120edo

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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	highly composite

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 2^1/120, 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, ↓9E♭
2	20	↑↑D, ↓8E♭	78/77
3	30	↑3D, ↓7E♭	56/55, 64/63, 65/64
4	40	↑4D, ↓6E♭	45/44
5	50	↑5D, ↓5E♭	33/32
6	60	↑6D, ↓4E♭
7	70	↑7D, ↓3E♭	80/77
8	80	↑8D, ↓↓E♭	22/21
9	90	↑9D, ↓E♭
10	100	D♯, E♭	52/49, 55/52
11	110	↑D♯, ↓9E	16/15
12	120	↑↑D♯, ↓8E	15/14, 77/72
13	130	↑3D♯, ↓7E	14/13
14	140	↑4D♯, ↓6E	13/12
15	150	↑5D♯, ↓5E	12/11, 49/45
16	160	↑6D♯, ↓4E
17	170	↑7D♯, ↓3E
18	180	↑8D♯, ↓↓E
19	190	↑9D♯, ↓E	49/44
20	200	E	9/8, 55/49
21	210	↑E, ↓9F	44/39
22	220	↑↑E, ↓8F
23	230	↑3E, ↓7F	8/7
24	240	↑4E, ↓6F
25	250	↑5E, ↓5F	15/13, 52/45
26	260	↑6E, ↓4F	64/55, 65/56
27	270	↑7E, ↓3F	7/6
28	280	↑8E, ↓↓F
29	290	↑9E, ↓F	13/11, 77/65
30	300	F
31	310	↑F, ↓9G♭
32	320	↑↑F, ↓8G♭	77/64
33	330	↑3F, ↓7G♭	63/52
34	340	↑4F, ↓6G♭	39/32
35	350	↑5F, ↓5G♭	11/9, 49/40, 60/49
36	360	↑6F, ↓4G♭	16/13
37	370	↑7F, ↓3G♭	26/21
38	380	↑8F, ↓↓G♭	56/45
39	390	↑9F, ↓G♭	5/4
40	400	F♯, G♭
41	410	↑F♯, ↓9G	33/26
42	420	↑↑F♯, ↓8G	14/11
43	430	↑3F♯, ↓7G	77/60
44	440	↑4F♯, ↓6G
45	450	↑5F♯, ↓5G
46	460	↑6F♯, ↓4G	64/49
47	470	↑7F♯, ↓3G	21/16, 55/42
48	480	↑8F♯, ↓↓G
49	490	↑9F♯, ↓G	65/49
50	500	G	4/3
51	510	↑G, ↓9A♭
52	520	↑↑G, ↓8A♭
53	530	↑3G, ↓7A♭
54	540	↑4G, ↓6A♭	15/11
55	550	↑5G, ↓5A♭	11/8
56	560	↑6G, ↓4A♭	18/13
57	570	↑7G, ↓3A♭	39/28
58	580	↑8G, ↓↓A♭	7/5
59	590	↑9G, ↓A♭	45/32
60	600	G♯, A♭
61	610	↑G♯, ↓9A	64/45
62	620	↑↑G♯, ↓8A	10/7, 63/44
63	630	↑3G♯, ↓7A	56/39
64	640	↑4G♯, ↓6A	13/9
65	650	↑5G♯, ↓5A	16/11
66	660	↑6G♯, ↓4A	22/15
67	670	↑7G♯, ↓3A
68	680	↑8G♯, ↓↓A	77/52
69	690	↑9G♯, ↓A
70	700	A	3/2
71	710	↑A, ↓9B♭
72	720	↑↑A, ↓8B♭
73	730	↑3A, ↓7B♭	32/21
74	740	↑4A, ↓6B♭	49/32, 75/49
75	750	↑5A, ↓5B♭
76	760	↑6A, ↓4B♭	65/42
77	770	↑7A, ↓3B♭
78	780	↑8A, ↓↓B♭	11/7
79	790	↑9A, ↓B♭	52/33
80	800	A♯, B♭
81	810	↑A♯, ↓9B	8/5
82	820	↑↑A♯, ↓8B	45/28, 77/48
83	830	↑3A♯, ↓7B	21/13
84	840	↑4A♯, ↓6B	13/8
85	850	↑5A♯, ↓5B	18/11, 49/30, 80/49
86	860	↑6A♯, ↓4B	64/39
87	870	↑7A♯, ↓3B
88	880	↑8A♯, ↓↓B
89	890	↑9A♯, ↓B
90	900	B
91	910	↑B, ↓9C	22/13
92	920	↑↑B, ↓8C
93	930	↑3B, ↓7C	12/7, 77/45
94	940	↑4B, ↓6C	55/32
95	950	↑5B, ↓5C	26/15, 45/26
96	960	↑6B, ↓4C
97	970	↑7B, ↓3C	7/4
98	980	↑8B, ↓↓C
99	990	↑9B, ↓C	39/22
100	1000	C	16/9
101	1010	↑C, ↓9D♭
102	1020	↑↑C, ↓8D♭
103	1030	↑3C, ↓7D♭
104	1040	↑4C, ↓6D♭
105	1050	↑5C, ↓5D♭	11/6
106	1060	↑6C, ↓4D♭	24/13
107	1070	↑7C, ↓3D♭	13/7
108	1080	↑8C, ↓↓D♭	28/15
109	1090	↑9C, ↓D♭	15/8
110	1100	C♯, D♭	49/26
111	1110	↑C♯, ↓9D
112	1120	↑↑C♯, ↓8D	21/11
113	1130	↑3C♯, ↓7D	77/40
114	1140	↑4C♯, ↓6D
115	1150	↑5C♯, ↓5D	64/33
116	1160	↑6C♯, ↓4D
117	1170	↑7C♯, ↓3D	55/28, 63/32
118	1180	↑8C♯, ↓↓D	77/39
119	1190	↑9C♯, ↓D
120	1200	D	2/1