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| The 97 equal temperament divides the octave into 97 equal parts of 12.371 cents each. It 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.
| | {{Infobox ET}} |
| | {{ED intro}} |
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| 97edo is the 25th prime edo. | | == Theory == |
| | 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. |
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| | === Odd harmonics === |
| | {{Harmonics in equal|97|columns=14}} |
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| | === Subsets and supersets === |
| | 97edo is the 25th [[prime edo]], following [[89edo]] and before [[101edo]]. |
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| | [[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. |
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| | == 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%. |
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| | 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. |
| | {{Q-odd-limit intervals|97}} |
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| == Theory == | | == Intervals == |
| Since 97edo has a step of 12.371 cents, it also allows one to use its MOS scales as circulating temperaments. 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.
| | {{Interval table}} |
| {| class="wikitable" | | |
| |+Circulating temperaments in 97edo
| | == Music == |
| !Tones
| | ; [[Bryan Deister]] |
| !Pattern
| | * [https://www.youtube.com/watch?v=hXDdKO3-RL4 ''microtonal improvisation in 97edo''] (2025) |
| !L:s
| | |
| |-
| | ; [[User:Francium|Francium]] |
| |5
| | * [https://www.youtube.com/watch?v=h7bT1oL8T0w ''Joyous Stellaris''] (2023) – [[semiquartal]] in 97edo tuning |
| |[[2L 3s]]
| | |
| |20:19
| | ; [[Mercury Amalgam]] |
| |-
| | * [https://www.youtube.com/watch?v=3JwH0gZmXHk ''Thanatonautical Tetrapharmacon''] (2023) |
| |6
| |
| |[[1L 5s]]
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| |17:16
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| |-
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| |7
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| |[[6L 1s]]
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| |14:13
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| |-
| |
| |8
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| |[[1L 7s]]
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| |13:12
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| |- | |
| |9
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| |[[7L 2s]]
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| |11:10
| |
| |-
| |
| |10
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| |[[7L 3s]]
| |
| |10:9
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| |-
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| |11
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| |[[9L 2s]]
| |
| | rowspan="2" |9:8
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| |-
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| |12
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| |[[1L 11s]]
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| |-
| |
| |13
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| |[[6L 7s]]
| |
| |8:7
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| |-
| |
| |14
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| |[[13L 1s]]
| |
| | rowspan="3" |7:6
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| |-
| |
| |15
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| |[[7L 8s]]
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| |-
| |
| |16
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| |1L 15s
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| |-
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| |17
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| |[[12L 5s]]
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| | rowspan="3" |6:5
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| |-
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| |18
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| |7L 11s
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| |-
| |
| |19
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| |[[2L 17s]]
| |
| |-
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| |20
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| |[[17L 3s]]
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| | rowspan="5" |5:4
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| |-
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| |21
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| |13L 8s
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| |-
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| |22
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| |[[9L 13s]]
| |
| |-
| |
| |23
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| |5L 18s
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| |-
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| |24
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| |1L 23s
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| |-
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| |25
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| |22L 3s
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| | rowspan="8" |4:3
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| |-
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| |26
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| |19L 7s
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| |-
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| |27
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| |16L 11s
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| |-
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| |28
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| |13L 15s
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| |-
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| |29
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| |10L 19s
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| |-
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| |30
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| |7L 23s
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| |-
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| |31
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| |4L 27s
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| |-
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| |32
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| |1L 31s
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| |-
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| |33
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| |31L 2s
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| | rowspan="16" |3:2
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| |-
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| |34
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| |29L 5s
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| |-
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| |35
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| |27L 8s
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| |-
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| |36
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| |25L 11s
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| |-
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| |37
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| |23L 14s
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| |-
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| |38
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| |21L 17s
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| |-
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| |39
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| |19L 20s
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| |-
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| |40
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| |17L 23s
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| |-
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| |41
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| |15L 26s
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| |-
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| |42
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| |13L 29s
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| |-
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| |43
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| |11L 32s
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| |-
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| |44
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| |9L 35s
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| |-
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| |45
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| |7L 38s
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| |-
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| |46
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| |5L 41s
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| |-
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| |47
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| |3L 44s
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| |-
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| |48
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| |1L 47s
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| |-
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| |49
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| |48L 1s
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| | rowspan="29" |2:1
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| |-
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| |50
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| |47L 3s
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| |-
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| |51
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| |46L 5s
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| |-
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| |52
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| |45L 7s
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| |-
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| |53
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| |44L 9s
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| |-
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| |54
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| |43L 11s
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| |-
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| |55
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| |42L 13s
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| |-
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| |56
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| |41L 15s
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| |-
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| |57
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| |40L 17s
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| |-
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| |58
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| |39L 19s
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| |-
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| |59
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| |38L 21s
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| |-
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| |60
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| |37L 23s
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| |-
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| |61
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| |36L 25s
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| |-
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| |62
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| |35L 27s
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| |-
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| |63
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| |34L 29s
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| |-
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| |64
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| |33L 31s
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| |-
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| |65
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| |32L 33s
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| |-
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| |66
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| |31L 35s
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| |-
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| |67
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| |30L 37s
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| |-
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| |68
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| |29L 39s
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| |-
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| |69
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| |28L 41s
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| |-
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| |70
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| |27L 43s
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| |-
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| |71
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| |26L 45s
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| |-
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| |72
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| |25L 47s
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| |-
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| |73
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| |24L 49s
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| |-
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| |74
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| |23L 51s
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| |-
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| |75
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| |22L 53s
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| |-
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| |76
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| |21L 55s
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| |-
| |
| |77
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| |20L 57s
| |
| |}
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| === Dissonance === | | == Instruments == |
| 97edo is one of the least harmonic EDOs within double digits or early hundreds, resulting in errors of well over 15%, or alternatively, above one standard deviation of 15.87% in intervals up to 15/14. 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]].
| | A [[Lumatone mapping for 97edo]] has now been demonstrated (see the Unnamed high-limit temperament mapping for full gamut coverage). |
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| 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.
| | [[Category:Listen]] |
| {| class="wikitable"
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| |+ Table of errors for superparticular intervals up to 17/16
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| |-
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| ! Interval (JI) !! Error ([[Relative cent|r¢]])
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| |-
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| | 3/2 || 25.9
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| |-
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| | 4/3 || 25.8
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| |-
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| | 5/4 || 22.7
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| |-
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| | 6/5 || 48.6
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| |-
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| | 7/6 || 42.8
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| |-
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| | 8/7 || 31.4
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| |-
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| | 9/8 || 48.2
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| |-
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| | 10/9 || 25.6
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| |-
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| | 11/10 || 33.7
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| |-
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| | 12/11 || 17.6
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| |-
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| | 13/12 || 20.1
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| |-
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| | 14/13 || 37.0
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| |-
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| | 15/14 || 34.6
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| |-
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| | 16/15 || 3.1
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| |-
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| | 17/16 || 48.3
| |
| |}
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| [[Category:Equal divisions of the octave]]
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| [[Category:Prime EDO]] | |