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

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 (%)	-45.7	-46.7	-35.9	+8.6	+8.3	-23.3	+7.6	+35.3	+15.8	+18.4	+20.9
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	Approximate ratios	Ups and downs notation (Dual flat fifth 44\76)	Ups and downs notation (Dual sharp fifth 45\76)
0	0	1/1	D	D
1	15.8		^D, ^E♭♭♭	^D, vvE♭
2	31.6		^^D, vvE♭♭	^^D, vE♭
3	47.4	36/35, 38/37	vD♯, vE♭♭	^3D, E♭
4	63.2	29/28	D♯, E♭♭	^4D, ^E♭
5	78.9	23/22	^D♯, ^E♭♭	^5D, ^^E♭
6	94.7		^^D♯, vvE♭	v5D♯, ^3E♭
7	110.5		vD𝄪, vE♭	v4D♯, ^4E♭
8	126.3	14/13	D𝄪, E♭	v3D♯, ^5E♭
9	142.1		^D𝄪, ^E♭	vvD♯, v5E
10	157.9	34/31	^^D𝄪, vvE	vD♯, v4E
11	173.7	32/29	vD♯𝄪, vE	D♯, v3E
12	189.5	19/17, 29/26, 39/35	E	^D♯, vvE
13	205.3		^E, ^F♭♭	^^D♯, vE
14	221.1	33/29	^^E, vvF♭	E
15	236.8		vE♯, vF♭	^E, vvF
16	252.6	22/19, 37/32	E♯, F♭	^^E, vF
17	268.4	7/6	^E♯, ^F♭	F
18	284.2	33/28	^^E♯, vvF	^F, vvG♭
19	300	19/16, 25/21	vE𝄪, vF	^^F, vG♭
20	315.8	6/5	F	^3F, G♭
21	331.6	23/19	^F, ^G♭♭♭	^4F, ^G♭
22	347.4		^^F, vvG♭♭	^5F, ^^G♭
23	363.2		vF♯, vG♭♭	v5F♯, ^3G♭
24	378.9		F♯, G♭♭	v4F♯, ^4G♭
25	394.7		^F♯, ^G♭♭	v3F♯, ^5G♭
26	410.5	33/26	^^F♯, vvG♭	vvF♯, v5G
27	426.3		vF𝄪, vG♭	vF♯, v4G
28	442.1		F𝄪, G♭	F♯, v3G
29	457.9		^F𝄪, ^G♭	^F♯, vvG
30	473.7		^^F𝄪, vvG	^^F♯, vG
31	489.5		vF♯𝄪, vG	G
32	505.3		G	^G, vvA♭
33	521.1	23/17	^G, ^A♭♭♭	^^G, vA♭
34	536.8		^^G, vvA♭♭	^3G, A♭
35	552.6	11/8	vG♯, vA♭♭	^4G, ^A♭
36	568.4	25/18, 32/23	G♯, A♭♭	^5G, ^^A♭
37	584.2	7/5	^G♯, ^A♭♭	v5G♯, ^3A♭
38	600		^^G♯, vvA♭	v4G♯, ^4A♭
39	615.8	10/7	vG𝄪, vA♭	v3G♯, ^5A♭
40	631.6	23/16, 36/25	G𝄪, A♭	vvG♯, v5A
41	647.4	16/11	^G𝄪, ^A♭	vG♯, v4A
42	663.2		^^G𝄪, vvA	G♯, v3A
43	678.9	34/23	vG♯𝄪, vA	^G♯, vvA
44	694.7		A	^^G♯, vA
45	710.5		^A, ^B♭♭♭	A
46	726.3		^^A, vvB♭♭	^A, vvB♭
47	742.1		vA♯, vB♭♭	^^A, vB♭
48	757.9		A♯, B♭♭	^3A, B♭
49	773.7		^A♯, ^B♭♭	^4A, ^B♭
50	789.5		^^A♯, vvB♭	^5A, ^^B♭
51	805.3		vA𝄪, vB♭	v5A♯, ^3B♭
52	821.1	37/23	A𝄪, B♭	v4A♯, ^4B♭
53	836.8		^A𝄪, ^B♭	v3A♯, ^5B♭
54	852.6		^^A𝄪, vvB	vvA♯, v5B
55	868.4	33/20, 38/23	vA♯𝄪, vB	vA♯, v4B
56	884.2	5/3	B	A♯, v3B
57	900	32/19, 37/22	^B, ^C♭♭	^A♯, vvB
58	915.8		^^B, vvC♭	^^A♯, vB
59	931.6	12/7	vB♯, vC♭	B
60	947.4	19/11	B♯, C♭	^B, vvC
61	963.2		^B♯, ^C♭	^^B, vC
62	978.9		^^B♯, vvC	C
63	994.7		vB𝄪, vC	^C, vvD♭
64	1010.5	34/19	C	^^C, vD♭
65	1026.3	29/16	^C, ^D♭♭♭	^3C, D♭
66	1042.1	31/17	^^C, vvD♭♭	^4C, ^D♭
67	1057.9		vC♯, vD♭♭	^5C, ^^D♭
68	1073.7	13/7	C♯, D♭♭	v5C♯, ^3D♭
69	1089.5		^C♯, ^D♭♭	v4C♯, ^4D♭
70	1105.3		^^C♯, vvD♭	v3C♯, ^5D♭
71	1121.1		vC𝄪, vD♭	vvC♯, v5D
72	1136.8		C𝄪, D♭	vC♯, v4D
73	1152.6	35/18, 37/19	^C𝄪, ^D♭	C♯, v3D
74	1168.4		^^C𝄪, vvD	^C♯, vvD
75	1184.2		vC♯𝄪, vD	^^C♯, vD
76	1200	2/1	D	D