1054.545 C♯, D♭♭♭♭ ^4C, v8D
59 1072.727 13/7 ^C♯, vD♭♭♭ ^5C, v7D
60 1090.909 15/8, 49/26 C𝄪, D♭♭♭ ^6C, v6D
61 1109.091 40/21 ^C𝄪, vD♭♭ ^7C, v5D
62 1127.273 25/13, 77/40 C♯𝄪, D♭♭ ^8C, v4D
63 1145.455 64/33 ^C♯𝄪, vD♭ C♯, v3D
64 1163.636 49/25, 55/28 C𝄪𝄪, D♭ ^C♯, vvD
65 1181.818 ^C𝄪𝄪, vD ^^C♯, vD
66 1200 2/1 D D