75edo
← 74edo | 75edo | 76edo → |
75 equal divisions of the octave (abbreviated 75edo or 75ed2), also called 75-tone equal temperament (75tet) or 75 equal temperament (75et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 75 equal parts of exactly 16 ¢ each. Each step represents a frequency ratio of 21/75, or the 75th root of 2.
Theory
75et tempers out 20000/19683 (tetracot comma) and 2109375/2097152 (semicomma) in the 5-limit, and provides a good tuning for the tetracot temperament. It tempers out 225/224 and 1728/1715 in the 7-limit, supporting bunya and orwell, and providing the optimal patent val for fog.
In the 11-limit, 75e val ⟨75 119 174 211 260] scores lower in error, and tempers 100/99 and 243/242, whereas the patent val ⟨75 119 174 211 259] tempers 99/98 and 121/120. In the 13-limit, it tempers 325/324 and 512/507, 17-limit 120/119 and 256/255 and 19-limit 190/189 and 250/247.
Since 75 is part of the Fibonacci sequence beginning with 5 and 12, it closely approximates the peppermint temperament. The size of its fifth is exactly 704 ¢, which is very close to the peppermint fifth of 704.096 ¢. This makes it suitable for neo-Gothic tunings. It also approximates the Carlos Beta scale well (4\75 ≈ 1\[Carlos Beta]
).
Odd harmonics
Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | absolute (¢) | +2.04 | -2.31 | +7.17 | +4.09 | -7.32 | +7.47 | -0.27 | +7.04 | +6.49 | -6.78 | -4.27 |
relative (%) | +13 | -14 | +45 | +26 | -46 | +47 | -2 | +44 | +41 | -42 | -27 | |
Steps (reduced) |
119 (44) |
174 (24) |
211 (61) |
238 (13) |
259 (34) |
278 (53) |
293 (68) |
307 (7) |
319 (19) |
329 (29) |
339 (39) |
Intervals
Steps | Cents | Ups and downs notation | Approximate ratios |
---|---|---|---|
0 | 0 | D | 1/1 |
1 | 16 | ↑D, ↓4E♭ | |
2 | 32 | ↑↑D, ↓3E♭ | 65/64 |
3 | 48 | ↑3D, ↓↓E♭ | 33/32, 36/35, 65/63, 77/75 |
4 | 64 | ↑4D, ↓E♭ | 27/26, 28/27, 80/77 |
5 | 80 | ↑5D, E♭ | |
6 | 96 | ↑6D, ↓7E | |
7 | 112 | ↑7D, ↓6E | 16/15, 77/72 |
8 | 128 | D♯, ↓5E | 14/13 |
9 | 144 | ↑D♯, ↓4E | 13/12 |
10 | 160 | ↑↑D♯, ↓3E | 11/10, 35/32 |
11 | 176 | ↑3D♯, ↓↓E | 72/65 |
12 | 192 | ↑4D♯, ↓E | 39/35 |
13 | 208 | E | 9/8 |
14 | 224 | ↑E, ↓4F | 25/22 |
15 | 240 | ↑↑E, ↓3F | |
16 | 256 | ↑3E, ↓↓F | 52/45, 65/56, 81/70 |
17 | 272 | ↑4E, ↓F | 7/6, 75/64 |
18 | 288 | F | 32/27, 77/65 |
19 | 304 | ↑F, ↓4G♭ | |
20 | 320 | ↑↑F, ↓3G♭ | 6/5, 65/54, 77/64 |
21 | 336 | ↑3F, ↓↓G♭ | 40/33, 63/52 |
22 | 352 | ↑4F, ↓G♭ | |
23 | 368 | ↑5F, G♭ | 26/21 |
24 | 384 | ↑6F, ↓7G | 5/4, 56/45, 81/65 |
25 | 400 | ↑7F, ↓6G | 49/39 |
26 | 416 | F♯, ↓5G | |
27 | 432 | ↑F♯, ↓4G | 9/7, 32/25, 77/60 |
28 | 448 | ↑↑F♯, ↓3G | 35/27 |
29 | 464 | ↑3F♯, ↓↓G | |
30 | 480 | ↑4F♯, ↓G | 33/25 |
31 | 496 | G | 4/3 |
32 | 512 | ↑G, ↓4A♭ | 35/26 |
33 | 528 | ↑↑G, ↓3A♭ | 65/48 |
34 | 544 | ↑3G, ↓↓A♭ | 48/35 |
35 | 560 | ↑4G, ↓A♭ | 18/13 |
36 | 576 | ↑5G, A♭ | 39/28 |
37 | 592 | ↑6G, ↓7A | 45/32 |
38 | 608 | ↑7G, ↓6A | 64/45, 77/54 |
39 | 624 | G♯, ↓5A | 56/39 |
40 | 640 | ↑G♯, ↓4A | 13/9, 81/56 |
41 | 656 | ↑↑G♯, ↓3A | 35/24 |
42 | 672 | ↑3G♯, ↓↓A | |
43 | 688 | ↑4G♯, ↓A | 52/35 |
44 | 704 | A | 3/2 |
45 | 720 | ↑A, ↓4B♭ | 50/33 |
46 | 736 | ↑↑A, ↓3B♭ | |
47 | 752 | ↑3A, ↓↓B♭ | 54/35, 65/42, 77/50 |
48 | 768 | ↑4A, ↓B♭ | 14/9, 25/16, 81/52 |
49 | 784 | ↑5A, B♭ | |
50 | 800 | ↑6A, ↓7B | 78/49 |
51 | 816 | ↑7A, ↓6B | 8/5, 45/28, 77/48 |
52 | 832 | A♯, ↓5B | 21/13 |
53 | 848 | ↑A♯, ↓4B | |
54 | 864 | ↑↑A♯, ↓3B | 33/20, 81/49 |
55 | 880 | ↑3A♯, ↓↓B | 5/3 |
56 | 896 | ↑4A♯, ↓B | |
57 | 912 | B | 27/16 |
58 | 928 | ↑B, ↓4C | 12/7, 75/44, 77/45 |
59 | 944 | ↑↑B, ↓3C | 45/26 |
60 | 960 | ↑3B, ↓↓C | |
61 | 976 | ↑4B, ↓C | 44/25 |
62 | 992 | C | 16/9 |
63 | 1008 | ↑C, ↓4D♭ | 70/39 |
64 | 1024 | ↑↑C, ↓3D♭ | 65/36 |
65 | 1040 | ↑3C, ↓↓D♭ | 20/11, 64/35 |
66 | 1056 | ↑4C, ↓D♭ | 24/13 |
67 | 1072 | ↑5C, D♭ | 13/7 |
68 | 1088 | ↑6C, ↓7D | 15/8 |
69 | 1104 | ↑7C, ↓6D | |
70 | 1120 | C♯, ↓5D | |
71 | 1136 | ↑C♯, ↓4D | 27/14, 52/27, 77/40 |
72 | 1152 | ↑↑C♯, ↓3D | 35/18, 64/33 |
73 | 1168 | ↑3C♯, ↓↓D | |
74 | 1184 | ↑4C♯, ↓D | |
75 | 1200 | D | 2/1 |
Regular temperament properties
Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
---|---|---|---|---|---|
Absolute (¢) | Relative (%) | ||||
2.3 | [119 -75⟩ | [⟨75 119]] | -0.645 | 0.645 | 4.03 |
2.3.5 | 20000/19683, 2109375/2097152 | [⟨75 119 174]] | -0.099 | 0.936 | 5.85 |
2.3.5.7 | 225/224, 1728/1715, 15625/15309 | [⟨75 119 174 211]] | -0.713 | 1.337 | 8.36 |