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| {{Infobox ET}} | | {{Infobox ET}} |
| {{EDO intro|93}} | | {{ED intro}} |
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| == Theory == | | == Theory == |
| | Since {{nowrap|93 {{=}} 3 × 31}}, 93edo is a [[contorted]] [[31edo]] through the [[7-limit]]. In the 11-limit the [[patent val]] [[tempering out|tempers out]] [[4000/3993]] and in the 13-limit [[144/143]], [[1188/1183]], and [[364/363]]. It provides the [[optimal patent val]] for the 11-limit [[31st-octave_temperaments#Prajapati|prajapati]] and 13-limit [[31st-octave_temperaments#Kumhar|kumhar]] temperaments and the 11- and 13-limit [[Meantone family#Trimean|trimean]] ({{nowrap|43 & 50}}) temperament, and is the 13th no-3s [[zeta peak edo]]. The 93bd val is close to the 9-odd limit minimax tuning for [[superpyth]] and approximates {{nowrap|{{frac|2|7}}-[[64/63|septimal comma]]}} superpyth very well. |
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| | Since 93edo has good approximations of [[13/1|13th]], [[17/1|17th]] and [[19/1|19th]] [[harmonic]]s unlike 31edo (as 838.710{{c}}, 103.226{{c}}, and 296.774{{c}} respectively, [[octave-reduced]]), it also allows one to give a clearer harmonic identity to [[31edo]]'s excellent approximation of 13:17:19. |
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| | === Odd harmonics === |
| {{Harmonics in equal|93}} | | {{Harmonics in equal|93}} |
| 93 = 3 * 31, and 93 is a [[contorted]] 31 through the 7 limit. In the 11-limit the patent val tempers out 4000/3993 and in the 13-limit 144/143, 1188/1183 and 364/363. It provides the optimal patent val for the 11-limit prajapati and 13-limit kumhar temperaments, and the 11 and 13 limit 43&50 temperament. It is the 13th no-3s zeta peak edo.
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| Since 93edo has good approximations of 13th, 17th and 19th harmonics unlike 31edo (as 838.710{{cent}}, 103.226{{cent}}, and 296.774{{cent}} respectively, [[octave-reduced]]), it also allows one to give a clearer harmonic identity to [[31edo]]'s excellent approximation of 13:17:19.
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| == Temperament properties == | | === No-3 approach === |
| Since 93edo has a step of 12.903{{cent}}, it also allows one to use its MOS scales as circulating temperaments, which it is the first edo to do. It is also the first edo to allow one to use a syntonic or Mavila MOS scale or a 17 tone MOS scale similar to a median between [http://www.neuroscience-of-music.se/pelog_main.htm Pelog] and the theories of Sundanese composer-musicologist-teacher [https://en.wikipedia.org/wiki/Raden_Machjar_Angga_Koesoemadinata Raden Machjar Angga Koesoemadinata] as a circulating temperament.
| | If prime 3 is ignored, 93edo represents the no-3 35-odd-limit consistently. 93edo is distinctly consistent within the no-3 19-integer-limit. |
| {| class="wikitable"
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| |+Circulating temperaments in 93edo
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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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| |[[3L 2s]]
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| |19:18
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| |-
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| |6
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| |[[3L 3s]]
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| |16:15
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| |-
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| |7
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| |[[2L 5s]]
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| |14:13
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| |-
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| |8
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| |[[5L 3s]]
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| |12:11
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| |-
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| |9
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| |[[3L 6s]]
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| |11:10
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| |-
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| |10
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| |[[3L 7s]]
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| |10:9
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| |-
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| |11
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| |[[5L 6s]]
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| |9:8
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| |-
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| |12
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| |[[9L 3s]]
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| | rowspan="2" |8:7
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| |-
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| |13
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| |[[2L 11s]]
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| |-
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| |14
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| |[[9L 5s]]
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| | rowspan="2" |7:6
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| |-
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| |15
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| |[[3L 12s]]
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| |-
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| |16
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| |13L 3s
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| | rowspan="3" |6:5
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| |-
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| |17
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| |[[8L 9s]]
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| |-
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| |18
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| |3L 15s
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| |-
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| |19
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| |[[17L 2s]]
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| | rowspan="5" |5:4
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| |-
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| |20
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| |[[13L 7s]]
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| |-
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| |21
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| |9L 12s
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| |-
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| |22
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| |[[5L 17s]]
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| |-
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| |23
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| |1L 22s
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| |-
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| |24
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| |21L 3s
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| | rowspan="7" |4:3
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| |-
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| |25
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| |18L 7s
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| |-
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| |26
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| |15L 11s
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| |-
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| |27
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| |12L 15s
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| |-
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| |28
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| |9L 19s
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| |-
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| |29
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| |6L 23s
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| |-
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| |30
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| |3L 27s
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| |-
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| |31
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| |[[31edo]]
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| |equal
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| |-
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| |32
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| |29L 3s
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| | rowspan="15" |3:2
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| |-
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| |33
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| |27L 6s
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| |-
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| |34
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| |25L 9s
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| |-
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| |35
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| |23L 12s
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| |-
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| |36
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| |21L 15s
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| |-
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| |37
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| |19L 18s
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| |-
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| |38
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| |17L 21s
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| |-
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| |39
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| |15L 24s
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| |-
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| |40
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| |13L 27s
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| |-
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| |41
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| |12L 29s
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| |-
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| |42
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| |9L 33s
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| |-
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| |43
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| |7L 36s
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| |-
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| |44
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| |5L 39s
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| |-
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| |45
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| |3L 42s
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| |-
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| |46
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| |1L 45s
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| |-
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| |47
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| |46L 1s
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| | rowspan="28" |2:1
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| |-
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| |48
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| |45L 3s
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| |-
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| |49
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| |44L 5s
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| |-
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| |50
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| |43L 7s
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| |-
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| |51
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| |42L 9s
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| |-
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| |52
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| |41L 11s
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| |-
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| |53
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| |40L 13s
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| |-
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| |54
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| |39L 15s
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| |-
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| |55
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| |38L 17s
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| |-
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| |56
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| |37L 19s
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| |-
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| |57
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| |36L 21s
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| |-
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| |58
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| |35L 23s
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| |-
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| |59
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| |34L 25s
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| |-
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| |60
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| |33L 27s
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| |-
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| |61
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| |32L 29s
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| |-
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| |62
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| |31L 31s
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| |-
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| |63
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| |30L 33s
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| |-
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| |64
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| |29L 35s
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| |-
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| |65
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| |28L 37s
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| |-
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| |66
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| |27L 39s
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| |-
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| |67
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| |26L 41s
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| |-
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| |68
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| |25L 43s
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| |-
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| |69
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| |24L 45s
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| |-
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| |70
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| |23L 47s
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| |-
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| |71
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| |22L 49s
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| |-
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| |72
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| |21L 51s
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| |-
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| |73
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| |20L 53s
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| |-
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| |74
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| |19L 55s
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| |}
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| [[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
| | == Intervals == |
| | {{Interval table}} |
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| == Scales == | | == Scales == |
| Shailaja
| | * Superpyth[5]: 21 17 17 21 17 ((21 38 55 76 93)\93) |
| | | * Superpyth[12]: 4 13 4 13 4 13 4 4 13 4 13 4 ((4 17 21 34 38 51 55 59 72 76 89 93)\93) |
| * 270.968 | | * Superpyth Shailaja: 21 34 4 17 17 ((21 55 59 76 93)\93) |
| * 709.677 | | * Superpyth Subminor Hexatonic: 17 4 17 17 21 17 ((17 21 38 55 76 93)\93) |
| * 761.290 | |
| * 980.645
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| * 1200.000 | |
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| Subminor Hexatonic | |
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| * 219.355
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| * 270.968
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| * 490.323
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| * 709.677
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| * 980.645
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| * 1200.000
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| Subminor Pentatonic
| | == Instruments == |
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| * 270.968
| | A [[Lumatone mapping for 93edo]] is available. |
| * 490.323
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| * 709.677
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| * 980.645
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| * 1200.000
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| == See Also == | | == Music == |
| | ; [[Bryan Deister]] |
| | * [https://www.youtube.com/shorts/eknKeDeRlQs ''microtonal improvisation in 93edo''] (2025) |
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| | == See also == |
| * [[93edo and stretched hemififths]] | | * [[93edo and stretched hemififths]] |