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'''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 ==
| | 11.6505
{{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
| |34.9515
| 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
| |116.5045
| |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
! rowspan="2" | [[Subgroup]]
| |198.058
! rowspan="2" | [[Comma list]]
| |Marvel and Benediction, Hemisecordite
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
| |18
! [[TE error|Absolute]] (¢)
| |209.708
! [[TE simple badness|Relative]] (%)
| |Hemisecordite
|-
|-
| |19
| 2.3
| |221.359
| {{monzo| -163 103 }}
| |Marvel and Benediction
| {{mapping| 103 166 }}
| +0.923
| 0.924
| 7.92
|-
|-
| |20
| 2.3.5
| |233.010
| 78732/78125, 34171875/33554432
| |Marvel and Benediction, Hemisecordite
| {{mapping| 103 166 239 }}
| +0.881
| 0.757
| 6.49
|-
|-
| |21
| 2.3.5.7
| |244.660
| 225/224, 1029/1024, 78732/78125
| |Marvel and Benediction
| {{mapping| 103 166 239 289 }}
| +0.824
| 0.663
| 5.68
|-
|-
| |22
| 2.3.5.7.11
| |256.311
| 225/224, 243/242, 385/384, 43923/43750
| |Hemisecordite
| {{mapping| 103 166 239 289 356 }}
| +0.876
| 0.602
| 5.16
|-
|-
| |23
| 2.3.5.7.11.13
| |267.961
| 225/224, 243/242, 351/350, 385/384, 847/845
| |Marvel and Benediction, Hemisecordite
| {{mapping| 103 166 239 289 356 381 }}
| +0.806
| 0.571
| 4.90
|-
|-
| |24
| 2.3.5.7.11.13.17
| |279.712
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| |Marvel and Benediction
| {{mapping| 103 166 239 289 356 381 421 }}
|-
| +0.694
| |25
| 0.595
| |291.262
| 5.10
| |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
| |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
|}
|}
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
|[[3L 2s]]
|21:20
|-
|6
|[[1L 5s]]
|18:17
|-
|7
|[[5L 2s]]
|15:14
|-
|8
|[[7L 1s]]
|13:12
|-
|9
|[[4L 5s]]
|12:11
|-
|10
|[[3L 7s]]
|11:10
|-
|11
|[[4L 7s]]
|10:9
|-
|12
|[[7L 5s]]
|9:8
|-
|13
|[[12L 1s]]
| rowspan="2" |8:7
|-
|14
|[[3L 11s]]
|-
|15
|[[13L 2s]]
| rowspan="3" |7:6
|-
|16
|[[7L 9s]]
|-
|17
|1L 16s
|-
|18
| 11L 7s
| rowspan="3" |6:5
|-
|19
|8L 11s
|-
|20
|[[3L 17s]]
|-
|21
| 19L 2s
| rowspan="5" |5:4
|-
|22
|[[15L 7s]]
|-
|23
|[[11L 12s]]
|-
|24
|[[7L 17s]]
|-
|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
! Periods<br>per 8ve
| 17L 26L
! Generator*
! Cents*
! Associated<br>ratio*
! Temperaments
|-
|-
|44
| 1
| 15L 29s
| 3\103
| 34.951
| 1990656/1953125
| [[Gammy]]
|-
|-
|45
| 1
| 13L 32s
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
|-
|46
| 1
| 11L 35s
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
|-
|47
| 1
|9L 38s
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
|-
|48
| 1
|7L 41s
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
|-
|49
| 1
|5L 44s
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
|-
|50
| 1
|3L 47s
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
|-
|51
| 1
|1L 50s
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
|-
|52
| 1
| 51L 1s
| 31\103
| rowspan="31" |2:1
| 361.165
| 16/13
| [[Phicordial]]
|-
|-
|53
| 1
| 50L 3s
| 37\103
| 431.06
| 77/60
| [[Lockerbie]]
|-
|-
|54
| 1
| 49L 5s
| 38\103
| 442.708
| 162/125
| [[Sensei]]
|-
|-
|55
| 1
| 48L 7s
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
|-
|56
| 1
| 47L 9s
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
|-
|57
| 1
| 46L 11s
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
|-
|58
| 1
| 45L 13s
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
|-
|59
| 1
| 44L 15s
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
|-
|60
| 1
| 43L 17s
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
|-
|61
| 1
| 42L 19s
| 50\103
|-
| 582.524
|62
| 7/5
| 41L 21s
| [[Neptune (temperament)|Neptune]]
|-
|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]


===13-limit temperaments===
{| class="wikitable"
|+
! colspan="3" |Marvel and Benediction
! colspan="3" |Hemisecordite
|-
!Degree
!cents
!Difference from 72edo
!Degree
!cents
!Difference from 62edo
|-
|1
|11.6505
| -5.016¢
|2
|23.301
|3.946¢
|-
|3
|34.9515
|1.618¢
|3
|34.9515
| -3.758¢
|-
|4
|46.602
| -3.398¢
|5
|58.252
|0.188¢
|-
|6
|69.903
|3.236¢
|7
|81.553
|4.134¢
|-
|7
|81.553
| -1.78¢
|8
|93.204
| -3.57¢
|-
|9
|104.854
|4.854¢
|10
|116.505
|0.376¢
|-
|10
|116.5045
| -0.162¢
|12
|139.806
|4.322¢
|-
|11
|128.155
| -5.178¢
|13
|151.456
| -3.382¢
|-
|13
|151.456¢
|1.456¢
|15
|174.757
|0.563¢
|-
|14
|163.107¢
| -3.56¢
|17
|198.058
|4.51¢
|-
|16
|186.408
|3.074¢
|18
|209.709
| -3.1945¢
|-
|17
|198.058
| -1.942¢
|20
|233.01
|0.751¢
|-
|19
|221.359
|4.693¢
|22
|256.311
|4.698¢
|-
|20
|233.01
| -0.324¢
|23
|267.961
| -3.007¢
|-
|21
|244.66
| -5.34¢
|25
|291.262
|0.94¢
|-
|23
|267.961
|1.2945¢
|27
|314.563
|4.886¢
|-
|24
|279.612
| -3.722¢
|28
|326.214
| -2.819¢
|-
|26
|302.913
|2.913¢
|30
|349.515
|1.1275¢
|-
|27
|314.563
| -2.104¢
|32
|372.8155
|5.074¢
|-
|29
|337.864
|4.531¢
|33
|384.466
| -2.631¢
|-
|30
|349.515
| -0.485¢
|35
|407.767
|1.315¢
|-
|31
|361.165
| -5.502¢
|37
|431.068
|5.2615¢
|-
|33
|384.466
|1.133¢
|38
|442.718
| -2.443¢
|-
|34
|396.1165
| -3.8835¢
|40
|466.0190
|1.503¢
|-
|36
|419.4175
|2.751¢
|42
|489.32
|5.449¢
|-
|37
|431.068
| -2.265¢
|43
|500.971
| -2.255¢
|-
|39
|454.369
|4.369¢
|45
|524.272
|1.691¢
|-
|40
|466.019
| -0.647¢
|47
|547.573
|5.637¢
|-
|41
|477.67
| -5.663¢
|48
|559.223
| -2.067¢
|-
|43
|500.971
|0.971¢
|50
|582.524
|1.879¢
|-
|44
|512.621¢
| -4.045¢
|52
|605.825
|5.825¢
|-
|46
|535.922¢
|2.589¢
|53
|617.476
| -1.879¢
|-
|47
|547.573¢
| -2.427¢
|55
|640.777¢
|2.067
|-
|49
|570.874¢
|4.207¢
|56
|652.427
| -5.637¢
|-
|50
|582.524
| -0.809¢
|58
|675.728
| -1.691
|-
|52
|605.825
|5.825¢
|60
|699.029
|2.255¢
|-
|53
|617.475
|0.809¢
|61
|710.68
| -5.449¢
|-
|54
|629.126¢
| -4.207¢
|63
|733.981
| -1.503¢
|-
|56
|652.427¢
|2.427¢
|65
|757.282
|2.443¢
|-
|57
|664.078
| -2.589¢
|66
|768.932
| -5.2615¢
|-
|59
|687.379
|4.045¢
|68
|792.233
| -1.315¢
|-
|60
|699.029
| -0.971¢
|70
|815.534
|2.631¢
|-
|62
|722.33
|5.663¢
|71
|827.1845
| -5.074¢
|-
|63
|733.981
|0.647¢
|73
|850.485
| -1.1275¢
|-
|64
|745.631
| -4.369¢
|75
|873.786
|2.819¢
|-
|66
|768.932
|2.265¢
|76
|885.437
| -4.886¢
|-
|67
|780.5825
| -2.751¢
|78
|908.738
| -0.94¢
|-
|69
|803.8835
|3.8835¢
|80
|932.039
|3.007¢
|-
|70
|815.534
| -1.133¢
|81
|943.689
| -4.698¢
|-
|72
|838.835
|5.501¢
|83
|966.99
| -0.752¢
|-
|73
|850.485
|0.485¢
|85
|990.291
|3.1945¢
|-
|74
|862.136
| -4.531¢
|86
|1001.942
| -4.51¢
|-
|76
|885.439
|2.104¢
|88
|1025.243
| -0.564¢
|-
|77
|897.087
| -2.913¢
|90
|1048.544
|3.382¢
|-
|79
|920.388
|3.722¢
|91
|1060.194
| -4.322¢
|-
|80
|932.039
| -1.2945¢
|93
|1083.495
| -0.376¢
|-
|82
|955.34
|5.34¢
|95
|1106.796
|3.57¢
|-
|83
|966.99
|0.324¢
|96
|1118.447¢
| -4.134¢
|-
|84
|978.641
| -4.693¢
|98
|1141.748
| -0.188¢
|-
|86
|1001.942
|1.942¢
|100
|1165.0485
|3.758¢
|-
|87
|1013.592
| -3.074¢
|101
|1176.699
| -3.946¢
|-
|89
|1036.893
|3.56¢
|
|
|
|-
|90
|1048.544
| -1.456¢
|
|
|
|-
|92
|1071.845
|5.178¢
|
|
|
|-
|93
|1083.495
|0.162¢
|
|
|
|-
|94
|1095.146
| -4.854¢
|
|
|
|-
|96
|1118.447
|1.78¢
|
|
|
|-
|97
|1130.097
| -3.236¢
|
|
|
|-
|99
|1153.398
|3.398¢
|
|
|
|-
|100
|1165.0485
| -1.618¢
|
|
|
|-
|102
|1188.3495
|5.016¢
|
|
|
|}
[[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"