97edo

From Xenharmonic Wiki
Jump to navigation Jump to search
← 96edo97edo98edo →
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 Approximate Ratios Ups and Downs Notation
0 0 1/1 D
1 12.371 ^D, v5E♭
2 24.742 ^^D, v4E♭
3 37.113 ^3D, v3E♭
4 49.485 35/34 ^4D, vvE♭
5 61.856 29/28 ^5D, vE♭
6 74.227 24/23 ^6D, E♭
7 86.598 20/19, 21/20, 41/39 ^7D, v10E
8 98.969 ^8D, v9E
9 111.34 16/15 ^9D, v8E
10 123.711 43/40, 44/41 ^10D, v7E
11 136.082 13/12, 40/37 D♯, v6E
12 148.454 12/11, 37/34 ^D♯, v5E
13 160.825 ^^D♯, v4E
14 173.196 21/19 ^3D♯, v3E
15 185.567 ^4D♯, vvE
16 197.938 28/25 ^5D♯, vE
17 210.309 26/23, 44/39 E
18 222.68 41/36 ^E, v5F
19 235.052 ^^E, v4F
20 247.423 15/13 ^3E, v3F
21 259.794 36/31, 43/37 ^4E, vvF
22 272.165 ^5E, vF
23 284.536 F
24 296.907 19/16 ^F, v5G♭
25 309.278 ^^F, v4G♭
26 321.649 ^3F, v3G♭
27 334.021 17/14 ^4F, vvG♭
28 346.392 11/9 ^5F, vG♭
29 358.763 16/13, 43/35 ^6F, G♭
30 371.134 26/21 ^7F, v10G
31 383.505 ^8F, v9G
32 395.876 39/31 ^9F, v8G
33 408.247 19/15, 43/34 ^10F, v7G
34 420.619 37/29 F♯, v6G
35 432.99 ^F♯, v5G
36 445.361 31/24 ^^F♯, v4G
37 457.732 30/23 ^3F♯, v3G
38 470.103 21/16, 38/29 ^4F♯, vvG
39 482.474 37/28, 41/31 ^5F♯, vG
40 494.845 G
41 507.216 ^G, v5A♭
42 519.588 ^^G, v4A♭
43 531.959 34/25 ^3G, v3A♭
44 544.33 26/19 ^4G, vvA♭
45 556.701 29/21, 40/29 ^5G, vA♭
46 569.072 ^6G, A♭
47 581.443 7/5 ^7G, v10A
48 593.814 31/22 ^8G, v9A
49 606.186 44/31 ^9G, v8A
50 618.557 10/7 ^10G, v7A
51 630.928 G♯, v6A
52 643.299 29/20, 42/29 ^G♯, v5A
53 655.67 19/13 ^^G♯, v4A
54 668.041 25/17 ^3G♯, v3A
55 680.412 37/25, 43/29 ^4G♯, vvA
56 692.784 ^5G♯, vA
57 705.155 A
58 717.526 ^A, v5B♭
59 729.897 29/19, 32/21 ^^A, v4B♭
60 742.268 23/15, 43/28 ^3A, v3B♭
61 754.639 ^4A, vvB♭
62 767.01 ^5A, vB♭
63 779.381 ^6A, B♭
64 791.753 30/19 ^7A, v10B
65 804.124 ^8A, v9B
66 816.495 ^9A, v8B
67 828.866 21/13 ^10A, v7B
68 841.237 13/8 A♯, v6B
69 853.608 18/11 ^A♯, v5B
70 865.979 28/17 ^^A♯, v4B
71 878.351 ^3A♯, v3B
72 890.722 ^4A♯, vvB
73 903.093 32/19 ^5A♯, vB
74 915.464 39/23 B
75 927.835 41/24 ^B, v5C
76 940.206 31/18, 43/25 ^^B, v4C
77 952.577 26/15 ^3B, v3C
78 964.948 ^4B, vvC
79 977.32 ^5B, vC
80 989.691 23/13, 39/22 C
81 1002.062 25/14, 41/23 ^C, v5D♭
82 1014.433 ^^C, v4D♭
83 1026.804 38/21 ^3C, v3D♭
84 1039.175 ^4C, vvD♭
85 1051.546 11/6 ^5C, vD♭
86 1063.918 24/13, 37/20 ^6C, D♭
87 1076.289 41/22 ^7C, v10D
88 1088.66 15/8 ^8C, v9D
89 1101.031 ^9C, v8D
90 1113.402 19/10, 40/21 ^10C, v7D
91 1125.773 23/12 C♯, v6D
92 1138.144 ^C♯, v5D
93 1150.515 ^^C♯, v4D
94 1162.887 ^3C♯, v3D
95 1175.258 ^4C♯, vvD
96 1187.629 ^5C♯, vD
97 1200 2/1 D

Music

Francium
Mercury Amalgam