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| {{Infobox ET}} | | {{Infobox ET}} |
| {{EDO intro|97}} | | {{ED intro}} |
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| == Theory == | | == Theory == |
| In the [[patent val]], 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 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 === | | === Odd harmonics === |
| {{Harmonics in equal|97}} | | {{Harmonics in equal|97|columns=14}} |
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| === Divisors === | | === Subsets and supersets === |
| 97edo is the 25th [[prime edo]]. | | 97edo is the 25th [[prime edo]], following [[89edo]] and before [[101edo]]. |
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| == JI approximation ==
| | [[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 has very poor approximation for [[superparticular]] intervals among edos up to 200. 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]]. | |
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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 in this tuning through JI-agnostic approaches.
| | == 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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| {| class="wikitable center-all"
| | 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. |
| |+style=white-space:nowrap| Superparticular intervals up to 17/16<br> by direct approximation (even if inconsistent)
| | {{Q-odd-limit intervals|97}} |
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| ! Interval !! Error (Relative, [[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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| |} | |
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| == Scales == | | == Intervals == |
| Since 97edo has a step of 12.371 cents, it also allows one to use its [[mos]] scales as [[circulating temperament]]s{{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.
| | {{Interval table}} |
| {| class="wikitable mw-collapsible mw-collapsed collapsible"
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| |+ style=white-space:nowrap | Circulating temperaments in 97edo
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| !Tones
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| !Pattern
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| !L:s
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| |-
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| |5
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| |[[2L 3s]]
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| |20:19
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| |-
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| |6
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| |[[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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| |-
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| |8
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| |[[1L 7s]]
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| |13:12
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| |-
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| |9
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| |[[7L 2s]]
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| |11:10
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| |-
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| |10
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| |[[7L 3s]]
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| |10:9
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| |-
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| |11
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| |[[9L 2s]]
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| | rowspan="2" |9:8
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| |-
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| |12
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| |[[1L 11s]]
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| |-
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| |13
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| |[[6L 7s]]
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| |8:7
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| |-
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| |14
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| |[[13L 1s]]
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| | rowspan="3" |7:6
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| |-
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| |15
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| |[[7L 8s]]
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| |-
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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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| |-
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| |19
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| |[[2L 17s]]
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| |-
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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]]
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| |-
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| |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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| |-
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| |77
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| |20L 57s
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| |}
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| == Music == | | == Music == |
| * [https://www.youtube.com/watch?v=3JwH0gZmXHk Thanatonautical Tetrapharmacon (Demo version, July 2021)] by [[Mercury Amalgam]] | | ; [[Bryan Deister]] |
| | * [https://www.youtube.com/watch?v=hXDdKO3-RL4 ''microtonal improvisation in 97edo''] (2025) |
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| | ; [[User:Francium|Francium]] |
| | * [https://www.youtube.com/watch?v=h7bT1oL8T0w ''Joyous Stellaris''] (2023) – [[semiquartal]] in 97edo tuning |
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| | ; [[Mercury Amalgam]] |
| | * [https://www.youtube.com/watch?v=3JwH0gZmXHk ''Thanatonautical Tetrapharmacon''] (2023) |
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| | == 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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| [[Category:Listen]] | | [[Category:Listen]] |