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'''103edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 103 steps of size 11.6505 [[cent|cent]]s each. It is a good [[Gamelismic_clan|miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limi|13-]] limits and [[Gamelismic_clan|benediction]] and [[Gamelismic_clan|hemisecordite]], two of the [[13-limit|13-limit]] extensions of miracle. It [[tempering_out|tempers out]] 78732/78125 in the [[5-limit|5-limit]]; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the [[11-limit|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 ==
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}}
 
=== Subsets and supersets ===
103edo is the 27th [[prime edo]], following [[101edo]] and before [[107edo]].
 
== Intervals ==
{{Main|Table of 103edo intervals}}
 
== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}
 
=== Zeta peak index ===
{{ZPI
| zpi = 596
| steps = 102.936629522070
| step size = 11.6576577800491
| tempered height = 8.543510
| pure height = 5.620365
| integral = 1.340775
| gap = 18.270998
| octave = 1200.73875134506
| consistent = 15
| distinct = 15
}}
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
|-
| | Degree
! rowspan="2" | [[Subgroup]]
| | cents
! rowspan="2" | [[Comma list]]
| | temperament
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
| | 1
! [[TE error|Absolute]] (¢)
| | 11.650
! [[TE simple badness|Relative]] (%)
| | Marvel and Benediction
|-
|-
| | 2
| 2.3
| | 23.301
| {{monzo| -163 103 }}
| | Hemisecordite
| {{mapping| 103 166 }}
| +0.923
| 0.924
| 7.92
|-
|-
| | 3
| 2.3.5
| | 34.951
| 78732/78125, 34171875/33554432
| | Marvel and Benediction, Hemisecordite
| {{mapping| 103 166 239 }}
| +0.881
| 0.757
| 6.49
|-
|-
| | 4
| 2.3.5.7
| | 46.602
| 225/224, 1029/1024, 78732/78125
| | Marvel and Benediction
| {{mapping| 103 166 239 289 }}
| +0.824
| 0.663
| 5.68
|-
|-
| | 5
| 2.3.5.7.11
| | 58.252
| 225/224, 243/242, 385/384, 43923/43750
| | Hemisecordite
| {{mapping| 103 166 239 289 356 }}
| +0.876
| 0.602
| 5.16
|-
|-
| | 6
| 2.3.5.7.11.13
| | 69.903
| 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
|-
|-
| | 7
| 2.3.5.7.11.13.17
| | 81.553
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| | Marvel and Benediction, Hemisecordite
| {{mapping| 103 166 239 289 356 381 421 }}
|-
| +0.694
| | 8
| 0.595
| | 93.204
| 5.10
| | Hemisecordite
|}
|-
 
| | 9
* 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.
| | 104.854
 
| | Marvel and Benediction
=== Rank-2 temperaments ===
|-
{| class="wikitable center-all left-5"
| | 10
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
| | 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
! Periods<br>per 8ve
| | 174.757
! Generator*
| | Hemisecordite
! Cents*
! Associated<br>ratio*
! Temperaments
|-
|-
| | 16
| 1
| | 186.408
| 3\103
| | Marvel and Benediction
| 34.951
| 1990656/1953125
| [[Gammy]]
|-
|-
| | 17
| 1
| | 198.058
| 5\103
| | Marvel and Benediction, Hemisecordite
| 58.252
| 27/26
| [[Hemisecordite]]
|-
|-
| | 18
| 1
| | 209.708
| 9\103
| | Hemisecordite
| 104.854
| 17/16
| [[Septendesemi]]
|-
|-
| | 19
| 1
| | 221.359
| 10\103
| | Marvel and Benediction
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
|-
| | 20
| 1
| | 233.010
| 16\103
| | Marvel and Benediction, Hemisecordite
| 186.408
| 10/9
| [[Mintone]]
|-
|-
| | 21
| 1
| | 244.660
| 20\103
| | Marvel and Benediction
| 233.010
| 8/7
| [[Slendric]]
|-
|-
| | 22
| 1
| | 256.311
| 21\103
| | Hemisecordite
| 244.660
| 15/13
| [[Subsemifourth]]
|-
|-
| | 23
| 1
| | 267.961
| 26\103
| | Marvel and Benediction, Hemisecordite
| 303.013
| 25/21
| [[Quinmite]]
|-
|-
| | 24
| 1
| | 279.712
| 31\103
| | Marvel and Benediction
| 361.165
| 16/13
| [[Phicordial]]
|-
|-
| | 25
| 1
| | 291.262
| 37\103
| | Hemisecordite
| 431.06
| 77/60
| [[Lockerbie]]
|-
|-
| | 26
| 1
| | 303.013
| 38\103
| | Marvel and Benediction
| 442.708
| 162/125
| [[Sensei]]
|-
|-
| | 27
| 1
| | 314.563
| 39\103
| | Marvel and Benediction, Hemisecordite
| 454.369
| 13/10
| [[Fibo]]
|-
|-
| | 28
| 1
| | 326.214
| 40\103
| | Hemisecordite
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
|-
| | 29
| 1
| | 337.864
| 42\103
| | Marvel and Benediction
| 489.320
| 65/49
| [[Catafourth]]
|-
|-
| | 30
| 1
| | 349.615
| 45\103
| | Marvel and Benediction, Hemisecordite
| 524.272
| 65/48
| [[Widefourth]]
|-
|-
| | 31
| 1
| | 361.165
| 47\103
| | Marvel and Benediction
| 547.573
| 11/8
| [[Heinz]]
|-
|-
| | 32
| 1
| | 372.816
| 48\103
| | Hemisecordite
| 559.223
| 242/175
| [[Tritriple]]
|-
|-
| | 33
| 1
| | 384.466
| 50\103
| | Marvel and Benediction, Hemisecordite
| 582.524
|-
| 7/5
| | 34
| [[Neptune]]
| | 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
| | 943.689
| | Hemisecordite
|-
| | 82
| | 955.34
| | Marvel and Benediction
|-
| | 83
| | 966.99
| | Marvel and Benediction, Hemisecordite
|-
| | 84
| | 978.641
| | Marvel and Benediction
|-
| | 85
| | 990.291
| | Hemisecordite
|-
| | 86
| | 1001.942
| | Marvel and Benediction, Hemisecordite
|-
| | 87
| | 1013.592
| | Marvel and Benediction
|-
| | 88
| | 1025.243
| | Hemisecordite
|-
| | 89
| | 1036.893
| | 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
|}
|}
[[Category:benediction]]
<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
[[Category:edo]]
 
[[Category:miracle]]
== Music ==
[[Category:prime_edo]]
; [[Francium]]
[[Category:theory]]
* "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:Listen]]
[[Category:Miracle]]
Retrieved from "https://en.xen.wiki/w/103edo"