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 (120edo), or 120-tone equal temperament (120tet), 120 equal temperament (120et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 120 equal parts of exactly 10 ¢ each.

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. 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

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 -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, v9Eb 81/80
2 20 ^^D, v8Eb 66/65, 78/77
3 30 ^3D, v7Eb 56/55, 64/63, 65/64
4 40 ^4D, v6Eb 36/35, 45/44, 49/48, 50/49, 55/54, 77/75
5 50 ^5D, v5Eb 33/32, 40/39
6 60 ^6D, v4Eb 26/25, 27/26, 65/63
7 70 ^7D, v3Eb 28/27, 80/77
8 80 ^8D, vvEb 21/20, 22/21, 25/24, 81/77
9 90 ^9D, vEb
10 100 D#, Eb 52/49, 55/52
11 110 ^D#, v9E 16/15, 35/33
12 120 ^^D#, v8E 15/14, 27/25, 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 11/10, 35/32, 54/49
17 170 ^7D#, v3E 72/65
18 180 ^8D#, vvE 39/35
19 190 ^9D#, vE 10/9, 28/25, 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, 25/22, 63/55
24 240 ^4E, v6F 55/48, 81/70
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 33/28, 75/64
29 290 ^9E, vF 13/11, 77/65
30 300 F 32/27
31 310 ^F, v9Gb 6/5, 25/21
32 320 ^^F, v8Gb 77/64
33 330 ^3F, v7Gb 63/52, 65/54
34 340 ^4F, v6Gb 39/32, 40/33
35 350 ^5F, v5Gb 11/9, 27/22, 49/40, 60/49
36 360 ^6F, v4Gb 16/13
37 370 ^7F, v3Gb 26/21, 81/65
38 380 ^8F, vvGb 56/45
39 390 ^9F, vGb 5/4, 44/35, 63/50
40 400 F#, Gb 49/39, 81/64
41 410 ^F#, v9G 33/26
42 420 ^^F#, v8G 14/11, 32/25, 80/63
43 430 ^3F#, v7G 9/7, 77/60
44 440 ^4F#, v6G 50/39
45 450 ^5F#, v5G 13/10
46 460 ^6F#, v4G 35/27, 64/49, 72/55
47 470 ^7F#, v3G 21/16, 33/25, 55/42
48 480 ^8F#, vvG
49 490 ^9F#, vG 65/49
50 500 G 4/3
51 510 ^G, v9Ab 27/20, 66/49, 75/56
52 520 ^^G, v8Ab 35/26
53 530 ^3G, v7Ab 65/48
54 540 ^4G, v6Ab 15/11, 48/35, 49/36
55 550 ^5G, v5Ab 11/8
56 560 ^6G, v4Ab 18/13
57 570 ^7G, v3Ab 39/28
58 580 ^8G, vvAb 7/5, 25/18
59 590 ^9G, vAb 45/32
60 600 G#, Ab 55/39, 78/55
61 610 ^G#, v9A 64/45
62 620 ^^G#, v8A 10/7, 36/25, 63/44, 77/54
63 630 ^3G#, v7A 56/39, 81/56
64 640 ^4G#, v6A 13/9, 75/52
65 650 ^5G#, v5A 16/11
66 660 ^6G#, v4A 22/15, 35/24, 72/49, 81/55
67 670 ^7G#, v3A
68 680 ^8G#, vvA 52/35, 65/44, 77/52
69 690 ^9G#, vA 40/27, 49/33
70 700 A 3/2
71 710 ^A, v9Bb
72 720 ^^A, v8Bb
73 730 ^3A, v7Bb 32/21, 50/33
74 740 ^4A, v6Bb 49/32, 54/35, 55/36, 75/49, 77/50
75 750 ^5A, v5Bb 20/13
76 760 ^6A, v4Bb 39/25, 65/42, 81/52
77 770 ^7A, v3Bb 14/9
78 780 ^8A, vvBb 11/7, 25/16, 63/40
79 790 ^9A, vBb 52/33
80 800 A#, Bb 78/49
81 810 ^A#, v9B 8/5, 35/22
82 820 ^^A#, v8B 45/28, 77/48, 81/50
83 830 ^3A#, v7B 21/13
84 840 ^4A#, v6B 13/8
85 850 ^5A#, v5B 18/11, 44/27, 49/30, 80/49
86 860 ^6A#, v4B 33/20, 64/39, 81/49
87 870 ^7A#, v3B
88 880 ^8A#, vvB
89 890 ^9A#, vB 5/3, 42/25
90 900 B 27/16
91 910 ^B, v9C 22/13
92 920 ^^B, v8C 56/33
93 930 ^3B, v7C 12/7, 75/44, 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, 44/25
98 980 ^8B, vvC
99 990 ^9B, vC 39/22
100 1000 C 16/9
101 1010 ^C, v9Db 9/5, 25/14
102 1020 ^^C, v8Db 70/39
103 1030 ^3C, v7Db 65/36
104 1040 ^4C, v6Db 20/11, 49/27, 64/35
105 1050 ^5C, v5Db 11/6, 81/44
106 1060 ^6C, v4Db 24/13
107 1070 ^7C, v3Db 13/7
108 1080 ^8C, vvDb 28/15, 50/27
109 1090 ^9C, vDb 15/8, 66/35
110 1100 C#, Db 49/26
111 1110 ^C#, v9D
112 1120 ^^C#, v8D 21/11, 40/21, 48/25
113 1130 ^3C#, v7D 27/14, 77/40
114 1140 ^4C#, v6D 25/13, 52/27
115 1150 ^5C#, v5D 39/20, 64/33
116 1160 ^6C#, v4D 35/18, 49/25
117 1170 ^7C#, v3D 55/28, 63/32
118 1180 ^8C#, vvD 65/33, 77/39
119 1190 ^9C#, vD
120 1200 D 2/1