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| '''116edo''' is the [[EDO|equal division of the octave]] into 116 parts of 10.3448 cents each. It tempers out 20000/19683 (tetracot comma) and 2197265625/2147483648 (wizard comma) in the 5-limit. Using the patent val, it tempers out 225/224, 15625/15309, and 51200/50421 in the 7-limit; 385/384, 540/539, 4000/3993, and 6655/6561 in the 11-limit; 169/168, 275/273, 352/351, and 640/637 in the 13-limit. 116edo provides the optimal patent val for [[Marvel temperaments|submajor temperament]].
| | {{Infobox ET}} |
| | {{ED intro}} |
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| Since 116edo has a step of 10.3448 cents, it also allows one to use its MOS scales as circulating temperaments.
| | 116edo is only [[consistent]] to the [[5-odd-limit]], and is not quite accurate for its size. It can be viewed as splitting [[58edo]]'s step in two, and the [[enfactoring|enfactored]] 116cef [[val]] comes out on top accuracy in the 7-, 11-, and 13-limit. In the 5-limit, however, the [[patent val]] {{val| 116 184 '''269''' }} beats the enfactored 116c val {{val| 116 184 '''270''' }} by a thin margin, and it [[Tempering out|tempers out]] 20000/19683 ([[tetracot comma]]) and 2197265625/2147483648 (wizard comma). |
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| {| class="wikitable"
| | In the 7-, 11- and 13-limit, the patent val {{val| 116 184 '''269''' 326 '''401''' '''429''' }} comes in second best after the enfactored 116cef val {{val| 116 184 '''270''' 326 '''402''' '''430''' }} , and it tempers out [[225/224]], 15625/15309, and 51200/50421 in the 7-limit; [[385/384]], [[540/539]], [[4000/3993]], and 6655/6561 in the 11-limit; [[169/168]], [[275/273]], [[352/351]], and [[640/637]] in the 13-limit. 116edo provides the [[optimal patent val]] for the [[submajor (temperament)|submajor]] temperament in the 11- and 13-limit. |
| |+Circulating temperaments in 116edo
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| !Tones
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| !Pattern
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| !L:s
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| |-
| |
| |5
| |
| |[[1L 4s]]
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| |24:23
| |
| |-
| |
| |6
| |
| |[[2L 4s]]
| |
| |20:19
| |
| |-
| |
| |7
| |
| |[[4L 3s]]
| |
| |17:16
| |
| |-
| |
| |8
| |
| |[[4L 4s]]
| |
| |15:14
| |
| |-
| |
| |9
| |
| |[[8L 1s]]
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| |13:12
| |
| |-
| |
| |10 | |
| |[[6L 4s]] | |
| |12:11
| |
| |-
| |
| |11
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| |[[6L 5s]]
| |
| |11:10
| |
| |-
| |
| |12
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| |[[8L 4s]]
| |
| |10:9
| |
| |-
| |
| |13
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| |[[12L 1s]]
| |
| | rowspan="2" |9:8
| |
| |-
| |
| |14
| |
| |[[4L 10s]]
| |
| |-
| |
| |15
| |
| |[[11L 4s]]
| |
| | rowspan="2" |8:7
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| |-
| |
| |16
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| |4L 12s
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| |-
| |
| |17
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| |[[14L 3s]]
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| | rowspan="3" |7:6
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| |-
| |
| |18
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| |8L 10s
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| |-
| |
| |19
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| |[[2L 17s]]
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| |-
| |
| |20
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| |16L 4s
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| | rowspan="4" |6:5
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| |-
| |
| |21
| |
| |11L 10s
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| |-
| |
| |22
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| |[[6L 16s]]
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| |-
| |
| |23
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| |1L 22s
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| |-
| |
| |24
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| |20L 4s
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| | rowspan="5" |5:4
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| |-
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| |25
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| |16L 9s
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| |-
| |
| |26
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| |12L 14s
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| |-
| |
| |27
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| |8L 19s
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| |-
| |
| |28
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| |4L 24s
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| |-
| |
| |29
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| |[[29edo]]
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| |equal
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| |-
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| |30
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| |26L 4s
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| | rowspan="9" |4:3
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| |-
| |
| |31
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| |23L 8s
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| |-
| |
| |32
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| |20L 12s
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| |-
| |
| |33
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| |17L 16s
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| |-
| |
| |34
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| |14L 20s
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| |-
| |
| |35
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| |11L 24s
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| |-
| |
| |36
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| |8L 28s
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| |-
| |
| |37
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| |5L 32s
| |
| |-
| |
| |38
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| |2L 36s
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| |-
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| |39
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| |38L 1s
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| | rowspan="19" |3:2
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| |-
| |
| |40
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| |36L 4s
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| |-
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| |41
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| |34L 7s
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| |-
| |
| |42
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| |32L 10s
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| |-
| |
| |43
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| |30L 13s
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| |-
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| |44
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| |28L 16s
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| |-
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| |45
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| |26L 19s
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| |-
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| |46
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| |24L 22s
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| |-
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| |47
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| |22L 25s
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| |-
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| |48
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| |20L 28s
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| |-
| |
| |49
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| |18L 31s
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| |-
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| |50
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| |16L 34s
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| |-
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| |51
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| |14L 37s
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| |-
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| |52
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| |12L 40s
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| |-
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| |53
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| |10L 43s
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| |-
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| |54
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| |8L 46s
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| |-
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| |55
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| |6L 49s
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| |-
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| |56
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| |4L 52s
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| |-
| |
| |57 | |
| |2L 55s
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| |-
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| |58
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| |[[58edo]]
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| |equal
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| |-
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| |59
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| |57L 2s
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| | rowspan="34" |2:1
| |
| |-
| |
| |60
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| |56L 4s
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| |-
| |
| |61
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| |55L 6s
| |
| |-
| |
| |62
| |
| |54L 8s
| |
| |-
| |
| |63
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| |53L 10s
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| |-
| |
| |64
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| |52L 12s
| |
| |-
| |
| |65
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| |51L 14s
| |
| |-
| |
| |66
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| |50L 16s
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| |-
| |
| |67
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| |49L 18s
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| |-
| |
| |68
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| |48L 20s
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| |-
| |
| |69
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| |47L 22s
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| |-
| |
| |70
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| |46L 24s
| |
| |-
| |
| |71
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| |45L 26s
| |
| |-
| |
| |72
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| |44L 28s
| |
| |-
| |
| |73
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| |43L 30s
| |
| |-
| |
| |74
| |
| |42L 32s
| |
| |-
| |
| |75
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| |41L 34s
| |
| |-
| |
| |76
| |
| |40L 36s
| |
| |-
| |
| |77
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| |39L 38s
| |
| |-
| |
| |78
| |
| |38L 40s
| |
| |-
| |
| |79
| |
| |37L 42s
| |
| |-
| |
| |80
| |
| |36L 44s
| |
| |-
| |
| |81
| |
| |35L 46s
| |
| |-
| |
| |82
| |
| |34L 48s
| |
| |-
| |
| |83
| |
| |33L 50s
| |
| |-
| |
| |84
| |
| |32L 52s
| |
| |-
| |
| |85
| |
| |31L 54s
| |
| |-
| |
| |86
| |
| |30L 56s
| |
| |-
| |
| |87
| |
| |29L 58s
| |
| |-
| |
| |88
| |
| |28L 60s
| |
| |-
| |
| |89
| |
| |27L 62s
| |
| |-
| |
| |90
| |
| |26L 64s
| |
| |-
| |
| |91
| |
| |25L 66s
| |
| |-
| |
| |92
| |
| |24L 68s
| |
| |}
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| [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | | === Prime harmonics === |
| | {{Harmonics in equal|116}} |
| | |
| | === Subsets and supersets === |
| | Since 116 factors into {{factorisation|116}}, 116edo has subset edos {{EDOs| 2, 4, 29, and 58 }}. [[232edo]], which doubles it, is a notable tuning. |
| | |
| | == Intervals == |
| | {{Interval table}} |
| | |
| | [[Category:Submajor (temperament)]] |