74edo
← 73edo | 74edo | 75edo → |
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.
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, 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 |