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| {{Infobox ET}} | | {{Infobox ET}} |
| '''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]].
| | {{EDO intro|116}} |
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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 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]]. |
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| {| class="wikitable"
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| |+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
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| |[[1L 4s]]
| |
| |24:23
| |
| |-
| |
| |6
| |
| |[[2L 4s]]
| |
| |20:19
| |
| |-
| |
| |7
| |
| |[[4L 3s]]
| |
| |17:16
| |
| |-
| |
| |8
| |
| |[[4L 4s]]
| |
| |15:14
| |
| |-
| |
| |9
| |
| |[[8L 1s]]
| |
| |13:12
| |
| |-
| |
| |10
| |
| |[[6L 4s]]
| |
| |12:11
| |
| |-
| |
| |11
| |
| |[[6L 5s]]
| |
| |11:10
| |
| |-
| |
| |12
| |
| |[[8L 4s]]
| |
| |10:9
| |
| |-
| |
| |13
| |
| |[[12L 1s]]
| |
| | rowspan="2" |9:8
| |
| |-
| |
| |14
| |
| |[[4L 10s]]
| |
| |-
| |
| |15 | |
| |[[11L 4s]]
| |
| | rowspan="2" |8:7
| |
| |-
| |
| |16
| |
| |4L 12s
| |
| |-
| |
| |17
| |
| |[[14L 3s]]
| |
| | rowspan="3" |7:6
| |
| |-
| |
| |18
| |
| |8L 10s
| |
| |-
| |
| |19
| |
| |[[2L 17s]]
| |
| |-
| |
| |20
| |
| |16L 4s
| |
| | rowspan="4" |6:5
| |
| |-
| |
| |21
| |
| |11L 10s
| |
| |-
| |
| |22
| |
| |[[6L 16s]]
| |
| |-
| |
| |23
| |
| |1L 22s
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| |-
| |
| |24
| |
| |20L 4s
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| | rowspan="5" |5:4
| |
| |-
| |
| |25
| |
| |16L 9s
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| |-
| |
| |26
| |
| |12L 14s
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| |-
| |
| |27
| |
| |8L 19s
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| |-
| |
| |28
| |
| |4L 24s
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| |-
| |
| |29
| |
| |[[29edo]]
| |
| |equal
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| |-
| |
| |30
| |
| |26L 4s
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| | rowspan="9" |4:3
| |
| |-
| |
| |31
| |
| |23L 8s
| |
| |-
| |
| |32
| |
| |20L 12s
| |
| |-
| |
| |33
| |
| |17L 16s
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| |-
| |
| |34
| |
| |14L 20s
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| |-
| |
| |35
| |
| |11L 24s
| |
| |-
| |
| |36
| |
| |8L 28s
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| |-
| |
| |37
| |
| |5L 32s
| |
| |-
| |
| |38
| |
| |2L 36s
| |
| |-
| |
| |39
| |
| |38L 1s
| |
| | rowspan="19" |3:2
| |
| |-
| |
| |40
| |
| |36L 4s
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| |-
| |
| |41
| |
| |34L 7s
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| |-
| |
| |42
| |
| |32L 10s
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| |-
| |
| |43
| |
| |30L 13s
| |
| |-
| |
| |44
| |
| |28L 16s
| |
| |-
| |
| |45
| |
| |26L 19s
| |
| |-
| |
| |46
| |
| |24L 22s
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| |-
| |
| |47
| |
| |22L 25s
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| |-
| |
| |48
| |
| |20L 28s
| |
| |-
| |
| |49
| |
| |18L 31s
| |
| |-
| |
| |50
| |
| |16L 34s
| |
| |-
| |
| |51
| |
| |14L 37s
| |
| |-
| |
| |52
| |
| |12L 40s
| |
| |-
| |
| |53
| |
| |10L 43s
| |
| |-
| |
| |54
| |
| |8L 46s
| |
| |-
| |
| |55
| |
| |6L 49s
| |
| |-
| |
| |56
| |
| |4L 52s
| |
| |-
| |
| |57
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| |2L 55s
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| |-
| |
| |58
| |
| |[[58edo]]
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| |equal
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| |-
| |
| |59
| |
| |57L 2s
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| | rowspan="34" |2:1
| |
| |-
| |
| |60
| |
| |56L 4s
| |
| |-
| |
| |61
| |
| |55L 6s
| |
| |-
| |
| |62
| |
| |54L 8s
| |
| |-
| |
| |63
| |
| |53L 10s
| |
| |-
| |
| |64
| |
| |52L 12s
| |
| |-
| |
| |65
| |
| |51L 14s
| |
| |-
| |
| |66
| |
| |50L 16s
| |
| |-
| |
| |67
| |
| |49L 18s
| |
| |-
| |
| |68
| |
| |48L 20s
| |
| |-
| |
| |69
| |
| |47L 22s
| |
| |-
| |
| |70
| |
| |46L 24s
| |
| |-
| |
| |71
| |
| |45L 26s
| |
| |-
| |
| |72
| |
| |44L 28s
| |
| |-
| |
| |73
| |
| |43L 30s
| |
| |-
| |
| |74
| |
| |42L 32s
| |
| |-
| |
| |75
| |
| |41L 34s
| |
| |-
| |
| |76
| |
| |40L 36s
| |
| |-
| |
| |77
| |
| |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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| | {{Harmonics in equal|116}} |
| [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | | [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> |
Revision as of 21:19, 30 May 2023
| Prime factorization
|
22 × 29
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| Step size
|
10.3448 ¢
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| Fifth
|
68\116 (703.448 ¢) (→ 17\29)
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| Semitones (A1:m2)
|
12:8 (124.1 ¢ : 82.76 ¢)
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| Consistency limit
|
5
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| Distinct consistency limit
|
5
|
Template:EDO intro
116edo 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 submajor temperament.
Approximation of prime harmonics in 116edo
| Harmonic
|
2
|
3
|
5
|
7
|
11
|
13
|
17
|
19
|
23
|
29
|
31
|
| Error
|
Absolute (¢)
|
+0.00
|
+1.49
|
-3.56
|
+3.59
|
-3.04
|
-2.60
|
-1.51
|
+2.49
|
+2.76
|
+4.91
|
+3.24
|
| Relative (%)
|
+0.0
|
+14.4
|
-34.4
|
+34.7
|
-29.4
|
-25.1
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-14.6
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+24.0
|
+26.7
|
+47.4
|
+31.3
|
Steps
(reduced)
|
116
(0)
|
184
(68)
|
269
(37)
|
326
(94)
|
401
(53)
|
429
(81)
|
474
(10)
|
493
(29)
|
525
(61)
|
564
(100)
|
575
(111)
|