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

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