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 10.000 ¢ 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

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
2 20 ^^D, v8Eb 78/77
3 30 ^3D, v7Eb 56/55, 64/63, 65/64
4 40 ^4D, v6Eb 45/44
5 50 ^5D, v5Eb 33/32
6 60 ^6D, v4Eb
7 70 ^7D, v3Eb 80/77
8 80 ^8D, vvEb 22/21
9 90 ^9D, vEb
10 100 D#, Eb 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, v9Gb
32 320 ^^F, v8Gb 77/64
33 330 ^3F, v7Gb 63/52
34 340 ^4F, v6Gb 39/32
35 350 ^5F, v5Gb 11/9, 49/40, 60/49
36 360 ^6F, v4Gb 16/13
37 370 ^7F, v3Gb 26/21
38 380 ^8F, vvGb 56/45
39 390 ^9F, vGb 5/4
40 400 F#, Gb
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, v9Ab
52 520 ^^G, v8Ab
53 530 ^3G, v7Ab
54 540 ^4G, v6Ab 15/11
55 550 ^5G, v5Ab 11/8
56 560 ^6G, v4Ab 18/13
57 570 ^7G, v3Ab 39/28
58 580 ^8G, vvAb 7/5
59 590 ^9G, vAb 45/32
60 600 G#, Ab
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, v9Bb
72 720 ^^A, v8Bb
73 730 ^3A, v7Bb 32/21
74 740 ^4A, v6Bb 49/32, 75/49
75 750 ^5A, v5Bb
76 760 ^6A, v4Bb 65/42
77 770 ^7A, v3Bb
78 780 ^8A, vvBb 11/7
79 790 ^9A, vBb 52/33
80 800 A#, Bb
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, v9Db
102 1020 ^^C, v8Db
103 1030 ^3C, v7Db
104 1040 ^4C, v6Db
105 1050 ^5C, v5Db 11/6
106 1060 ^6C, v4Db 24/13
107 1070 ^7C, v3Db 13/7
108 1080 ^8C, vvDb 28/15
109 1090 ^9C, vDb 15/8
110 1100 C#, Db 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