74edo

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← 73edo74edo75edo →
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 21/74, 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 (%) -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)

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, vE♭♭ 78/77
2 32.4324 ^^D, E♭♭ 45/44, 50/49, 56/55, 64/63, 65/64
3 48.6486 ^3D, v4E♭ 33/32, 40/39, 77/75
4 64.8649 ^4D, v3E♭ 26/25, 80/77
5 81.0811 D♯, vvE♭ 21/20, 22/21
6 97.2973 ^D♯, vE♭ 52/49, 55/52
7 113.514 ^^D♯, E♭ 15/14, 16/15
8 129.73 ^3D♯, v4E 14/13
9 145.946 ^4D♯, v3E 12/11
10 162.162 D𝄪, vvE 11/10
11 178.378 ^D𝄪, vE
12 194.595 E 28/25, 55/49
13 210.811 ^E, vF♭ 44/39
14 227.027 ^^E, F♭ 8/7, 25/22
15 243.243 ^3E, v4F 15/13
16 259.459 ^4E, v3F 64/55, 65/56
17 275.676 E♯, vvF 75/64
18 291.892 ^E♯, vF 13/11, 77/65
19 308.108 F 25/21
20 324.324 ^F, vG♭♭ 77/64
21 340.541 ^^F, G♭♭ 39/32
22 356.757 ^3F, v4G♭ 16/13, 49/40
23 372.973 ^4F, v3G♭ 26/21
24 389.189 F♯, vvG♭ 5/4
25 405.405 ^F♯, vG♭
26 421.622 ^^F♯, G♭ 14/11, 32/25
27 437.838 ^3F♯, v4G 77/60
28 454.054 ^4F♯, v3G 13/10
29 470.27 F𝄪, vvG 21/16, 55/42
30 486.486 ^F𝄪, vG 65/49
31 502.703 G 4/3, 75/56
32 518.919 ^G, vA♭♭ 35/26
33 535.135 ^^G, A♭♭ 15/11
34 551.351 ^3G, v4A♭ 11/8
35 567.568 ^4G, v3A♭ 39/28
36 583.784 G♯, vvA♭ 7/5, 45/32
37 600 ^G♯, vA♭ 55/39, 78/55
38 616.216 ^^G♯, A♭ 10/7, 63/44, 64/45
39 632.432 ^3G♯, v4A 56/39, 75/52
40 648.649 ^4G♯, v3A 16/11
41 664.865 G𝄪, vvA 22/15
42 681.081 ^G𝄪, vA 52/35, 65/44, 77/52
43 697.297 A 3/2
44 713.514 ^A, vB♭♭
45 729.73 ^^A, B♭♭ 32/21
46 745.946 ^3A, v4B♭ 20/13, 77/50
47 762.162 ^4A, v3B♭ 65/42
48 778.378 A♯, vvB♭ 11/7, 25/16
49 794.595 ^A♯, vB♭
50 810.811 ^^A♯, B♭ 8/5
51 827.027 ^3A♯, v4B 21/13
52 843.243 ^4A♯, v3B 13/8, 80/49
53 859.459 A𝄪, vvB 64/39
54 875.676 ^A𝄪, vB
55 891.892 B 42/25
56 908.108 ^B, vC♭ 22/13
57 924.324 ^^B, C♭ 75/44
58 940.541 ^3B, v4C 55/32
59 956.757 ^4B, v3C 26/15
60 972.973 B♯, vvC 7/4, 44/25
61 989.189 ^B♯, vC 39/22
62 1005.41 C 25/14
63 1021.62 ^C, vD♭♭
64 1037.84 ^^C, D♭♭ 20/11
65 1054.05 ^3C, v4D♭ 11/6
66 1070.27 ^4C, v3D♭ 13/7
67 1086.49 C♯, vvD♭ 15/8, 28/15
68 1102.7 ^C♯, vD♭ 49/26
69 1118.92 ^^C♯, D♭ 21/11, 40/21
70 1135.14 ^3C♯, v4D 25/13, 77/40
71 1151.35 ^4C♯, v3D 39/20, 64/33
72 1167.57 C𝄪, vvD 49/25, 55/28, 63/32
73 1183.78 ^C𝄪, vD 77/39
74 1200 D 2/1

Music