Xenharmonic Wiki

103edo: Difference between revisions

← Older edit
VisualWikitext
No edit summary
m Update link
 
(41 intermediate revisions by 10 users not shown)
Line 1: Line 1:
'''103edo''' is the [[EDO|equal division of the octave]] into 103 steps of size 11.6505 [[cent]]s each. It is a good [[Gamelismic_clan|miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limit|13-]] limits and [[Gamelismic_clan|benediction]] and [[Gamelismic_clan|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering_out|tempers out]] 78732/78125 in the [[5-limit]]; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the [[11-limit]]; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal patent val for [[Marvel_family#Marvel|marvel temperament]] as well as [[Gamelismic_clan#Miracle|benediction]] and [[Gamelismic_clan#Hemisecordite|hemisecordite]].
{{Infobox ET}}
{{ED intro}}


103edo is the 27th [[prime_numbers|prime]] edo.
== Theory ==
{{Primes in edo|103}}
In 103edo, all intervals within the [[17-odd-limit]] are [[consistent]], with the sole exception of [[9/8]] and its octave complement [[16/9]], which barely miss (relative error 50.2%). Its closest [[zeta peak index]], [[596zpi]], [[stretched and compressed tuning|stretches the octave]] by +0.739 cents. This expansion is uniquely consistent within the 15-integer-limit.


===13-limit temperaments===
103edo is a good [[miracle]] tuning, especially for the [[7-limit]], and for [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]], and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]], and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.


{| class="wikitable"
=== Prime harmonics ===
|-
{{Harmonics in equal|103|intervals=prime}}
| | Degree
 
| | cents
=== Subsets and supersets ===
| | temperament
103edo is the 27th [[prime edo]], following [[101edo]] and before [[107edo]].
|-
 
| | 1
== Intervals ==
| | 11.650
{{Main|Table of 103edo intervals}}
| | Marvel and Benediction
 
|-
== Approximation to JI ==
| | 2
=== Interval mappings ===
| | 23.301
{{Q-odd-limit intervals}}
| | Hemisecordite
 
|-
=== Zeta peak index ===
| | 3
{{ZPI
| | 34.951
| zpi = 596
| | Marvel and Benediction, Hemisecordite
| steps = 102.936629522070
|-
| step size = 11.6576577800491
| | 4
| tempered height = 8.543510
| | 46.602
| pure height = 5.620365
| | Marvel and Benediction
| integral = 1.340775
|-
| gap = 18.270998
| | 5
| octave = 1200.73875134506
| | 58.252
| consistent = 15
| | Hemisecordite
| distinct = 15
|-
}}
| | 6
 
| | 69.903
== Regular temperament properties ==
| | Marvel and Benediction
{| class="wikitable center-4 center-5 center-6"
|-
| | 7
| | 81.553
| | Marvel and Benediction, Hemisecordite
|-
| | 8
| | 93.204
| | Hemisecordite
|-
| | 9
| | 104.854
| | Marvel and Benediction
|-
| | 10
| | 116.505
| | Marvel and Benediction, Hemisecordite
|-
| | 11
| | 128.155
| | Marvel and Benediction
|-
| | 12
| | 139.806
| | Hemisecordite
|-
| | 13
| | 151.456
| | Marvel and Benediction, Hemisecordite
|-
| | 14
| | 163.107
| | Marvel and Benediction
|-
| | 15
| | 174.757
| | Hemisecordite
|-
| | 16
| | 186.408
| | Marvel and Benediction
|-
| | 17
| | 198.058
| | Marvel and Benediction, Hemisecordite
|-
| | 18
| | 209.708
| | Hemisecordite
|-
| | 19
| | 221.359
| | Marvel and Benediction
|-
| | 20
| | 233.010
| | Marvel and Benediction, Hemisecordite
|-
| | 21
| | 244.660
| | Marvel and Benediction
|-
| | 22
| | 256.311
| | Hemisecordite
|-
| | 23
| | 267.961
| | Marvel and Benediction, Hemisecordite
|-
| | 24
| | 279.712
| | Marvel and Benediction
|-
| | 25
| | 291.262
| | Hemisecordite
|-
| | 26
| | 303.013
| | Marvel and Benediction
|-
| | 27
| | 314.563
| | Marvel and Benediction, Hemisecordite
|-
| | 28
| | 326.214
| | Hemisecordite
|-
| | 29
| | 337.864
| | Marvel and Benediction
|-
| | 30
| | 349.615
| | Marvel and Benediction, Hemisecordite
|-
| | 31
| | 361.165
| | Marvel and Benediction
|-
| | 32
| | 372.816
| | Hemisecordite
|-
| | 33
| | 384.466
| | Marvel and Benediction, Hemisecordite
|-
| | 34
| | 396.117
| | Marvel and Benediction
|-
| | 35
| | 407.767
| | Hemisecordite
|-
| | 36
| | 419.4175
| | Marvel and Benediction
|-
| | 37
| | 431.068
| | Marvel and Benediction, Hemisecordite
|-
| | 38
| | 442.708
| | Hemisecordite
|-
| | 39
| | 454.369
| | Marvel and Benediction
|-
| | 40
| | 466.019
| | Marvel and Benediction, Hemisecordite
|-
| | 41
| | 477.670
| | Marvel and Benediction
|-
| | 42
| | 489.320
| | Hemisecordite
|-
| | 43
| | 500.971
| | Marvel and Benediction, Hemisecordite
|-
| | 44
| | 512.621
| | Marvel and Benediction
|-
| | 45
| | 524.272
| | Hemisecordite
|-
| | 46
| | 535.922
| | Marvel and Benediction
|-
| | 47
| | 547.573
| | Marevel and Benediction, Hemisecordite
|-
| | 48
| | 559.223
| | Hemisecordite
|-
| | 49
| | 570.874
| | Marvel and Benediction
|-
| | 50
| | 582.524
| | Marvel and Benediction, Hemisecordite
|-
| | 52
| | 605.825
| | Marvel and Benediction, Hemisecordite
|-
| | 53
| | 617.476
| | Marvel and Benediction, Hemisecordite
|-
| | 54
| | 629.126
| | Marvel and Benediction
|-
| | 55
| | 640.777
| | Hemisecordite
|-
| | 56
| | 652.427
| | Marvel and Benediction, Hemisecordite
|-
| | 57
| | 664.078
| | Marvel and Benediction
|-
| | 58
| | 675.728
| | Hemisecordite
|-
| | 59
| | 687.379
| | Marvel and Benediction
|-
| | 60
| | 699.029
| | Marvel and Benediction, Hemisecordite
|-
| | 61
| | 710.677
| | Hemisecordite
|-
| | 62
| | 722.330
| | Marvel and Benediction
|-
| | 63
| | 733.981
| | Marvel and Benediction, Hemisecordite
|-
| | 64
| | 745.631
| | Marvel and Benediction
|-
| | 65
| | 757.282
| | Hemisecordite
|-
| | 66
| | 768.932
| | Marvel and Benediction, Hemisecordite
|-
| | 67
| | 780.583
| | Marvel and Benediction
|-
| | 68
| | 792.233
| | Hemisecordite
|-
| | 69
| | 803.883
| | Marvel and Benediction
|-
| | 70
| | 815.534
| | Marvel and Benediction, Hemisecordite
|-
| | 71
| | 827.185
| | Hemisecordite
|-
| | 72
| | 838.835
| | Marvel and Benediction
|-
| | 73
| | 850.385
| | Marvel and Benediction. Hemisecordite
|-
| | 74
| | 862.156
| | Marvel and Benediction
|-
| | 75
| | 873.786
| | Hemisecordite
|-
| | 76
| | 885.537
| | Marvel and Benediction, Hemisecordite
|-
| | 77
| | 896.987
| | Marvel and Benediction
|-
| | 78
| | 908.638
| | Hemisecordite
|-
| | 79
| | 920.388
| | Marvel and Benediction, Hemisecordite
|-
| | 80
| | 932.039
| | Marvel and Benediction
|-
|-
| | 81
! rowspan="2" | [[Subgroup]]
| | 943.689
! rowspan="2" | [[Comma list]]
| | Hemisecordite
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
| | 82
! [[TE error|Absolute]] (¢)
| | 955.34
! [[TE simple badness|Relative]] (%)
| | Marvel and Benediction
|-
|-
| | 83
| 2.3
| | 966.99
| {{monzo| -163 103 }}
| | Marvel and Benediction, Hemisecordite
| {{mapping| 103 166 }}
| +0.923
| 0.924
| 7.92
|-
|-
| | 84
| 2.3.5
| | 978.641
| 78732/78125, 34171875/33554432
| | Marvel and Benediction
| {{mapping| 103 166 239 }}
| +0.881
| 0.757
| 6.49
|-
|-
| | 85
| 2.3.5.7
| | 990.291
| 225/224, 1029/1024, 78732/78125
| | Hemisecordite
| {{mapping| 103 166 239 289 }}
| +0.824
| 0.663
| 5.68
|-
|-
| | 86
| 2.3.5.7.11
| | 1001.942
| 225/224, 243/242, 385/384, 43923/43750
| | Marvel and Benediction, Hemisecordite
| {{mapping| 103 166 239 289 356 }}
| +0.876
| 0.602
| 5.16
|-
|-
| | 87
| 2.3.5.7.11.13
| | 1013.592
| 225/224, 243/242, 351/350, 385/384, 847/845
| | Marvel and Benediction
| {{mapping| 103 166 239 289 356 381 }}
| +0.806
| 0.571
| 4.90
|-
|-
| | 88
| 2.3.5.7.11.13.17
| | 1025.243
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| | Hemisecordite
| {{mapping| 103 166 239 289 356 381 421 }}
|-
| +0.694
| | 89
| 0.595
| | 1036.893
| 5.10
| | Marvel and Benediction
|-
| | 90
| | 1048.544
| | Marvel and Benediction, Hemisecordite
|-
| | 91
| | 1060.194
| | Hemisecordite
|-
| | 92
| | 1071.845
| | Marvel and Benediction
|-
| | 93
| | 1083.495
| | Marvel and Benediction, Hemisecordite
|-
| | 94
| | 1095.146
| | Marvel and Benediction
|-
| | 95
| | 1106.796
| | Hemisecordite
|-
| | 96
| | 1118.447
| | Marvel and Benediction, Hemisecordite
|-
| | 97
| | 1130.097
| | Marvel and Benediction
|-
| | 98
| | 1141.748
| | Hemisecordite
|-
| | 99
| | 1153.398
| | Marvel and Benediction
|-
| | 100
| | 1165.0485
| | Marvel and Benediction, Hemisecordite
|-
| | 101
| | 1176.609
| | Hemisecordite
|-
| | 102
| | 1188.3495
| | Marvel and Benediction
|}
|}
Since 103edo has a step of 11.6505 cents, it also allows one to use its MOS scales as circulating temperaments.
 
{| class="wikitable"
* 103et (103h val) has lower absolute errors than any smaller equal temperaments in the [[13-limit|13-]], [[17-limit|17-]], and [[19-limit]]s, being beaten by [[111edo|111]] in all of them.
|+Circulating temperaments in 103edo
 
!Tones
=== Rank-2 temperaments ===
!Pattern
{| class="wikitable center-all left-5"
!L:s
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
|5
! Periods<br>per 8ve
|[[3L 2s]]
! Generator*
|21:20
! Cents*
! Associated<br>ratio*
! Temperaments
|-
|-
|6
| 1
|[[1L 5s]]
| 3\103
|18:17
| 34.951
| 1990656/1953125
| [[Gammy]]
|-
|-
|7
| 1
|[[5L 2s]]
| 5\103
|15:14
| 58.252
| 27/26
| [[Hemisecordite]]
|-
|-
|8
| 1
|[[7L 1s]]
| 9\103
|13:12
| 104.854
| 17/16
| [[Septendesemi]]
|-
|-
|9
| 1
|[[4L 5s]]
| 10\103
|12:11
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
|-
|10
| 1
|[[3L 7s]]
| 16\103
|11:10
| 186.408
| 10/9
| [[Mintone]]
|-
|-
|11
| 1
|[[4L 7s]]
| 20\103
|10:9
| 233.010
| 8/7
| [[Slendric]]
|-
|-
|12
| 1
|[[7L 5s]]
| 21\103
|9:8
| 244.660
| 15/13
| [[Subsemifourth]]
|-
|-
|13
| 1
|[[12L 1s]]
| 26\103
| rowspan="2" |8:7
| 303.013
| 25/21
| [[Quinmite]]
|-
|-
|14
| 1
|[[3L 11s]]
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
|-
|15
| 1
|[[13L 2s]]
| 37\103
| rowspan="3" |7:6
| 431.06
| 77/60
| [[Lockerbie]]
|-
|-
|16
| 1
|[[7L 9s]]
| 38\103
| 442.708
| 162/125
| [[Sensei]]
|-
|-
|17
| 1
|1L 16s
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
|-
|18
| 1
|11L 7s
| 40\103
| rowspan="3" |6:5
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
|-
|19
| 1
|8L 11s
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
|-
|20
| 1
|[[3L 17s]]
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
|-
|21
| 1
|19L 2s
| 47\103
| rowspan="5" |5:4
| 547.573
| 11/8
| [[Heinz]]
|-
|-
|22
| 1
|[[15L 7s]]
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
|-
|23
| 1
|[[11L 12s]]
| 50\103
|-
| 582.524
|24
| 7/5
|[[7L 17s]]
| [[Neptune (temperament)|Neptune]]
|-
|25
|3L 22s
|-
|26
|25L 1s
| rowspan="9" |4:3
|-
|27
|[[22L 5s]]
|-
|28
|19L 9s
|-
|29
|16L 13s
|-
|30
|13L 17s
|-
|31
|10L 21s
|-
|32
|7L 25s
|-
|33
|4L 29s
|-
|34
|1L 33s
|-
|35
|33L 2s
| rowspan="17" |3:2
|-
|36
|31L 5s
|-
|37
|29L 8s
|-
|38
|27L 11s
|-
|39
|25L 14s
|-
|40
|23L 17s
|-
|41
|21L 20s
|-
|42
|19L 23s
|-
|43
|17L 26L
|-
|44
|15L 29s
|-
|45
|13L 32s
|-
|46
|11L 35s
|-
|47
|9L 38s
|-
|48
|7L 41s
|-
|49
|5L 44s
|-
|50
|3L 47s
|-
|51
|1L 50s
|-
|52
|51L 1s
| rowspan="31" |2:1
|-
|53
|50L 3s
|-
|54
|49L 5s
|-
|55
|48L 7s
|-
|56
|47L 9s
|-
|57
|46L 11s
|-
|58
|45L 13s
|-
|59
|44L 15s
|-
|60
|43L 17s
|-
|61
|42L 19s
|-
|62
|41L 21s
|-
|63
|40L 23s
|-
|64
|39L 25s
|-
|65
|38L 27s
|-
|66
|37L 29s
|-
|67
|36L 31s
|-
|68
|35L 33s
|-
|69
|34L 35s
|-
|70
|33L 37s
|-
|71
|32L 39s
|-
|72
|31L 41s
|-
|73
|30L 43s
|-
|74
|29L 45s
|-
|75
|28L 47s
|-
|76
|27L 49s
|-
|77
|26L 51s
|-
|78
|25L 53s
|-
|79
|24L 55s
|-
|80
|23L 57s
|-
|81
|22L 59s
|-
|82
|21L 61s
|}
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
== Music ==
; [[Francium]]
* "Forest Tribe Dance" from ''Mysteries'' (2023) – [https://open.spotify.com/track/0lPUfgduKoJliGbU3kcow0 Spotify] | [https://francium223.bandcamp.com/track/forest-tribe-dance Bandcamp] | [https://www.youtube.com/watch?v=5pbnmzXAFcM YouTube]
[[Category:Benediction]]
[[Category:Benediction]]
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Miracle]]
[[Category:Miracle]]
[[Category:Prime EDO]]
[[Category:Theory]]
Retrieved from "https://en.xen.wiki/w/103edo"