75edo

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← 74edo75edo76edo →
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] (corresponding to 401zpi) 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. It tempers out 325/324 and 512/507 in the 13-limit, 120/119 and 256/255 in the 17-limit, and 190/189 and 250/247 in the 19-limit.

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)

Riemann zeta function

The Riemann zeta function includes two peaks of similar magnitude around 75edo: 400zpi and 401zpi, corresponding to the 75dfghk and 75eij vals, with differing mappings for all primes above 5. 400zpi tempers out 686/675, 875/864, and 5120/5103 in the 7-limit, 121/120 and 441/440 in the 11-limit, 91/90, 352/351, and 2080/2079 in the 13-limit, 136/135 in the 17-limit, 190/189 in the 19-limit, and 161/160 in the 23-limit. 401zpi tempers out 20000/19683, 1728/1715, and 225/224 in the 7-limit, 100/99 and 2200/2187 in the 11-limit, 144/143 and 275/273 in the 13-limit, 120/119 and 1225/1224 in the 17-limit, 190/189 in the 19-limit, and 162/161 in the 23-limit. Its step is mapped to 49/48 (the slendro diesis) in 400zpi, but 64/63 (Archytas' comma) in 401zpi and 75p.

The Riemann zeta function around 75edo, showing 400zpi and 401zpi

Compare how prime harmonics are mapped in each zeta peak:

Approximation of harmonics in 400zpi
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +1.59 +4.57 +1.37 -4.37 -1.83 -2.66 -2.47 -2.77 +2.91 +2.14 -1.17
Relative (%) +9.9 +28.5 +8.6 -27.3 -11.4 -16.6 -15.4 -17.3 +18.2 +13.4 -7.3
Steps 75 119 174 210 259 277 306 318 339 364 371
Approximation of harmonics in 401zpi
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -1.46 -0.27 -5.69 +3.08 +3.63 +2.07 +1.08 +0.29 +5.12 +3.34 -0.26
Relative (%) -9.1 -1.7 -35.6 +19.3 +22.7 +13.0 +6.8 +1.8 +32.1 +20.9 -1.6
Steps 75 119 174 211 260 278 307 319 340 365 372

Intervals

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

Music

Claudi Meneghin