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

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 13-limit meantone, a.k.a. 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.

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	16.2		^D, vE♭♭
2	32.4		^^D, E♭♭
3	48.6	34/33, 37/36, 38/37	vvD♯, ^E♭♭
4	64.9	26/25	vD♯, ^^E♭♭
5	81.1	21/20, 22/21	D♯, vvE♭
6	97.3	18/17	^D♯, vE♭
7	113.5	16/15	^^D♯, E♭
8	129.7	14/13	vvD𝄪, ^E♭
9	145.9	25/23, 37/34	vD𝄪, ^^E♭
10	162.2	11/10	D𝄪, vvE
11	178.4	31/28	^D𝄪, vE
12	194.6	19/17, 28/25, 37/33	E
13	210.8	26/23, 35/31	^E, vF♭
14	227	33/29	^^E, F♭
15	243.2	23/20, 38/33	vvE♯, ^F♭
16	259.5		vE♯, ^^F♭
17	275.7	34/29	E♯, vvF
18	291.9	13/11	^E♯, vF
19	308.1		F
20	324.3	29/24	^F, vG♭♭
21	340.5	28/23	^^F, G♭♭
22	356.8	16/13	vvF♯, ^G♭♭
23	373	26/21, 31/25, 36/29	vF♯, ^^G♭♭
24	389.2	5/4	F♯, vvG♭
25	405.4	24/19	^F♯, vG♭
26	421.6	37/29	^^F♯, G♭
27	437.8		vvF𝄪, ^G♭
28	454.1	13/10	vF𝄪, ^^G♭
29	470.3	21/16, 38/29	F𝄪, vvG
30	486.5		^F𝄪, vG
31	502.7		G
32	518.9	31/23	^G, vA♭♭
33	535.1	15/11	^^G, A♭♭
34	551.4	11/8	vvG♯, ^A♭♭
35	567.6		vG♯, ^^A♭♭
36	583.8	7/5	G♯, vvA♭
37	600	17/12, 24/17	^G♯, vA♭
38	616.2	10/7	^^G♯, A♭
39	632.4		vvG𝄪, ^A♭
40	648.6	16/11	vG𝄪, ^^A♭
41	664.9	22/15	G𝄪, vvA
42	681.1		^G𝄪, vA
43	697.3		A
44	713.5		^A, vB♭♭
45	729.7	29/19, 32/21, 35/23	^^A, B♭♭
46	745.9	20/13, 37/24	vvA♯, ^B♭♭
47	762.2	31/20	vA♯, ^^B♭♭
48	778.4		A♯, vvB♭
49	794.6	19/12	^A♯, vB♭
50	810.8	8/5	^^A♯, B♭
51	827	21/13, 29/18	vvA𝄪, ^B♭
52	843.2	13/8	vA𝄪, ^^B♭
53	859.5	23/14	A𝄪, vvB
54	875.7		^A𝄪, vB
55	891.9		B
56	908.1	22/13	^B, vC♭
57	924.3	29/17	^^B, C♭
58	940.5		vvB♯, ^C♭
59	956.8	33/19	vB♯, ^^C♭
60	973		B♯, vvC
61	989.2	23/13	^B♯, vC
62	1005.4	25/14, 34/19	C
63	1021.6		^C, vD♭♭
64	1037.8	20/11	^^C, D♭♭
65	1054.1		vvC♯, ^D♭♭
66	1070.3	13/7	vC♯, ^^D♭♭
67	1086.5	15/8	C♯, vvD♭
68	1102.7	17/9	^C♯, vD♭
69	1118.9	21/11	^^C♯, D♭
70	1135.1	25/13	vvC𝄪, ^D♭
71	1151.4	33/17, 37/19	vC𝄪, ^^D♭
72	1167.6		C𝄪, vvD
73	1183.8		^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