76edo

← 75edo

76edo

77edo →

Prime factorization	22 × 19
Step size	15.7895¢
Fifth	44\76 (694.737¢) (→11\19)
Semitones (A1:m2)	4:8 (63.16¢ : 126.3¢)
Dual sharp fifth	45\76 (710.526¢)
Dual flat fifth	44\76 (694.737¢) (→11\19)
Dual major 2nd	13\76 (205.263¢)
Consistency limit	7
Distinct consistency limit	7

76 equal divisions of the octave (abbreviated 76edo or 76ed2), also called 76-tone equal temperament (76tet) or 76 equal temperament (76et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 76 equal parts of about 15.8 ¢ each. Each step represents a frequency ratio of 2^1/76, or the 76th root of 2.

Theory

Approximation of odd harmonics in 76edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	absolute (¢)	-7.22	-7.37	-5.67	+1.35	+1.31	-3.69	+1.20	+5.57	+2.49	+2.90	+3.30
	relative (%)	-46	-47	-36	+9	+8	-23	+8	+35	+16	+18	+21
Steps (reduced)		120 (44)	176 (24)	213 (61)	241 (13)	263 (35)	281 (53)	297 (69)	311 (7)	323 (19)	334 (30)	344 (40)

This tuning's 5-limit patent val is contorted in the 5-limit, reflecting the fact that 76 = 4 * 19. In the 7-limit it tempers out 2401/2400 as well as 81/80, and so supports squares temperament. In the 11-limit, it tempers out 245/242 and 385/384, and supports the 24&26 temperament. In the 13-limit, it tempers out 105/104, 144/143, 351/350 and 364/363. While the 44\76 = 11\19 fifth is already flat, the 43\76 fifth, even flatter, is an almost perfect approximation to the hornbostel temperament POTE fifth, whereas its sharp fifth, 45\76, makes for an excellent superpyth fifth. Hence you can do hornbostel/mavila, squares/meantone, and superpyth all with the same equal division.

Using non-patent vals, 76edo provides an excellent tuning for teff temperament, a low complexity, medium accuracy, and high limit (17 or 19) temperament.

Intervals

Steps	Cents	Ups and downs notation (dual flat fifth 44\76)	Ups and downs notation (dual sharp fifth 45\76)	Approximate ratios
0	0	D	D	1/1
1	15.7895	↑D, ↓3E♭♭	↑D, ↓↓E♭
2	31.5789	↑↑D, ↓↓E♭♭	↑↑D, ↓E♭	49/48, 50/49, 56/55, 66/65
3	47.3684	↑3D, ↓E♭♭	↑3D, E♭	33/32, 36/35, 40/39
4	63.1579	D♯, E♭♭	↑4D, ↓10E	80/77
5	78.9474	↑D♯, ↓3E♭	↑5D, ↓9E	21/20
6	94.7368	↑↑D♯, ↓↓E♭	↑6D, ↓8E	55/52
7	110.526	↑3D♯, ↓E♭	↑7D, ↓7E
8	126.316	D𝄪, E♭	↑8D, ↓6E	14/13
9	142.105	↑D𝄪, ↓3E	↑9D, ↓5E	13/12
10	157.895	↑↑D𝄪, ↓↓E	↑10D, ↓4E
11	173.684	↑3D𝄪, ↓E	D♯, ↓3E	72/65
12	189.474	E	↑D♯, ↓↓E	39/35
13	205.263	↑E, ↓3F♭	↑↑D♯, ↓E	55/49
14	221.053	↑↑E, ↓↓F♭	E
15	236.842	↑3E, ↓F♭	↑E, ↓↓F	8/7, 55/48
16	252.632	E♯, F♭	↑↑E, ↓F	65/56
17	268.421	↑E♯, ↓3F	F	7/6, 64/55
18	284.211	↑↑E♯, ↓↓F	↑F, ↓↓G♭	13/11, 33/28
19	300	↑3E♯, ↓F	↑↑F, ↓G♭	25/21
20	315.789	F	↑3F, G♭	6/5, 77/64
21	331.579	↑F, ↓3G♭♭	↑4F, ↓10G	40/33
22	347.368	↑↑F, ↓↓G♭♭	↑5F, ↓9G	49/40, 60/49
23	363.158	↑3F, ↓G♭♭	↑6F, ↓8G	16/13
24	378.947	F♯, G♭♭	↑7F, ↓7G
25	394.737	↑F♯, ↓3G♭	↑8F, ↓6G	49/39, 63/50
26	410.526	↑↑F♯, ↓↓G♭	↑9F, ↓5G	33/26
27	426.316	↑3F♯, ↓G♭	↑10F, ↓4G	50/39
28	442.105	F𝄪, G♭	F♯, ↓3G
29	457.895	↑F𝄪, ↓3G	↑F♯, ↓↓G	13/10
30	473.684	↑↑F𝄪, ↓↓G	↑↑F♯, ↓G
31	489.474	↑3F𝄪, ↓G	G	65/49
32	505.263	G	↑G, ↓↓A♭
33	521.053	↑G, ↓3A♭♭	↑↑G, ↓A♭	65/48, 66/49
34	536.842	↑↑G, ↓↓A♭♭	↑3G, A♭	49/36
35	552.632	↑3G, ↓A♭♭	↑4G, ↓10A	11/8, 48/35
36	568.421	G♯, A♭♭	↑5G, ↓9A	25/18, 39/28
37	584.211	↑G♯, ↓3A♭	↑6G, ↓8A	7/5
38	600	↑↑G♯, ↓↓A♭	↑7G, ↓7A	55/39, 78/55
39	615.789	↑3G♯, ↓A♭	↑8G, ↓6A	10/7
40	631.579	G𝄪, A♭	↑9G, ↓5A	36/25, 56/39
41	647.368	↑G𝄪, ↓3A	↑10G, ↓4A	16/11, 35/24
42	663.158	↑↑G𝄪, ↓↓A	G♯, ↓3A	72/49
43	678.947	↑3G𝄪, ↓A	↑G♯, ↓↓A	49/33, 77/52
44	694.737	A	↑↑G♯, ↓A
45	710.526	↑A, ↓3B♭♭	A
46	726.316	↑↑A, ↓↓B♭♭	↑A, ↓↓B♭
47	742.105	↑3A, ↓B♭♭	↑↑A, ↓B♭	20/13
48	757.895	A♯, B♭♭	↑3A, B♭	65/42
49	773.684	↑A♯, ↓3B♭	↑4A, ↓10B	39/25
50	789.474	↑↑A♯, ↓↓B♭	↑5A, ↓9B	52/33
51	805.263	↑3A♯, ↓B♭	↑6A, ↓8B	78/49
52	821.053	A𝄪, B♭	↑7A, ↓7B	77/48
53	836.842	↑A𝄪, ↓3B	↑8A, ↓6B	13/8
54	852.632	↑↑A𝄪, ↓↓B	↑9A, ↓5B	49/30, 80/49
55	868.421	↑3A𝄪, ↓B	↑10A, ↓4B	33/20
56	884.211	B	A♯, ↓3B	5/3
57	900	↑B, ↓3C♭	↑A♯, ↓↓B	42/25
58	915.789	↑↑B, ↓↓C♭	↑↑A♯, ↓B	22/13, 56/33
59	931.579	↑3B, ↓C♭	B	12/7, 55/32
60	947.368	B♯, C♭	↑B, ↓↓C
61	963.158	↑B♯, ↓3C	↑↑B, ↓C	7/4
62	978.947	↑↑B♯, ↓↓C	C
63	994.737	↑3B♯, ↓C	↑C, ↓↓D♭
64	1010.53	C	↑↑C, ↓D♭	70/39
65	1026.32	↑C, ↓3D♭♭	↑3C, D♭	65/36
66	1042.11	↑↑C, ↓↓D♭♭	↑4C, ↓10D
67	1057.89	↑3C, ↓D♭♭	↑5C, ↓9D	24/13
68	1073.68	C♯, D♭♭	↑6C, ↓8D	13/7
69	1089.47	↑C♯, ↓3D♭	↑7C, ↓7D
70	1105.26	↑↑C♯, ↓↓D♭	↑8C, ↓6D
71	1121.05	↑3C♯, ↓D♭	↑9C, ↓5D	40/21
72	1136.84	C𝄪, D♭	↑10C, ↓4D	77/40
73	1152.63	↑C𝄪, ↓3D	C♯, ↓3D	35/18, 39/20, 64/33
74	1168.42	↑↑C𝄪, ↓↓D	↑C♯, ↓↓D	49/25, 55/28, 65/33
75	1184.21	↑3C𝄪, ↓D	↑↑C♯, ↓D
76	1200	D	D	2/1