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 shares the perfect fifth with 12edo, tempering out the Pythagorean comma. 120edo is an excellent tuning in the 2.3.7.11.13.23.29 subgroup. In the no-5's 11-limit, it tempers out 243/242. In the patent val 120edo is also a tuning for the 7-limit decoid temperament.
The 120bdd val is a tuning for superpyth where 3/2 is tuned to exactly 710 cents. It may be used as a de facto dual fifth in newcome temperament.
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 (%) | +0.0 | -19.6 | +36.9 | +11.7 | -13.2 | -5.3 | -49.6 | +24.9 | +17.3 | +4.2 | +49.6 | |
Steps (reduced) |
120 (0) |
190 (70) |
279 (39) |
337 (97) |
415 (55) |
444 (84) |
490 (10) |
510 (30) |
543 (63) |
583 (103) |
595 (115) |
Subsets and supersets
120edo is the 10th highly composite edo and the 5th factorial edo (120 = 5! = 1 × 2 × 3 × 4 × 5). It has many subsets: 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 30, 40, and 60.
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.
JI approximation
The following tables show how 15-odd-limit intervals are represented in 120edo. Prime harmonics are in bold; inconsistent intervals are in italics.
Interval and complement | Error (abs, ¢) | Error (rel, %) |
---|---|---|
1/1, 2/1 | 0.000 | 0.0 |
13/8, 16/13 | 0.528 | 5.3 |
15/14, 28/15 | 0.557 | 5.6 |
11/6, 12/11 | 0.637 | 6.4 |
13/11, 22/13 | 0.790 | 7.9 |
7/4, 8/7 | 1.174 | 11.7 |
11/8, 16/11 | 1.318 | 13.2 |
13/12, 24/13 | 1.427 | 14.3 |
13/7, 14/13 | 1.702 | 17.0 |
15/8, 16/15 | 1.731 | 17.3 |
3/2, 4/3 | 1.955 | 19.6 |
15/13, 26/15 | 2.259 | 22.6 |
9/5, 10/9 | 2.404 | 24.0 |
11/7, 14/11 | 2.492 | 24.9 |
7/5, 10/7 | 2.512 | 25.1 |
11/9, 18/11 | 2.592 | 25.9 |
15/11, 22/15 | 3.049 | 30.5 |
7/6, 12/7 | 3.129 | 31.3 |
13/9, 18/13 | 3.382 | 33.8 |
5/4, 8/5 | 3.686 | 36.9 |
9/8, 16/9 | 3.910 | 39.1 |
13/10, 20/13 | 4.214 | 42.1 |
5/3, 6/5 | 4.359 | 43.6 |
9/7, 14/9 | 4.916 | 49.2 |
11/10, 20/11 | 4.996 | 50.0 |
Interval and complement | Error (abs, ¢) | Error (rel, %) |
---|---|---|
1/1, 2/1 | 0.000 | 0.0 |
13/8, 16/13 | 0.528 | 5.3 |
15/14, 28/15 | 0.557 | 5.6 |
11/6, 12/11 | 0.637 | 6.4 |
13/11, 22/13 | 0.790 | 7.9 |
7/4, 8/7 | 1.174 | 11.7 |
11/8, 16/11 | 1.318 | 13.2 |
13/12, 24/13 | 1.427 | 14.3 |
13/7, 14/13 | 1.702 | 17.0 |
15/8, 16/15 | 1.731 | 17.3 |
3/2, 4/3 | 1.955 | 19.6 |
15/13, 26/15 | 2.259 | 22.6 |
11/7, 14/11 | 2.492 | 24.9 |
7/5, 10/7 | 2.512 | 25.1 |
11/9, 18/11 | 2.592 | 25.9 |
15/11, 22/15 | 3.049 | 30.5 |
7/6, 12/7 | 3.129 | 31.3 |
13/9, 18/13 | 3.382 | 33.8 |
5/4, 8/5 | 3.686 | 36.9 |
9/8, 16/9 | 3.910 | 39.1 |
13/10, 20/13 | 4.214 | 42.1 |
11/10, 20/11 | 5.004 | 50.0 |
9/7, 14/9 | 5.084 | 50.8 |
5/3, 6/5 | 5.641 | 56.4 |
9/5, 10/9 | 7.596 | 76.0 |
Intervals
Steps | Cents | Approximate Ratios | Ups and Downs Notation |
---|---|---|---|
0 | 0 | 1/1 | D |
1 | 10 | ^D, v9E♭ | |
2 | 20 | 78/77 | ^^D, v8E♭ |
3 | 30 | 56/55, 64/63, 65/64 | ^3D, v7E♭ |
4 | 40 | 45/44 | ^4D, v6E♭ |
5 | 50 | 33/32 | ^5D, v5E♭ |
6 | 60 | ^6D, v4E♭ | |
7 | 70 | 80/77 | ^7D, v3E♭ |
8 | 80 | 22/21 | ^8D, vvE♭ |
9 | 90 | ^9D, vE♭ | |
10 | 100 | 52/49, 55/52 | D♯, E♭ |
11 | 110 | 16/15 | ^D♯, v9E |
12 | 120 | 15/14, 77/72 | ^^D♯, v8E |
13 | 130 | 14/13 | ^3D♯, v7E |
14 | 140 | 13/12 | ^4D♯, v6E |
15 | 150 | 12/11, 49/45 | ^5D♯, v5E |
16 | 160 | ^6D♯, v4E | |
17 | 170 | ^7D♯, v3E | |
18 | 180 | ^8D♯, vvE | |
19 | 190 | 49/44 | ^9D♯, vE |
20 | 200 | 9/8, 55/49 | E |
21 | 210 | 44/39 | ^E, v9F |
22 | 220 | ^^E, v8F | |
23 | 230 | 8/7 | ^3E, v7F |
24 | 240 | ^4E, v6F | |
25 | 250 | 15/13, 52/45 | ^5E, v5F |
26 | 260 | 64/55, 65/56 | ^6E, v4F |
27 | 270 | 7/6 | ^7E, v3F |
28 | 280 | ^8E, vvF | |
29 | 290 | 13/11, 77/65 | ^9E, vF |
30 | 300 | F | |
31 | 310 | ^F, v9G♭ | |
32 | 320 | 77/64 | ^^F, v8G♭ |
33 | 330 | 63/52 | ^3F, v7G♭ |
34 | 340 | 39/32 | ^4F, v6G♭ |
35 | 350 | 11/9, 49/40, 60/49 | ^5F, v5G♭ |
36 | 360 | 16/13 | ^6F, v4G♭ |
37 | 370 | 26/21 | ^7F, v3G♭ |
38 | 380 | 56/45 | ^8F, vvG♭ |
39 | 390 | 5/4 | ^9F, vG♭ |
40 | 400 | F♯, G♭ | |
41 | 410 | 33/26 | ^F♯, v9G |
42 | 420 | 14/11 | ^^F♯, v8G |
43 | 430 | 77/60 | ^3F♯, v7G |
44 | 440 | ^4F♯, v6G | |
45 | 450 | ^5F♯, v5G | |
46 | 460 | 64/49 | ^6F♯, v4G |
47 | 470 | 21/16, 55/42 | ^7F♯, v3G |
48 | 480 | ^8F♯, vvG | |
49 | 490 | 65/49 | ^9F♯, vG |
50 | 500 | 4/3 | G |
51 | 510 | ^G, v9A♭ | |
52 | 520 | ^^G, v8A♭ | |
53 | 530 | ^3G, v7A♭ | |
54 | 540 | 15/11 | ^4G, v6A♭ |
55 | 550 | 11/8 | ^5G, v5A♭ |
56 | 560 | 18/13 | ^6G, v4A♭ |
57 | 570 | 39/28 | ^7G, v3A♭ |
58 | 580 | 7/5 | ^8G, vvA♭ |
59 | 590 | 45/32 | ^9G, vA♭ |
60 | 600 | G♯, A♭ | |
61 | 610 | 64/45 | ^G♯, v9A |
62 | 620 | 10/7, 63/44 | ^^G♯, v8A |
63 | 630 | 56/39 | ^3G♯, v7A |
64 | 640 | 13/9 | ^4G♯, v6A |
65 | 650 | 16/11 | ^5G♯, v5A |
66 | 660 | 22/15 | ^6G♯, v4A |
67 | 670 | ^7G♯, v3A | |
68 | 680 | 77/52 | ^8G♯, vvA |
69 | 690 | ^9G♯, vA | |
70 | 700 | 3/2 | A |
71 | 710 | ^A, v9B♭ | |
72 | 720 | ^^A, v8B♭ | |
73 | 730 | 32/21 | ^3A, v7B♭ |
74 | 740 | 49/32, 75/49 | ^4A, v6B♭ |
75 | 750 | ^5A, v5B♭ | |
76 | 760 | 65/42 | ^6A, v4B♭ |
77 | 770 | ^7A, v3B♭ | |
78 | 780 | 11/7 | ^8A, vvB♭ |
79 | 790 | 52/33 | ^9A, vB♭ |
80 | 800 | A♯, B♭ | |
81 | 810 | 8/5 | ^A♯, v9B |
82 | 820 | 45/28, 77/48 | ^^A♯, v8B |
83 | 830 | 21/13 | ^3A♯, v7B |
84 | 840 | 13/8 | ^4A♯, v6B |
85 | 850 | 18/11, 49/30, 80/49 | ^5A♯, v5B |
86 | 860 | 64/39 | ^6A♯, v4B |
87 | 870 | ^7A♯, v3B | |
88 | 880 | ^8A♯, vvB | |
89 | 890 | ^9A♯, vB | |
90 | 900 | B | |
91 | 910 | 22/13 | ^B, v9C |
92 | 920 | ^^B, v8C | |
93 | 930 | 12/7, 77/45 | ^3B, v7C |
94 | 940 | 55/32 | ^4B, v6C |
95 | 950 | 26/15, 45/26 | ^5B, v5C |
96 | 960 | ^6B, v4C | |
97 | 970 | 7/4 | ^7B, v3C |
98 | 980 | ^8B, vvC | |
99 | 990 | 39/22 | ^9B, vC |
100 | 1000 | 16/9 | C |
101 | 1010 | ^C, v9D♭ | |
102 | 1020 | ^^C, v8D♭ | |
103 | 1030 | ^3C, v7D♭ | |
104 | 1040 | ^4C, v6D♭ | |
105 | 1050 | 11/6 | ^5C, v5D♭ |
106 | 1060 | 24/13 | ^6C, v4D♭ |
107 | 1070 | 13/7 | ^7C, v3D♭ |
108 | 1080 | 28/15 | ^8C, vvD♭ |
109 | 1090 | 15/8 | ^9C, vD♭ |
110 | 1100 | 49/26 | C♯, D♭ |
111 | 1110 | ^C♯, v9D | |
112 | 1120 | 21/11 | ^^C♯, v8D |
113 | 1130 | 77/40 | ^3C♯, v7D |
114 | 1140 | ^4C♯, v6D | |
115 | 1150 | 64/33 | ^5C♯, v5D |
116 | 1160 | ^6C♯, v4D | |
117 | 1170 | 55/28, 63/32 | ^7C♯, v3D |
118 | 1180 | 77/39 | ^8C♯, vvD |
119 | 1190 | ^9C♯, vD | |
120 | 1200 | 2/1 | D |