74edo

← 73edo

74edo

75edo →

Prime factorization	2 × 37
Step size	16.2162¢
Fifth	43\74 (697.297¢)
Semitones (A1:m2)	5:7 (81.08¢ : 113.5¢)
Consistency limit	5
Distinct consistency limit	5

74 equal divisions of the octave (abbreviated 74edo or 74ed2), also called 74-tone equal temperament (74tet) or 74 equal temperament (74et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 74 equal parts of about 16.2 ¢ each. Each step represents a frequency ratio of 2^1/74, or the 74th root of 2.

It is most notable as a meantone tuning, tempering out 81/80 in the 5-limit; 81/80 and 126/125 (and hence 225/224) in the 7-limit; 99/98, 176/175 and 441/440 in the 11-limit; and 144/143 and 847/845 in the 13-limit. Discarding 847/845 from that gives 13-limit meantone, aka 13-limit huygens, for which 74edo gives the optimal patent val; and discarding 144/143 gives a 13-limit 62&74 temperament with half-octave period and two parallel tracks of meantone.

Approximation of odd harmonics in 74edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	absolute (¢)	-4.66	+2.88	+4.15	+6.90	+0.03	+2.72	-1.78	-7.66	-5.62	-0.51	+4.16
	relative (%)	-29	+18	+26	+43	+0	+17	-11	-47	-35	-3	+26
Steps (reduced)		117 (43)	172 (24)	208 (60)	235 (13)	256 (34)	274 (52)	289 (67)	302 (6)	314 (18)	325 (29)	335 (39)

74 tunes 11 only 1/30 of a cent sharp, and 13 2.7 cents sharp, making it a distinctly interesting choice for higher-limit meantone.

Intervals

Steps	Cents	Ups and downs notation	Approximate ratios
0	0	D	1/1
1	16.2162	↑D, ↓E♭♭	78/77
2	32.4324	↑↑D, E♭♭	45/44, 50/49, 56/55, 64/63, 65/64
3	48.6486	↑3D, ↓4E♭	33/32, 40/39, 77/75
4	64.8649	↑4D, ↓3E♭	26/25, 80/77
5	81.0811	D♯, ↓↓E♭	21/20, 22/21
6	97.2973	↑D♯, ↓E♭	52/49, 55/52
7	113.514	↑↑D♯, E♭	15/14, 16/15
8	129.73	↑3D♯, ↓4E	14/13
9	145.946	↑4D♯, ↓3E	12/11
10	162.162	D𝄪, ↓↓E	11/10
11	178.378	↑D𝄪, ↓E
12	194.595	E	28/25, 55/49
13	210.811	↑E, ↓F♭	44/39
14	227.027	↑↑E, F♭	8/7, 25/22
15	243.243	↑3E, ↓4F	15/13
16	259.459	↑4E, ↓3F	64/55, 65/56
17	275.676	E♯, ↓↓F	75/64
18	291.892	↑E♯, ↓F	13/11, 77/65
19	308.108	F	25/21
20	324.324	↑F, ↓G♭♭	77/64
21	340.541	↑↑F, G♭♭	39/32
22	356.757	↑3F, ↓4G♭	16/13, 49/40
23	372.973	↑4F, ↓3G♭	26/21
24	389.189	F♯, ↓↓G♭	5/4
25	405.405	↑F♯, ↓G♭
26	421.622	↑↑F♯, G♭	14/11, 32/25
27	437.838	↑3F♯, ↓4G	77/60
28	454.054	↑4F♯, ↓3G	13/10
29	470.27	F𝄪, ↓↓G	21/16, 55/42
30	486.486	↑F𝄪, ↓G	65/49
31	502.703	G	4/3, 75/56
32	518.919	↑G, ↓A♭♭	35/26
33	535.135	↑↑G, A♭♭	15/11
34	551.351	↑3G, ↓4A♭	11/8
35	567.568	↑4G, ↓3A♭	39/28
36	583.784	G♯, ↓↓A♭	7/5, 45/32
37	600	↑G♯, ↓A♭	55/39, 78/55
38	616.216	↑↑G♯, A♭	10/7, 63/44, 64/45
39	632.432	↑3G♯, ↓4A	56/39, 75/52
40	648.649	↑4G♯, ↓3A	16/11
41	664.865	G𝄪, ↓↓A	22/15
42	681.081	↑G𝄪, ↓A	52/35, 65/44, 77/52
43	697.297	A	3/2
44	713.514	↑A, ↓B♭♭
45	729.73	↑↑A, B♭♭	32/21
46	745.946	↑3A, ↓4B♭	20/13, 77/50
47	762.162	↑4A, ↓3B♭	65/42
48	778.378	A♯, ↓↓B♭	11/7, 25/16
49	794.595	↑A♯, ↓B♭
50	810.811	↑↑A♯, B♭	8/5
51	827.027	↑3A♯, ↓4B	21/13
52	843.243	↑4A♯, ↓3B	13/8, 80/49
53	859.459	A𝄪, ↓↓B	64/39
54	875.676	↑A𝄪, ↓B
55	891.892	B	42/25
56	908.108	↑B, ↓C♭	22/13
57	924.324	↑↑B, C♭	75/44
58	940.541	↑3B, ↓4C	55/32
59	956.757	↑4B, ↓3C	26/15
60	972.973	B♯, ↓↓C	7/4, 44/25
61	989.189	↑B♯, ↓C	39/22
62	1005.41	C	25/14
63	1021.62	↑C, ↓D♭♭
64	1037.84	↑↑C, D♭♭	20/11
65	1054.05	↑3C, ↓4D♭	11/6
66	1070.27	↑4C, ↓3D♭	13/7
67	1086.49	C♯, ↓↓D♭	15/8, 28/15
68	1102.7	↑C♯, ↓D♭	49/26
69	1118.92	↑↑C♯, D♭	21/11, 40/21
70	1135.14	↑3C♯, ↓4D	25/13, 77/40
71	1151.35	↑4C♯, ↓3D	39/20, 64/33
72	1167.57	C𝄪, ↓↓D	49/25, 55/28, 63/32
73	1183.78	↑C𝄪, ↓D	77/39
74	1200	D	2/1

Music

• Twinkle canon – 74 edo by Claudi Meneghin ^{[dead link]}