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114edo: Difference between revisions

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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|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.
'''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.


===Period of 19-limit Shrutar===
===Period of 19-limit Shrutar===
Line 5: Line 5:
{| class="wikitable"
{| class="wikitable"
|-
|-
! | Degree
! |Degree
! | Cents
! |Cents
!Difference from 68edo
|-
|-
| | 2
| |2
| | 21.05263
| |21.05263
|3.40557¢
|-
|-
| | 3
| |3
| | 31.57895
| |31.57895
| -3.71517¢
|-
|-
| | 5
| | 5
| | 52.63158
| | 52.63158
| -0.3096¢
|-
|-
| | 7
| |7
| | 73.68421
| |73.68421
|3.096¢
|-
|-
| | 8
| |8
| | 84.21053
| |84.21053
| -4.02477¢
|-
|-
| | 10
| |10
| | 105.26316
| |105.26316
|  -0.619195¢
|-
|-
| | 12
| |12
| | 126.31579
| | 126.31579
|2.78638¢
|-
|-
| | 13
| |13
| | 136.842105
| |136.842105
|  -4.334365¢
|-
|-
| | 15
| |15
| | 157.89474
| |157.89474
| -0.9288¢
|-
|-
| | 17
| | 17
| | 178.94737
| |178.94737
|2.47678¢
|-
|-
| | 18
| |18
| | 189.47369
| | 189.47369
| -4.644¢
|-
|-
| | 20
| |20
| | 210.52632
| |210.52632
| -1.23839¢
|-
|-
| | 22
| |22
| | 231.57895
| |231.57895
|2.16718¢
|-
|-
| | 23
| |23
| | 242.10526
| |242.10526
|  -4.953560372
|-
|-
| | 25
| |25
| | 263.157895
| |263.157895
| -1.548¢
|-
|-
| | 27
| |27
| | 284.21053
| |284.21053
|1.857585¢
|-
|-
| | 29
| |29
| | 305.26316
| |305.26316
|5.26316¢
|-
|-
| | 30
| |30
| | 315.78947
| |315.78947
| -1.857585¢
|-
|-
| | 32
| |32
| | 336.842105
| |336.842105
|1.548¢
|-
|-
| | 34
| |34
| | 357.89474
| |357.89474
|4.95356¢
|-
|-
| | 35
| |35
| | 368.42105
| | 368.42105
| -2.16718¢
|-
|-
| | 37
| |37
| | 389.47368
| |389.47368
|1.23839¢
|-
|-
| | 39
| |39
| | 410.52632
| |410.52632
|4.64396¢
|-
|-
| | 40
| |40
| | 421.05263
| |421.05263
| -2.47678¢
|-
|-
| | 42
| |42
| | 442.10526
| |442.10526
|0.92879¢
|-
|-
| | 44
| |44
| | 463.157895
| |463.157895
|4.334365¢
|-
|-
| | 45
| |45
| | 473.68421
| |473.68421
| -2.78638¢
|-
|-
| | 47
| |47
| | 494.73684
| |494.73684
|0.619195¢
|-
|-
| | 49
| |49
| | 515.78947
| |515.78947
|4.02477¢
|-
|-
| | 50
| |50
| | 526.31579
| |526.31579
| -3.095975¢
|-
|-
| | 52
| |52
| | 547.36842
| |547.36842
|0.3096¢
|-
|-
| | 54
| |54
| | 568.42105
| |568.42105
|3.71517¢
|-
|-
| | 55
| |55
| | 578.94737
| |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. It is the first edo which allows one to use an MOS scale of 90 tones or more as a circulating temperament.
{| 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:Equal divisions of the octave]]
[[Category:shrutar]]
[[Category:shrutar]]
[[Category:theory]]
[[Category:theory]]
Retrieved from "https://en.xen.wiki/w/114edo"