# 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 21/74, 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

Scott Joplin

Claudi Meneghin