75edo

 ← 74edo 75edo 76edo →
Prime factorization 3 × 52
Step size 16¢
Fifth 44\75 (704¢)
Semitones (A1:m2) 8:5 (128¢ : 80¢)
Consistency limit 5
Distinct consistency limit 5

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 cents, which is very close to the peppermint fifth of 704.096 cents. This makes it suitable for neo-Gothic tunings. It also approximates the Carlos Beta scale well (4\75 ≈ 1\Carlos Beta).

Odd harmonics

Approximation of odd harmonics in 75edo
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 (%) +12.8 -14.5 +44.8 +25.6 -45.7 +46.7 -1.7 +44.0 +40.5 -42.4 -26.7
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 Approximate Ratios Ups and Downs Notation
0 0 1/1 D
1 16 ^D, v4E♭
2 32 65/64 ^^D, v3E♭
3 48 33/32, 36/35, 65/63, 77/75 ^3D, vvE♭
4 64 27/26, 28/27, 80/77 ^4D, vE♭
5 80 ^5D, E♭
6 96 ^6D, v7E
7 112 16/15, 77/72 ^7D, v6E
8 128 14/13 D♯, v5E
9 144 13/12 ^D♯, v4E
10 160 11/10, 35/32 ^^D♯, v3E
11 176 72/65 ^3D♯, vvE
12 192 39/35 ^4D♯, vE
13 208 9/8 E
14 224 25/22 ^E, v4F
15 240 ^^E, v3F
16 256 52/45, 65/56, 81/70 ^3E, vvF
17 272 7/6, 75/64 ^4E, vF
18 288 32/27, 77/65 F
19 304 ^F, v4G♭
20 320 6/5, 65/54, 77/64 ^^F, v3G♭
21 336 40/33, 63/52 ^3F, vvG♭
22 352 ^4F, vG♭
23 368 26/21 ^5F, G♭
24 384 5/4, 56/45, 81/65 ^6F, v7G
25 400 49/39 ^7F, v6G
26 416 F♯, v5G
27 432 9/7, 32/25, 77/60 ^F♯, v4G
28 448 35/27 ^^F♯, v3G
29 464 ^3F♯, vvG
30 480 33/25 ^4F♯, vG
31 496 4/3 G
32 512 35/26 ^G, v4A♭
33 528 65/48 ^^G, v3A♭
34 544 48/35 ^3G, vvA♭
35 560 18/13 ^4G, vA♭
36 576 39/28 ^5G, A♭
37 592 45/32 ^6G, v7A
38 608 64/45, 77/54 ^7G, v6A
39 624 56/39 G♯, v5A
40 640 13/9, 81/56 ^G♯, v4A
41 656 35/24 ^^G♯, v3A
42 672 ^3G♯, vvA
43 688 52/35 ^4G♯, vA
44 704 3/2 A
45 720 50/33 ^A, v4B♭
46 736 ^^A, v3B♭
47 752 54/35, 65/42, 77/50 ^3A, vvB♭
48 768 14/9, 25/16, 81/52 ^4A, vB♭
49 784 ^5A, B♭
50 800 78/49 ^6A, v7B
51 816 8/5, 45/28, 77/48 ^7A, v6B
52 832 21/13 A♯, v5B
53 848 ^A♯, v4B
54 864 33/20, 81/49 ^^A♯, v3B
55 880 5/3 ^3A♯, vvB
56 896 ^4A♯, vB
57 912 27/16 B
58 928 12/7, 75/44, 77/45 ^B, v4C
59 944 45/26 ^^B, v3C
60 960 ^3B, vvC
61 976 44/25 ^4B, vC
62 992 16/9 C
63 1008 70/39 ^C, v4D♭
64 1024 65/36 ^^C, v3D♭
65 1040 20/11, 64/35 ^3C, vvD♭
66 1056 24/13 ^4C, vD♭
67 1072 13/7 ^5C, D♭
68 1088 15/8 ^6C, v7D
69 1104 ^7C, v6D
70 1120 C♯, v5D
71 1136 27/14, 52/27, 77/40 ^C♯, v4D
72 1152 35/18, 64/33 ^^C♯, v3D
73 1168 ^3C♯, vvD
74 1184 ^4C♯, vD
75 1200 2/1 D

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

Claudi Meneghin