120edo
| ← 119edo | 120edo | 121edo → |
120 equal divisions of the octave (abbreviated 120edo or 120ed2), also called 120-tone equal temperament (120tet) or 120 equal temperament (120et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 120 equal parts of exactly 10 ¢ each. Each step represents a frequency ratio of 21/120, or the 120th root of 2.
Theory
120edo is the 10th highly composite EDO and the 5th factorial EDO (120 = 1*2*3*4*5 = 5!).
120edo is an excellent tuning in the 2.3.7.11.13.23.29 subgroup. In the no-5s 11-limit, it tempers out 243/242.
120edo shares the perfect fifth with 12edo, tempering out the Pythagorean comma. The sharp fifth of 710 cents also has regular temperament interpretations. It is used in the 120b val for tuning the 5-limit superpyth temperament where it represents 3/2, and in the 120g val as a tuning for the 19-limit surmarvelpyth temperament where it represents 675/448, which is marvel comma sharp of 3/2. It may be used as a de facto dual fifth in newcome temperament. In the patent val 120edo is also a tuning for the 7-limit decoid temperament.
The step size of this EDO is near the upper boundary of the just noticeable difference.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.00 | -1.96 | +3.69 | +1.17 | -1.32 | -0.53 | -4.96 | +2.49 | +1.73 | +0.42 | +4.96 |
| relative (%) | +0 | -20 | +37 | +12 | -13 | -5 | -50 | +25 | +17 | +4 | +50 | |
| Steps (reduced) |
120 (0) |
190 (70) |
279 (39) |
337 (97) |
415 (55) |
444 (84) |
490 (10) |
510 (30) |
543 (63) |
583 (103) |
595 (115) | |
Miscellaneous properties
Being the simplest division of the octave by the Germanic long hundred, it has a unit step which is the fine relative cent of 1edo.
120edo also has a concoctic generator that resembles the leap day excess of earth, 29\120 corresponding to 5 hours and 48 minutes.
Interval list
| Steps | Cents | Ups and downs notation | Approximate ratios |
|---|---|---|---|
| 0 | 0 | D | 1/1 |
| 1 | 10 | ^D, v9E♭ | |
| 2 | 20 | ^^D, v8E♭ | 78/77 |
| 3 | 30 | ^3D, v7E♭ | 56/55, 64/63, 65/64 |
| 4 | 40 | ^4D, v6E♭ | 45/44 |
| 5 | 50 | ^5D, v5E♭ | 33/32 |
| 6 | 60 | ^6D, v4E♭ | |
| 7 | 70 | ^7D, v3E♭ | 80/77 |
| 8 | 80 | ^8D, vvE♭ | 22/21 |
| 9 | 90 | ^9D, vE♭ | |
| 10 | 100 | D♯, E♭ | 52/49, 55/52 |
| 11 | 110 | ^D♯, v9E | 16/15 |
| 12 | 120 | ^^D♯, v8E | 15/14, 77/72 |
| 13 | 130 | ^3D♯, v7E | 14/13 |
| 14 | 140 | ^4D♯, v6E | 13/12 |
| 15 | 150 | ^5D♯, v5E | 12/11, 49/45 |
| 16 | 160 | ^6D♯, v4E | |
| 17 | 170 | ^7D♯, v3E | |
| 18 | 180 | ^8D♯, vvE | |
| 19 | 190 | ^9D♯, vE | 49/44 |
| 20 | 200 | E | 9/8, 55/49 |
| 21 | 210 | ^E, v9F | 44/39 |
| 22 | 220 | ^^E, v8F | |
| 23 | 230 | ^3E, v7F | 8/7 |
| 24 | 240 | ^4E, v6F | |
| 25 | 250 | ^5E, v5F | 15/13, 52/45 |
| 26 | 260 | ^6E, v4F | 64/55, 65/56 |
| 27 | 270 | ^7E, v3F | 7/6 |
| 28 | 280 | ^8E, vvF | |
| 29 | 290 | ^9E, vF | 13/11, 77/65 |
| 30 | 300 | F | |
| 31 | 310 | ^F, v9G♭ | |
| 32 | 320 | ^^F, v8G♭ | 77/64 |
| 33 | 330 | ^3F, v7G♭ | 63/52 |
| 34 | 340 | ^4F, v6G♭ | 39/32 |
| 35 | 350 | ^5F, v5G♭ | 11/9, 49/40, 60/49 |
| 36 | 360 | ^6F, v4G♭ | 16/13 |
| 37 | 370 | ^7F, v3G♭ | 26/21 |
| 38 | 380 | ^8F, vvG♭ | 56/45 |
| 39 | 390 | ^9F, vG♭ | 5/4 |
| 40 | 400 | F♯, G♭ | |
| 41 | 410 | ^F♯, v9G | 33/26 |
| 42 | 420 | ^^F♯, v8G | 14/11 |
| 43 | 430 | ^3F♯, v7G | 77/60 |
| 44 | 440 | ^4F♯, v6G | |
| 45 | 450 | ^5F♯, v5G | |
| 46 | 460 | ^6F♯, v4G | 64/49 |
| 47 | 470 | ^7F♯, v3G | 21/16, 55/42 |
| 48 | 480 | ^8F♯, vvG | |
| 49 | 490 | ^9F♯, vG | 65/49 |
| 50 | 500 | G | 4/3 |
| 51 | 510 | ^G, v9A♭ | |
| 52 | 520 | ^^G, v8A♭ | |
| 53 | 530 | ^3G, v7A♭ | |
| 54 | 540 | ^4G, v6A♭ | 15/11 |
| 55 | 550 | ^5G, v5A♭ | 11/8 |
| 56 | 560 | ^6G, v4A♭ | 18/13 |
| 57 | 570 | ^7G, v3A♭ | 39/28 |
| 58 | 580 | ^8G, vvA♭ | 7/5 |
| 59 | 590 | ^9G, vA♭ | 45/32 |
| 60 | 600 | G♯, A♭ | |
| 61 | 610 | ^G♯, v9A | 64/45 |
| 62 | 620 | ^^G♯, v8A | 10/7, 63/44 |
| 63 | 630 | ^3G♯, v7A | 56/39 |
| 64 | 640 | ^4G♯, v6A | 13/9 |
| 65 | 650 | ^5G♯, v5A | 16/11 |
| 66 | 660 | ^6G♯, v4A | 22/15 |
| 67 | 670 | ^7G♯, v3A | |
| 68 | 680 | ^8G♯, vvA | 77/52 |
| 69 | 690 | ^9G♯, vA | |
| 70 | 700 | A | 3/2 |
| 71 | 710 | ^A, v9B♭ | |
| 72 | 720 | ^^A, v8B♭ | |
| 73 | 730 | ^3A, v7B♭ | 32/21 |
| 74 | 740 | ^4A, v6B♭ | 49/32, 75/49 |
| 75 | 750 | ^5A, v5B♭ | |
| 76 | 760 | ^6A, v4B♭ | 65/42 |
| 77 | 770 | ^7A, v3B♭ | |
| 78 | 780 | ^8A, vvB♭ | 11/7 |
| 79 | 790 | ^9A, vB♭ | 52/33 |
| 80 | 800 | A♯, B♭ | |
| 81 | 810 | ^A♯, v9B | 8/5 |
| 82 | 820 | ^^A♯, v8B | 45/28, 77/48 |
| 83 | 830 | ^3A♯, v7B | 21/13 |
| 84 | 840 | ^4A♯, v6B | 13/8 |
| 85 | 850 | ^5A♯, v5B | 18/11, 49/30, 80/49 |
| 86 | 860 | ^6A♯, v4B | 64/39 |
| 87 | 870 | ^7A♯, v3B | |
| 88 | 880 | ^8A♯, vvB | |
| 89 | 890 | ^9A♯, vB | |
| 90 | 900 | B | |
| 91 | 910 | ^B, v9C | 22/13 |
| 92 | 920 | ^^B, v8C | |
| 93 | 930 | ^3B, v7C | 12/7, 77/45 |
| 94 | 940 | ^4B, v6C | 55/32 |
| 95 | 950 | ^5B, v5C | 26/15, 45/26 |
| 96 | 960 | ^6B, v4C | |
| 97 | 970 | ^7B, v3C | 7/4 |
| 98 | 980 | ^8B, vvC | |
| 99 | 990 | ^9B, vC | 39/22 |
| 100 | 1000 | C | 16/9 |
| 101 | 1010 | ^C, v9D♭ | |
| 102 | 1020 | ^^C, v8D♭ | |
| 103 | 1030 | ^3C, v7D♭ | |
| 104 | 1040 | ^4C, v6D♭ | |
| 105 | 1050 | ^5C, v5D♭ | 11/6 |
| 106 | 1060 | ^6C, v4D♭ | 24/13 |
| 107 | 1070 | ^7C, v3D♭ | 13/7 |
| 108 | 1080 | ^8C, vvD♭ | 28/15 |
| 109 | 1090 | ^9C, vD♭ | 15/8 |
| 110 | 1100 | C♯, D♭ | 49/26 |
| 111 | 1110 | ^C♯, v9D | |
| 112 | 1120 | ^^C♯, v8D | 21/11 |
| 113 | 1130 | ^3C♯, v7D | 77/40 |
| 114 | 1140 | ^4C♯, v6D | |
| 115 | 1150 | ^5C♯, v5D | 64/33 |
| 116 | 1160 | ^6C♯, v4D | |
| 117 | 1170 | ^7C♯, v3D | 55/28, 63/32 |
| 118 | 1180 | ^8C♯, vvD | 77/39 |
| 119 | 1190 | ^9C♯, vD | |
| 120 | 1200 | D | 2/1 |