66edo
← 65edo | 66edo | 67edo → |
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
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 | -25 | -29 | -22 | -32 | -23 | +15 | +23 | -36 | +11 | +44 | |
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 |