97edo

 ← 96edo 97edo 98edo →
Prime factorization 97 (prime)
Step size 12.3711¢
Fifth 57\97 (705.155¢)
Semitones (A1:m2) 11:6 (136.1¢ : 74.23¢)
Consistency limit 5
Distinct consistency limit 5

97 equal divisions of the octave (abbreviated 97edo or 97ed2), also called 97-tone equal temperament (97tet) or 97 equal temperament (97et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 97 equal parts of about 12.4 ¢ each. Each step represents a frequency ratio of 21/97, or the 97th root of 2.

Theory

97edo is only consistent to the 5-odd-limit. The patent val of 97edo tempers out 875/864, 1029/1024, and 4000/3969 in the 7-limit, 100/99, 245/242, 385/384 and 441/440 in the 11-limit, and 196/195, 352/351 and 676/675 in the 13-limit. It provides the optimal patent val for the 13-limit 41 & 97 temperament tempering out 100/99, 196/195, 245/242 and 385/384.

Odd harmonics

Approximation of odd harmonics in 97edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +3.20 -2.81 -3.88 -5.97 +5.38 +0.71 +0.39 -5.99 -0.61 -0.68 +2.65
Relative (%) +25.9 -22.7 -31.3 -48.3 +43.5 +5.7 +3.2 -48.4 -4.9 -5.5 +21.4
Steps
(reduced)
154
(57)
225
(31)
272
(78)
307
(16)
336
(45)
359
(68)
379
(88)
396
(8)
412
(24)
426
(38)
439
(51)

Subsets and supersets

97edo is the 25th prime edo, following 89edo and before 101edo.

388edo and 2619edo, which contain 97edo as a subset, have very high consistency limits – 37 and 33 respectively. 3395edo, which divides the edostep in 35, is a zeta edo. The berkelium temperament realizes some relationships between them through a regular temperament perspective.

Approximation to JI

97edo has very poor direct approximation for superparticular intervals among edos up to 200, and the worst for intervals up to 9/8 among edos up to 100. It has errors of well above one standard deviation (about 15.87%) in superparticular intervals with denominators up to 14. The first good approximation is the 16/15 semitone using the 9th note, with an error of 3%, meaning 97edo can be used as a rough version of 16/15 equal-step tuning.

Since 97edo is a prime edo, it lacks specific modulation circles, symmetrical chords or sub-edos that are present in composite edos. When notable equal divisions like 19, 31, 41, or 53 have strong JI-based harmony, 97edo does not have easily representable modulation because of its inability to represent superparticulars. However, this might result in interest in this tuning through JI-agnostic approaches. The following tables show how 15-odd-limit intervals are represented in 97edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 97edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
15/13, 26/15 0.318 2.6
15/8, 16/15 0.391 3.2
13/8, 16/13 0.709 5.7
11/9, 18/11 1.016 8.2
7/5, 10/7 1.069 8.6
9/7, 14/9 2.094 16.9
11/6, 12/11 2.183 17.6
13/12, 24/13 2.490 20.1
5/4, 8/5 2.809 22.7
11/7, 14/11 3.111 25.1
9/5, 10/9 3.163 25.6
3/2, 4/3 3.200 25.9
13/10, 20/13 3.518 28.4
7/4, 8/7 3.877 31.3
11/10, 20/11 4.179 33.8
15/14, 28/15 4.269 34.5
13/7, 14/13 4.587 37.1
13/11, 22/13 4.674 37.8
15/11, 22/15 4.992 40.4
7/6, 12/7 5.294 42.8
11/8, 16/11 5.383 43.5
13/9, 18/13 5.690 46.0
9/8, 16/9 5.972 48.3
5/3, 6/5 6.008 48.6
15-odd-limit intervals in 97edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
15/13, 26/15 0.318 2.6
15/8, 16/15 0.391 3.2
13/8, 16/13 0.709 5.7
11/9, 18/11 1.016 8.2
7/5, 10/7 1.069 8.6
11/6, 12/11 2.183 17.6
13/12, 24/13 2.490 20.1
5/4, 8/5 2.809 22.7
3/2, 4/3 3.200 25.9
13/10, 20/13 3.518 28.4
7/4, 8/7 3.877 31.3
15/14, 28/15 4.269 34.5
13/7, 14/13 4.587 37.1
13/11, 22/13 4.674 37.8
15/11, 22/15 4.992 40.4
11/8, 16/11 5.383 43.5
13/9, 18/13 5.690 46.0
5/3, 6/5 6.008 48.6
9/8, 16/9 6.399 51.7
7/6, 12/7 7.077 57.2
11/10, 20/11 8.192 66.2
9/5, 10/9 9.208 74.4
11/7, 14/11 9.261 74.9
9/7, 14/9 10.277 83.1

Intervals

Steps Cents Ups and Downs Notation Approximate Ratios
0 0 D 1/1
1 12.3711 ^D, v5E♭
2 24.7423 ^^D, v4E♭ 64/63, 65/64, 78/77
3 37.1134 ^3D, v3E♭ 45/44, 50/49
4 49.4845 ^4D, vvE♭ 65/63, 77/75
5 61.8557 ^5D, vE♭ 80/77
6 74.2268 ^6D, E♭
7 86.5979 ^7D, v10E 21/20
8 98.9691 ^8D, v9E 55/52
9 111.34 ^9D, v8E 16/15, 77/72
10 123.711 ^10D, v7E 14/13, 15/14
11 136.082 D♯, v6E 13/12
12 148.454 ^D♯, v5E 12/11
13 160.825 ^^D♯, v4E
14 173.196 ^3D♯, v3E
15 185.567 ^4D♯, vvE
16 197.938 ^5D♯, vE 28/25
17 210.309 E 44/39
18 222.68 ^E, v5F
19 235.052 ^^E, v4F 8/7, 55/48, 63/55
20 247.423 ^3E, v3F 15/13, 52/45
21 259.794 ^4E, vvF 64/55, 65/56
22 272.165 ^5E, vF 75/64
23 284.536 F 13/11
24 296.907 ^F, v5G♭ 25/21, 77/65
25 309.278 ^^F, v4G♭
26 321.649 ^3F, v3G♭ 77/64
27 334.021 ^4F, vvG♭ 63/52
28 346.392 ^5F, vG♭ 11/9, 39/32, 49/40
29 358.763 ^6F, G♭ 16/13, 27/22
30 371.134 ^7F, v10G 26/21
31 383.505 ^8F, v9G 5/4
32 395.876 ^9F, v8G
33 408.247 ^10F, v7G
34 420.619 F♯, v6G
35 432.99 ^F♯, v5G 77/60
36 445.361 ^^F♯, v4G
37 457.732 ^3F♯, v3G 13/10
38 470.103 ^4F♯, vvG 21/16, 55/42, 72/55
39 482.474 ^5F♯, vG
40 494.845 G 4/3
41 507.216 ^G, v5A♭ 75/56
42 519.588 ^^G, v4A♭
43 531.959 ^3G, v3A♭
44 544.33 ^4G, vvA♭
45 556.701 ^5G, vA♭
46 569.072 ^6G, A♭
47 581.443 ^7G, v10A 7/5
48 593.814 ^8G, v9A 45/32, 55/39
49 606.186 ^9G, v8A 64/45, 78/55
50 618.557 ^10G, v7A 10/7, 63/44
51 630.928 G♯, v6A 75/52
52 643.299 ^G♯, v5A
53 655.67 ^^G♯, v4A
54 668.041 ^3G♯, v3A
55 680.412 ^4G♯, vvA 77/52
56 692.784 ^5G♯, vA
57 705.155 A 3/2
58 717.526 ^A, v5B♭
59 729.897 ^^A, v4B♭ 32/21, 55/36
60 742.268 ^3A, v3B♭ 20/13
61 754.639 ^4A, vvB♭ 65/42
62 767.01 ^5A, vB♭
63 779.381 ^6A, B♭
64 791.753 ^7A, v10B
65 804.124 ^8A, v9B
66 816.495 ^9A, v8B 8/5, 77/48
67 828.866 ^10A, v7B 21/13
68 841.237 A♯, v6B 13/8, 44/27
69 853.608 ^A♯, v5B 18/11, 64/39, 80/49
70 865.979 ^^A♯, v4B
71 878.351 ^3A♯, v3B
72 890.722 ^4A♯, vvB
73 903.093 ^5A♯, vB 42/25
74 915.464 B 22/13
75 927.835 ^B, v5C 77/45
76 940.206 ^^B, v4C 55/32
77 952.577 ^3B, v3C 26/15, 45/26
78 964.948 ^4B, vvC 7/4
79 977.32 ^5B, vC
80 989.691 C 39/22
81 1002.06 ^C, v5D♭ 25/14
82 1014.43 ^^C, v4D♭
83 1026.8 ^3C, v3D♭
84 1039.18 ^4C, vvD♭
85 1051.55 ^5C, vD♭ 11/6
86 1063.92 ^6C, D♭ 24/13
87 1076.29 ^7C, v10D 13/7, 28/15
88 1088.66 ^8C, v9D 15/8
89 1101.03 ^9C, v8D
90 1113.4 ^10C, v7D 40/21
91 1125.77 C♯, v6D
92 1138.14 ^C♯, v5D 77/40
93 1150.52 ^^C♯, v4D
94 1162.89 ^3C♯, v3D 49/25
95 1175.26 ^4C♯, vvD 63/32, 77/39
96 1187.63 ^5C♯, vD
97 1200 D 2/1

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