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'''114edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 114 parts, each of 10.52632 [[cent]]s. In the [[5-limit|5-limit]] it [[tempering_out|tempers out]] 2048/2025, in the [[7-limit|7-limit]] 245/243, in the [[11-limit|11-limit]] 121/120, 176/175 and [[Quartisma|117440512/117406179]], in the [[13-limit|13-limit]] 196/195 and 325/324, in the [[17-limit|17-limit]] 136/135 and 154/153, in the [[19-limit|19-limit]] 286/285 and 343/342. These commas make for 114edo being an excellent tuning for [[Diaschismic_family|shrutar temperament]]; it is in fact the [[Optimal_patent_val|optimal patent val]] for [[Shrutar|shrutar]] in the 11- 13- 17- and 19-limit, as well as the rank three bisector temperament.
{{Infobox ET}}
{{ED intro}}


===Period of 19-limit Shrutar===
== Theory ==
In the [[5-limit]] the equal temperament [[tempering out|tempers out]] 2048/2025 ([[diaschisma]]), in the [[7-limit]] [[245/243]], in the [[11-limit]] [[121/120]], [[176/175]] and notably the [[quartisma]], in the [[13-limit]] [[196/195]] and [[325/324]], in the [[17-limit]] [[136/135]] and [[154/153]], in the [[19-limit]] [[286/285]] and [[343/342]]. These commas make for 114edo being an excellent tuning for the [[shrutar]] temperament; it is in fact the [[optimal patent val]] for shrutar in the 11-, 13-, 17-, and 19-limit, as well as the rank-3 [[bisector]] temperament.


{| class="wikitable"
=== Odd harmonics ===
|-
{{Harmonics in equal|114}}
! |Degree
 
! |Cents
=== Subsets and supersets ===
!Difference from 68edo
Since 114 factors into {{factorization|114}}, 114 edo has subset edos {{EDOs| 2, 3, 6, 19, 38, and 57 }}.
|-
 
| |2
== Intervals ==
| |21.05263
{{Interval table}}
|3.40557¢
 
|-
[[Category:Shrutar]]
| |3
[[Category:Bisector]]
| |31.57895
| -3.71517¢
|-
| | 5
| | 52.63158
| -0.3096¢
|-
| |7
| |73.68421
|3.096¢
|-
| |8
| |84.21053
| -4.02477¢
|-
| |10
| |105.26316
|  -0.619195¢
|-
| |12
| | 126.31579
|2.78638¢
|-
| |13
| |136.842105
|  -4.334365¢
|-
| |15
| |157.89474
| -0.9288¢
|-
| | 17
| |178.94737
|2.47678¢
|-
| |18
| | 189.47369
| -4.644¢
|-
| |20
| |210.52632
| -1.23839¢
|-
| |22
| |231.57895
|2.16718¢
|-
| |23
| |242.10526
|  -4.953560372
|-
| |25
| |263.157895
| -1.548¢
|-
| |27
| |284.21053
|1.857585¢
|-
| |29
| |305.26316
|5.26316¢
|-
| |30
| |315.78947
| -1.857585¢
|-
| |32
| |336.842105
|1.548¢
|-
| |34
| |357.89474
|4.95356¢
|-
| |35
| | 368.42105
| -2.16718¢
|-
| |37
| |389.47368
|1.23839¢
|-
| |39
| |410.52632
|4.64396¢
|-
| |40
| |421.05263
| -2.47678¢
|-
| |42
| |442.10526
|0.92879¢
|-
| |44
| |463.157895
|4.334365¢
|-
| |45
| |473.68421
| -2.78638¢
|-
| |47
| |494.73684
|0.619195¢
|-
| |49
| |515.78947
|4.02477¢
|-
| |50
| |526.31579
| -3.095975¢
|-
| |52
| |547.36842
|0.3096¢
|-
| |54
| |568.42105
|3.71517¢
|-
| |55
| |578.94737
| -3.40557¢
|}
Since 114edo has a step of 10.52632 cents, it also allows one to use its MOS scales as circulating temperaments.
{| class="wikitable"
|+Circulating temperaments in 114edo
!Tones
!Pattern
!L:s
|-
|5
|[[4L 1s]]
|23:22
|-
|6
|[[6edo]]
|equal
|-
|7
|[[2L 5s]]
|17:16
|-
|8
|[[2L 6s]]
|15:14
|-
|9
|[[6L 3s]]
|13:12
|-
|10
|[[4L 6s]]
|12:11
|-
|11
|[[4L 7s]]
|11:10
|-
|12
|[[6L 6s]]
|10:9
|-
|13
|[[10L 3s]]
| rowspan="2" |9:8
|-
|14
|[[2L 12s]]
|-
|15
|[[9L 6s]]
| rowspan="2" |8:7
|-
|16
|2L 14s
|-
|17
|[[12L 5s]]
| rowspan="2" |7:6
|-
|18
|6L 12s
|-
|19
|[[19edo]]
|equal
|-
|20
|14L 6s
| rowspan="3" |6:5
|-
|21
|9L 12s
|-
|22
|4L 18s
|-
|23
|22L 1s
| rowspan="6" |5:4
|-
|24
|18L 6s
|-
|25
|14L 11s
|-
|26
|10L 16s
|-
|27
|6L 21s
|-
|28
|2L 26s
|-
|29
|27L 2s
| rowspan="9" |4:3
|-
|30
|24L 6s
|-
|31
|21L 10s
|-
|32
|18L 14s
|-
|33
|15L 18s
|-
|34
|12L 22s
|-
|35
|9L 26s
|-
|36
|6L 30s
|-
|37
|3L 34s
|-
|38
|[[38edo]]
|equal
|-
|39
|36L 3s
| rowspan="18" |3:2
|-
|40
|34L 6s
|-
|41
|32L 9s
|-
|42
|30L 12s
|-
|43
|28L 15s
|-
|44
|26L 18s
|-
|45
|24L 21s
|-
|46
|22L 24s
|-
|47
|20L 27s
|-
|48
|18L 30L
|-
|49
|16L 33s
|-
|50
|14L 36s
|-
|51
|12L 39s
|-
|52
|10L 42s
|-
|53
|8L 45s
|-
|54
|6L 48s
|-
|55
|4L 52s
|-
|56
|2L 54s
|-
|57
|[[57edo]]
|equal
|-
|58
|56L 2s
| rowspan="34" |2:1
|-
|59
|55L 4s
|-
|60
|54L 6s
|-
|61
|53L 8s
|-
|62
|52L 10s
|-
|63
|51L 12s
|-
|64
|50L 14s
|-
|65
|49L 16s
|-
|66
|48L 18s
|-
|67
|47L 20s
|-
|68
|46L 22s
|-
|69
|45L 24s
|-
|70
|44L 26s
|-
|71
|43L 28s
|-
|72
|42L 30s
|-
|73
|41L 32s
|-
|74
|40L 34s
|-
|75
|39L 36s
|-
|76
|38L 38s
|-
|77
|37L 40s
|-
|78
|36L 42s
|-
|79
|35L 44s
|-
|80
|34L 46s
|-
|81
|33L 48s
|-
|82
|32L 50s
|-
|83
|31L 52s
|-
|84
|30L 54s
|-
|85
|29L 56s
|-
|86
|28L 58s
|-
|87
|27L 60s
|-
|88
|26L 62s
|-
|89
|25L 64s
|-
|90
|24L 66s
|-
|91
|23L 68s
|}
[[Category:Equal divisions of the octave]]
[[Category:shrutar]]
[[Category:theory]]
Retrieved from "https://en.xen.wiki/w/114edo"