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 18.182 ¢ 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, vEbbbb | ^D, vvEb | |
| 2 | 36.3636 | D#, Ebbbb | ^^D, vEb | 50/49, 56/55 |
| 3 | 54.5455 | ^D#, vEbbb | ^3D, Eb | 33/32 |
| 4 | 72.7273 | Dx, Ebbb | ^4D, v8E | 26/25, 80/77 |
| 5 | 90.9091 | ^Dx, vEbb | ^5D, v7E | 21/20, 55/52 |
| 6 | 109.091 | D#x, Ebb | ^6D, v6E | 16/15, 52/49 |
| 7 | 127.273 | ^D#x, vEb | ^7D, v5E | 14/13 |
| 8 | 145.455 | Dxx, Eb | ^8D, v4E | |
| 9 | 163.636 | ^Dxx, vE | D#, v3E | 11/10 |
| 10 | 181.818 | E | ^D#, vvE | 49/44 |
| 11 | 200 | ^E, vFbbb | ^^D#, vE | 28/25, 55/49 |
| 12 | 218.182 | E#, Fbbb | E | 25/22 |
| 13 | 236.364 | ^E#, vFbb | ^E, vvF | 8/7 |
| 14 | 254.545 | Ex, Fbb | ^^E, vF | 15/13, 65/56 |
| 15 | 272.727 | ^Ex, vFb | F | 75/64 |
| 16 | 290.909 | E#x, Fb | ^F, vvGb | 13/11, 33/28, 77/65 |
| 17 | 309.091 | ^E#x, vF | ^^F, vGb | |
| 18 | 327.273 | F | ^3F, Gb | 40/33 |
| 19 | 345.455 | ^F, vGbbbb | ^4F, v8G | 39/32, 49/40 |
| 20 | 363.636 | F#, Gbbbb | ^5F, v7G | 16/13, 26/21 |
| 21 | 381.818 | ^F#, vGbbb | ^6F, v6G | 5/4 |
| 22 | 400 | Fx, Gbbb | ^7F, v5G | 44/35 |
| 23 | 418.182 | ^Fx, vGbb | ^8F, v4G | 14/11, 33/26 |
| 24 | 436.364 | F#x, Gbb | F#, v3G | |
| 25 | 454.545 | ^F#x, vGb | ^F#, vvG | 13/10 |
| 26 | 472.727 | Fxx, Gb | ^^F#, vG | 21/16 |
| 27 | 490.909 | ^Fxx, vG | G | 4/3, 65/49 |
| 28 | 509.091 | G | ^G, vvAb | 35/26, 75/56 |
| 29 | 527.273 | ^G, vAbbbb | ^^G, vAb | |
| 30 | 545.455 | G#, Abbbb | ^3G, Ab | 11/8 |
| 31 | 563.636 | ^G#, vAbbb | ^4G, v8A | |
| 32 | 581.818 | Gx, Abbb | ^5G, v7A | 7/5 |
| 33 | 600 | ^Gx, vAbb | ^6G, v6A | |
| 34 | 618.182 | G#x, Abb | ^7G, v5A | 10/7 |
| 35 | 636.364 | ^G#x, vAb | ^8G, v4A | 75/52 |
| 36 | 654.545 | Gxx, Ab | G#, v3A | 16/11 |
| 37 | 672.727 | ^Gxx, vA | ^G#, vvA | 65/44, 77/52 |
| 38 | 690.909 | A | ^^G#, vA | 52/35 |
| 39 | 709.091 | ^A, vBbbbb | A | 3/2 |
| 40 | 727.273 | A#, Bbbbb | ^A, vvBb | 32/21 |
| 41 | 745.455 | ^A#, vBbbb | ^^A, vBb | 20/13, 77/50 |
| 42 | 763.636 | Ax, Bbbb | ^3A, Bb | |
| 43 | 781.818 | ^Ax, vBbb | ^4A, v8B | 11/7, 52/33 |
| 44 | 800 | A#x, Bbb | ^5A, v7B | 35/22 |
| 45 | 818.182 | ^A#x, vBb | ^6A, v6B | 8/5 |
| 46 | 836.364 | Axx, Bb | ^7A, v5B | 13/8, 21/13 |
| 47 | 854.545 | ^Axx, vB | ^8A, v4B | 64/39, 80/49 |
| 48 | 872.727 | B | A#, v3B | 33/20 |
| 49 | 890.909 | ^B, vCbbb | ^A#, vvB | |
| 50 | 909.091 | B#, Cbbb | ^^A#, vB | 22/13, 56/33 |
| 51 | 927.273 | ^B#, vCbb | B | 75/44 |
| 52 | 945.455 | Bx, Cbb | ^B, vvC | 26/15 |
| 53 | 963.636 | ^Bx, vCb | ^^B, vC | 7/4 |
| 54 | 981.818 | B#x, Cb | C | 44/25 |
| 55 | 1000 | ^B#x, vC | ^C, vvDb | 25/14 |
| 56 | 1018.18 | C | ^^C, vDb | |
| 57 | 1036.36 | ^C, vDbbbb | ^3C, Db | 20/11 |
| 58 | 1054.55 | C#, Dbbbb | ^4C, v8D | |
| 59 | 1072.73 | ^C#, vDbbb | ^5C, v7D | 13/7 |
| 60 | 1090.91 | Cx, Dbbb | ^6C, v6D | 15/8, 49/26 |
| 61 | 1109.09 | ^Cx, vDbb | ^7C, v5D | 40/21 |
| 62 | 1127.27 | C#x, Dbb | ^8C, v4D | 25/13, 77/40 |
| 63 | 1145.45 | ^C#x, vDb | C#, v3D | 64/33 |
| 64 | 1163.64 | Cxx, Db | ^C#, vvD | 49/25, 55/28 |
| 65 | 1181.82 | ^Cxx, vD | ^^C#, vD | |
| 66 | 1200 | D | D | 2/1 |