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
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
| 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 |