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.

Theory

74edo is most notable as a meantone tuning, tempering out 81/80 in the 5-limit; 126/125 and 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 the 13-limit meantone extension grosstone, for which 74edo gives the optimal patent val; and discarding 144/143 gives semimeantone, a 13-limit 62 & 74 temperament with half-octave period and two parallel tracks of meantone.

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

Odd harmonics

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 (%)	-28.7	+17.7	+25.6	+42.6	+0.2	+16.7	-11.0	-47.2	-34.7	-3.1	+25.6
Steps (reduced)		117 (43)	172 (24)	208 (60)	235 (13)	256 (34)	274 (52)	289 (67)	302 (6)	314 (18)	325 (29)	335 (39)

Subsets and supersets

Since 74 factors into 2 × 37, 74edo contains 2edo and 37edo as its subsets.

Intervals

Steps	Cents	Approximate Ratios	Ups and Downs Notation
0	0	1/1	D
1	16.216	78/77	^D, vE♭♭
2	32.432	45/44, 50/49, 56/55, 64/63, 65/64	^^D, E♭♭
3	48.649	33/32, 40/39, 77/75	^3D, v4E♭
4	64.865	26/25, 80/77	^4D, v3E♭
5	81.081	21/20, 22/21	D♯, vvE♭
6	97.297	52/49, 55/52	^D♯, vE♭
7	113.514	15/14, 16/15	^^D♯, E♭
8	129.73	14/13	^3D♯, v4E
9	145.946	12/11	^4D♯, v3E
10	162.162	11/10	D𝄪, vvE
11	178.378		^D𝄪, vE
12	194.595	28/25, 55/49	E
13	210.811	44/39	^E, vF♭
14	227.027	8/7, 25/22	^^E, F♭
15	243.243	15/13	^3E, v4F
16	259.459	64/55, 65/56	^4E, v3F
17	275.676	75/64	E♯, vvF
18	291.892	13/11, 77/65	^E♯, vF
19	308.108	25/21	F
20	324.324	77/64	^F, vG♭♭
21	340.541	39/32	^^F, G♭♭
22	356.757	16/13, 49/40	^3F, v4G♭
23	372.973	26/21	^4F, v3G♭
24	389.189	5/4	F♯, vvG♭
25	405.405		^F♯, vG♭
26	421.622	14/11, 32/25	^^F♯, G♭
27	437.838	77/60	^3F♯, v4G
28	454.054	13/10	^4F♯, v3G
29	470.27	21/16, 55/42	F𝄪, vvG
30	486.486	65/49	^F𝄪, vG
31	502.703	4/3, 75/56	G
32	518.919	35/26	^G, vA♭♭
33	535.135	15/11	^^G, A♭♭
34	551.351	11/8	^3G, v4A♭
35	567.568	39/28	^4G, v3A♭
36	583.784	7/5, 45/32	G♯, vvA♭
37	600	55/39, 78/55	^G♯, vA♭
38	616.216	10/7, 63/44, 64/45	^^G♯, A♭
39	632.432	56/39, 75/52	^3G♯, v4A
40	648.649	16/11	^4G♯, v3A
41	664.865	22/15	G𝄪, vvA
42	681.081	52/35, 65/44, 77/52	^G𝄪, vA
43	697.297	3/2	A
44	713.514		^A, vB♭♭
45	729.73	32/21	^^A, B♭♭
46	745.946	20/13, 77/50	^3A, v4B♭
47	762.162	65/42	^4A, v3B♭
48	778.378	11/7, 25/16	A♯, vvB♭
49	794.595		^A♯, vB♭
50	810.811	8/5	^^A♯, B♭
51	827.027	21/13	^3A♯, v4B
52	843.243	13/8, 80/49	^4A♯, v3B
53	859.459	64/39	A𝄪, vvB
54	875.676		^A𝄪, vB
55	891.892	42/25	B
56	908.108	22/13	^B, vC♭
57	924.324	75/44	^^B, C♭
58	940.541	55/32	^3B, v4C
59	956.757	26/15	^4B, v3C
60	972.973	7/4, 44/25	B♯, vvC
61	989.189	39/22	^B♯, vC
62	1005.405	25/14	C
63	1021.622		^C, vD♭♭
64	1037.838	20/11	^^C, D♭♭
65	1054.054	11/6	^3C, v4D♭
66	1070.27	13/7	^4C, v3D♭
67	1086.486	15/8, 28/15	C♯, vvD♭
68	1102.703	49/26	^C♯, vD♭
69	1118.919	21/11, 40/21	^^C♯, D♭
70	1135.135	25/13, 77/40	^3C♯, v4D
71	1151.351	39/20, 64/33	^4C♯, v3D
72	1167.568	49/25, 55/28, 63/32	C𝄪, vvD
73	1183.784	77/39	^C𝄪, vD
74	1200	2/1	D

Music

Modern renderings

Scott Joplin

• Maple Leaf Rag (1899) – rendered by Francium (2024)

21st century

Claudi Meneghin

• Twinkle canon (2012) – detail | play