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The '''97 equal temperament''' divides the octave into 97 equal parts of 12.371 cents each.
{{Infobox ET}}
{{ED intro}}


== Theory ==
== Theory ==
{{primes in edo|97}}
97edo is only [[consistent]] to the [[5-odd-limit]]. The [[patent val]] of 97edo [[tempering out|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 {{nowrap|41 & 97}} temperament tempering out 100/99, 196/195, 245/242 and 385/384.


97edo tempers out 875/864, 4000/3969 and 1029/1024 in the 7-limit, 245/242, 100/99, 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. 97edo is the 25th prime edo.
=== Odd harmonics ===
{{Harmonics in equal|97|columns=14}}


Since 97edo has a step of 12.371 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}. It is the first prime edo which does this and the first edo which allows one to use an MOS scale with a step 20 degrees or larger as a circulating temperament.
=== Subsets and supersets ===
{| class="wikitable"
97edo is the 25th [[prime edo]], following [[89edo]] and before [[101edo]].
|+Circulating temperaments in 97edo
!Tones
!Pattern
!L:s
|-
|5
|[[2L 3s]]
|20:19
|-
|6
|[[1L 5s]]
|17:16
|-
|7
|[[6L 1s]]
|14:13
|-
|8
|[[1L 7s]]
|13:12
|-
|9
|[[7L 2s]]
|11:10
|-
|10
|[[7L 3s]]
|10:9
|-
|11
|[[9L 2s]]
| rowspan="2" |9:8
|-
|12
|[[1L 11s]]
|-
|13
|[[6L 7s]]
|8:7
|-
|14
|[[13L 1s]]
| rowspan="3" |7:6
|-
|15
|[[7L 8s]]
|-
|16
|1L 15s
|-
|17
|[[12L 5s]]
| rowspan="3" |6:5
|-
|18
|7L 11s
|-
|19
|[[2L 17s]]
|-
|20
|[[17L 3s]]
| rowspan="5" |5:4
|-
|21
|13L 8s
|-
|22
|[[9L 13s]]
|-
|23
|5L 18s
|-
|24
|1L 23s
|-
|25
|22L 3s
| rowspan="8" |4:3
|-
|26
|19L 7s
|-
|27
|16L 11s
|-
|28
|13L 15s
|-
|29
|10L 19s
|-
|30
|7L 23s
|-
|31
|4L 27s
|-
|32
|1L 31s
|-
|33
|31L 2s
| rowspan="16" |3:2
|-
|34
|29L 5s
|-
|35
|27L 8s
|-
|36
|25L 11s
|-
|37
|23L 14s
|-
|38
|21L 17s
|-
|39
|19L 20s
|-
|40
|17L 23s
|-
|41
|15L 26s
|-
|42
|13L 29s
|-
|43
|11L 32s
|-
|44
|9L 35s
|-
|45
|7L 38s
|-
|46
|5L 41s
|-
|47
|3L 44s
|-
|48
|1L 47s
|-
|49
|48L 1s
| rowspan="29" |2:1
|-
|50
|47L 3s
|-
|51
|46L 5s
|-
|52
|45L 7s
|-
|53
|44L 9s
|-
|54
|43L 11s
|-
|55
|42L 13s
|-
|56
|41L 15s
|-
|57
|40L 17s
|-
|58
|39L 19s
|-
|59
|38L 21s
|-
|60
|37L 23s
|-
|61
|36L 25s
|-
|62
|35L 27s
|-
|63
|34L 29s
|-
|64
|33L 31s
|-
|65
|32L 33s
|-
|66
|31L 35s
|-
|67
|30L 37s
|-
|68
|29L 39s
|-
|69
|28L 41s
|-
|70
|27L 43s
|-
|71
|26L 45s
|-
|72
|25L 47s
|-
|73
|24L 49s
|-
|74
|23L 51s
|-
|75
|22L 53s
|-
|76
|21L 55s
|-
|77
|20L 57s
|}


=== Dissonance ===
[[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 [[The Riemann zeta function and tuning|zeta edo]]. The [[berkelium]] temperament realizes some relationships between them through a regular temperament perspective.
97edo is one of the least harmonic EDOs within double digits or early hundreds, resulting in 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 [[16/15ths equal temperament]].


Since 97edo is a prime EDO, it lacks specific modulation circles, symmetrical chords or sub-EDOs that are present in composite EDOs. When edos like [[19edo|19]], [[29edo|29]], [[31edo|31]], [[41edo|41]], or [[53edo|53]] have mathematically justified harmony, 97edo is essentially "irredeemable" in terms of either modulation or approximation rationales. However, this might result in interest towards this tuning through emancipation of the dissonance.
== Approximation to JI ==
{| class="wikitable"
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%.
|+ Table of errors for superparticular intervals up to 17/16
|-
! Interval (JI) !! Error ([[Relative cent|r¢]])
|-
| 3/2 || 25.9
|-
| 4/3 || 25.8 
|-
| 5/4 || 22.7 
|-
| 6/5 || 48.6
|-
| 7/6 || 42.8
|-
| 8/7 || 31.4 
|-
| 9/8 || 48.2 
|-
| 10/9 || 25.6
|-
| 11/10 || 33.7 
|-
| 12/11 || 17.6 
|-
| 13/12 || 20.1 
|-
| 14/13 || 37.0 
|-
| 15/14 || 34.6
|-
| 16/15 || 3.1 
|-
| 17/16 || 48.3 
|}


[[Category:Equal divisions of the octave]]
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 {{EDOs|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.
[[Category:Prime EDO]]
{{Q-odd-limit intervals|97}}
 
== Intervals ==
{{Interval table}}
 
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=hXDdKO3-RL4 ''microtonal improvisation in 97edo''] (2025)
 
; [[User:Francium|Francium]]
* [https://www.youtube.com/watch?v=h7bT1oL8T0w ''Joyous Stellaris''] (2023) – [[semiquartal]] in 97edo tuning
 
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=3JwH0gZmXHk ''Thanatonautical Tetrapharmacon''] (2023)
 
== Instruments ==
A [[Lumatone mapping for 97edo]] has now been demonstrated (see the Unnamed high-limit temperament mapping for full gamut coverage).
 
[[Category:Listen]]

Latest revision as of 02:05, 6 August 2025

← 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 25 27 29
Error Absolute (¢) +3.20 -2.81 -3.88 -5.97 +5.38 +0.71 +0.39 -5.99 -0.61 -0.68 +2.65 -5.62 -2.77 -2.77
Relative (%) +25.9 -22.7 -31.3 -48.3 +43.5 +5.7 +3.2 -48.4 -4.9 -5.5 +21.4 -45.4 -22.4 -22.4
Steps
(reduced) 		154
(57) 		225
(31) 		272
(78) 		307
(16) 		336
(45) 		359
(68) 		379
(88) 		396
(8) 		412
(24) 		426
(38) 		439
(51) 		450
(62) 		461
(73) 		471
(83)

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%.

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 12.4 ^D, v5E♭
2 24.7 ^^D, v4E♭
3 37.1 ^3D, v3E♭
4 49.5 35/34 ^4D, vvE♭
5 61.9 29/28 ^5D, vE♭
6 74.2 24/23 v5D♯, E♭
7 86.6 20/19, 21/20, 41/39 v4D♯, ^E♭
8 99 v3D♯, ^^E♭
9 111.3 16/15 vvD♯, ^3E♭
10 123.7 43/40, 44/41 vD♯, ^4E♭
11 136.1 13/12, 40/37 D♯, ^5E♭
12 148.5 12/11, 37/34 ^D♯, v5E
13 160.8 ^^D♯, v4E
14 173.2 21/19 ^3D♯, v3E
15 185.6 ^4D♯, vvE
16 197.9 28/25 ^5D♯, vE
17 210.3 26/23, 44/39 E
18 222.7 41/36 ^E, v5F
19 235.1 ^^E, v4F
20 247.4 15/13 ^3E, v3F
21 259.8 36/31, 43/37 ^4E, vvF
22 272.2 ^5E, vF
23 284.5 F
24 296.9 19/16 ^F, v5G♭
25 309.3 ^^F, v4G♭
26 321.6 ^3F, v3G♭
27 334 17/14 ^4F, vvG♭
28 346.4 11/9 ^5F, vG♭
29 358.8 16/13, 43/35 v5F♯, G♭
30 371.1 26/21 v4F♯, ^G♭
31 383.5 v3F♯, ^^G♭
32 395.9 39/31 vvF♯, ^3G♭
33 408.2 19/15, 43/34 vF♯, ^4G♭
34 420.6 37/29 F♯, ^5G♭
35 433 ^F♯, v5G
36 445.4 31/24 ^^F♯, v4G
37 457.7 30/23 ^3F♯, v3G
38 470.1 21/16, 38/29 ^4F♯, vvG
39 482.5 37/28, 41/31 ^5F♯, vG
40 494.8 G
41 507.2 ^G, v5A♭
42 519.6 ^^G, v4A♭
43 532 34/25 ^3G, v3A♭
44 544.3 26/19 ^4G, vvA♭
45 556.7 29/21, 40/29 ^5G, vA♭
46 569.1 v5G♯, A♭
47 581.4 7/5 v4G♯, ^A♭
48 593.8 31/22 v3G♯, ^^A♭
49 606.2 44/31 vvG♯, ^3A♭
50 618.6 10/7 vG♯, ^4A♭
51 630.9 G♯, ^5A♭
52 643.3 29/20, 42/29 ^G♯, v5A
53 655.7 19/13 ^^G♯, v4A
54 668 25/17 ^3G♯, v3A
55 680.4 37/25, 43/29 ^4G♯, vvA
56 692.8 ^5G♯, vA
57 705.2 A
58 717.5 ^A, v5B♭
59 729.9 29/19, 32/21 ^^A, v4B♭
60 742.3 23/15, 43/28 ^3A, v3B♭
61 754.6 ^4A, vvB♭
62 767 ^5A, vB♭
63 779.4 v5A♯, B♭
64 791.8 30/19 v4A♯, ^B♭
65 804.1 v3A♯, ^^B♭
66 816.5 vvA♯, ^3B♭
67 828.9 21/13 vA♯, ^4B♭
68 841.2 13/8 A♯, ^5B♭
69 853.6 18/11 ^A♯, v5B
70 866 28/17 ^^A♯, v4B
71 878.4 ^3A♯, v3B
72 890.7 ^4A♯, vvB
73 903.1 32/19 ^5A♯, vB
74 915.5 39/23 B
75 927.8 41/24 ^B, v5C
76 940.2 31/18, 43/25 ^^B, v4C
77 952.6 26/15 ^3B, v3C
78 964.9 ^4B, vvC
79 977.3 ^5B, vC
80 989.7 23/13, 39/22 C
81 1002.1 25/14, 41/23 ^C, v5D♭
82 1014.4 ^^C, v4D♭
83 1026.8 38/21 ^3C, v3D♭
84 1039.2 ^4C, vvD♭
85 1051.5 11/6 ^5C, vD♭
86 1063.9 24/13, 37/20 v5C♯, D♭
87 1076.3 41/22 v4C♯, ^D♭
88 1088.7 15/8 v3C♯, ^^D♭
89 1101 vvC♯, ^3D♭
90 1113.4 19/10, 40/21 vC♯, ^4D♭
91 1125.8 23/12 C♯, ^5D♭
92 1138.1 ^C♯, v5D
93 1150.5 ^^C♯, v4D
94 1162.9 ^3C♯, v3D
95 1175.3 ^4C♯, vvD
96 1187.6 ^5C♯, vD
97 1200 2/1 D

Music

Bryan Deister
Francium
Mercury Amalgam

Instruments

A Lumatone mapping for 97edo has now been demonstrated (see the Unnamed high-limit temperament mapping for full gamut coverage).

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