# 120edo

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 ← 119edo 120edo 121edo →
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