120edo

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← 119edo120edo121edo →
Prime factorization 23 × 3 × 5
Step size 10¢ 
Fifth 70\120 (700¢) (→7\12)
Semitones (A1:m2) 10:10 (100¢ : 100¢)
Consistency limit 3
Distinct consistency limit 3
Special properties

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

Approximation of prime harmonics in 120edo
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.

15-odd-limit intervals in 120edo (direct approximation, even if inconsistent)
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
15-odd-limit intervals in 120edo (patent val mapping)
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
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