66edo

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← 65edo66edo67edo →
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

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)

The patent val is contorted in the 5-limit, tempering out the same commas 250/243, 2048/2025 and 3125/3072 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 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 Pi with only +0.023 cents of error.

Interval table

Steps Cents Ups and Downs Notation
(Dual Flat Fifth 38\66)
Ups and Downs Notation
(Dual Sharp Fifth 39\66)
Approximate Ratios
0 0 D D 1/1
1 18.1818 ^D, vE♭♭♭♭ ^D, vvE♭
2 36.3636 D♯, E♭♭♭♭ ^^D, vE♭ 50/49, 56/55
3 54.5455 ^D♯, vE♭♭♭ ^3D, E♭ 33/32
4 72.7273 D𝄪, E♭♭♭ ^4D, v8E 26/25, 80/77
5 90.9091 ^D𝄪, vE♭♭ ^5D, v7E 21/20, 55/52
6 109.091 D♯𝄪, E♭♭ ^6D, v6E 16/15, 52/49
7 127.273 ^D♯𝄪, vE♭ ^7D, v5E 14/13
8 145.455 D𝄪𝄪, E♭ ^8D, v4E
9 163.636 ^D𝄪𝄪, vE D♯, v3E 11/10
10 181.818 E ^D♯, vvE 49/44
11 200 ^E, vF♭♭♭ ^^D♯, vE 28/25, 55/49
12 218.182 E♯, F♭♭♭ E 25/22
13 236.364 ^E♯, vF♭♭ ^E, vvF 8/7
14 254.545 E𝄪, F♭♭ ^^E, vF 15/13, 65/56
15 272.727 ^E𝄪, vF♭ F 75/64
16 290.909 E♯𝄪, F♭ ^F, vvG♭ 13/11, 33/28, 77/65
17 309.091 ^E♯𝄪, vF ^^F, vG♭
18 327.273 F ^3F, G♭ 40/33
19 345.455 ^F, vG♭♭♭♭ ^4F, v8G 39/32, 49/40
20 363.636 F♯, G♭♭♭♭ ^5F, v7G 16/13, 26/21
21 381.818 ^F♯, vG♭♭♭ ^6F, v6G 5/4
22 400 F𝄪, G♭♭♭ ^7F, v5G 44/35
23 418.182 ^F𝄪, vG♭♭ ^8F, v4G 14/11, 33/26
24 436.364 F♯𝄪, G♭♭ F♯, v3G
25 454.545 ^F♯𝄪, vG♭ ^F♯, vvG 13/10
26 472.727 F𝄪𝄪, G♭ ^^F♯, vG 21/16
27 490.909 ^F𝄪𝄪, vG G 4/3, 65/49
28 509.091 G ^G, vvA♭ 35/26, 75/56
29 527.273 ^G, vA♭♭♭♭ ^^G, vA♭
30 545.455 G♯, A♭♭♭♭ ^3G, A♭ 11/8
31 563.636 ^G♯, vA♭♭♭ ^4G, v8A
32 581.818 G𝄪, A♭♭♭ ^5G, v7A 7/5
33 600 ^G𝄪, vA♭♭ ^6G, v6A
34 618.182 G♯𝄪, A♭♭ ^7G, v5A 10/7
35 636.364 ^G♯𝄪, vA♭ ^8G, v4A 75/52
36 654.545 G𝄪𝄪, A♭ G♯, v3A 16/11
37 672.727 ^G𝄪𝄪, vA ^G♯, vvA 65/44, 77/52
38 690.909 A ^^G♯, vA 52/35
39 709.091 ^A, vB♭♭♭♭ A 3/2
40 727.273 A♯, B♭♭♭♭ ^A, vvB♭ 32/21
41 745.455 ^A♯, vB♭♭♭ ^^A, vB♭ 20/13, 77/50
42 763.636 A𝄪, B♭♭♭ ^3A, B♭
43 781.818 ^A𝄪, vB♭♭ ^4A, v8B 11/7, 52/33
44 800 A♯𝄪, B♭♭ ^5A, v7B 35/22
45 818.182 ^A♯𝄪, vB♭ ^6A, v6B 8/5
46 836.364 A𝄪𝄪, B♭ ^7A, v5B 13/8, 21/13
47 854.545 ^A𝄪𝄪, vB ^8A, v4B 64/39, 80/49
48 872.727 B A♯, v3B 33/20
49 890.909 ^B, vC♭♭♭ ^A♯, vvB
50 909.091 B♯, C♭♭♭ ^^A♯, vB 22/13, 56/33
51 927.273 ^B♯, vC♭♭ B 75/44
52 945.455 B𝄪, C♭♭ ^B, vvC 26/15
53 963.636 ^B𝄪, vC♭ ^^B, vC 7/4
54 981.818 B♯𝄪, C♭ C 44/25
55 1000 ^B♯𝄪, vC ^C, vvD♭ 25/14
56 1018.18 C ^^C, vD♭
57 1036.36 ^C, vD♭♭♭♭ ^3C, D♭ 20/11
58 1054.55 C♯, D♭♭♭♭ ^4C, v8D
59 1072.73 ^C♯, vD♭♭♭ ^5C, v7D 13/7
60 1090.91 C𝄪, D♭♭♭ ^6C, v6D 15/8, 49/26
61 1109.09 ^C𝄪, vD♭♭ ^7C, v5D 40/21
62 1127.27 C♯𝄪, D♭♭ ^8C, v4D 25/13, 77/40
63 1145.45 ^C♯𝄪, vD♭ C♯, v3D 64/33
64 1163.64 C𝄪𝄪, D♭ ^C♯, vvD 49/25, 55/28
65 1181.82 ^C𝄪𝄪, vD ^^C♯, vD
66 1200 D D 2/1