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103edo: Difference between revisions

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'''103edo''' is the [[EDO|equal division of the octave]] into 103 steps of size 11.6505 [[cent]]s each.
{{Infobox ET}}
{{ED intro}}


== Theory ==
== Theory ==
103edo is a good [[miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limit|13-]]limit and [[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.
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.


103edo is the 27th [[prime numbers|prime]] edo.
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.


=== Prime harmonics ===
=== Prime harmonics ===
{{Primes in edo|103}}
{{Harmonics in equal|103|intervals=prime}}
 
=== Subsets and supersets ===
103edo is the 27th [[prime edo]], following [[101edo]] and before [[107edo]].


== Intervals ==
== 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
}}


{| class="wikitable center-1 right-2"
== Regular temperament properties ==
|-
{| class="wikitable center-4 center-5 center-6"
! Degree
! Cents
! Approximate Ratios
|-
| 1
| 11.650
| 81/80, 126/125
|-
| 2
| 23.301
| 65/64, 66/65, 78/77
|-
| 3
| 34.951
| 49/48, 50/49, 64/63
|-
| 4
| 46.602
| 33/32, 35/34, 36/35
|-
| 5
| 58.252
| 27/26, 34/33
|-
| 6
| 69.903
| 25/24, 26/25, 28/27
|-
| 7
| 81.553
| 21/20, 22/21
|-
| 8
| 93.204
| 18/17
|-
| 9
| 104.854
| 17/16
|-
| 10
| 116.505
| 15/14, 16/15
|-
| 11
| 128.155
| 14/13
|-
| 12
| 139.806
| 13/12
|-
| 13
| 151.456
| 12/11
|-
| 14
| 163.107
| 11/10
|-
| 15
| 174.757
| 72/65
|-
| 16
| 186.408
| 10/9
|-
| 17
| 198.058
| 9/8
|-
| 18
| 209.708
|
|-
| 19
| 221.359
| 17/15, 25/22
|-
| 20
| 233.010
| 8/7
|-
| 21
| 244.660
| 15/13
|-
| 22
| 256.311
|
|-
| 23
| 267.961
| 7/6
|-
| 24
| 279.712
| 20/17
|-
| 25
| 291.262
| 13/11
|-
| 26
| 303.013
| 25/21
|-
| 27
| 314.563
| 6/5
|-
|-
| 28
! rowspan="2" | [[Subgroup]]
| 326.214
! rowspan="2" | [[Comma list]]
| 63/52, 65/54
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
| 29
! [[TE error|Absolute]] (¢)
| 337.864
! [[TE simple badness|Relative]] (%)
| 17/14, 39/32
|-
|-
| 30
| 2.3
| 349.615
| {{monzo| -163 103 }}
| 11/9, 27/22
| {{mapping| 103 166 }}
| +0.923
| 0.924
| 7.92
|-
|-
| 31
| 2.3.5
| 361.165
| 78732/78125, 34171875/33554432
| 16/13, 21/17
| {{mapping| 103 166 239 }}
| +0.881
| 0.757
| 6.49
|-
|-
| 32
| 2.3.5.7
| 372.816
| 225/224, 1029/1024, 78732/78125
| 26/21, 81/65
| {{mapping| 103 166 239 289 }}
| +0.824
| 0.663
| 5.68
|-
|-
| 33
| 2.3.5.7.11
| 384.466
| 225/224, 243/242, 385/384, 43923/43750
| 5/4
| {{mapping| 103 166 239 289 356 }}
| +0.876
| 0.602
| 5.16
|-
|-
| 34
| 2.3.5.7.11.13
| 396.117
| 225/224, 243/242, 351/350, 385/384, 847/845
| 44/35
| {{mapping| 103 166 239 289 356 381 }}
| +0.806
| 0.571
| 4.90
|-
|-
| 35
| 2.3.5.7.11.13.17
| 407.767
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| 33/26
| {{mapping| 103 166 239 289 356 381 421 }}
|-
| +0.694
| 36
| 0.595
| 419.417
| 5.10
| 14/11
|-
| 37
| 431.068
| 9/7
|-
| 38
| 442.708
| 22/17
|-
| 39
| 454.369
| 13/10
|-
| 40
| 466.019
| 17/13, 21/16
|-
| 41
| 477.670
|
|-
| 42
| 489.320
| 65/49
|-
| 43
| 500.971
| 4/3
|-
| 44
| 512.621
| 27/20
|-
| 45
| 524.272
| 65/48
|-
| 46
| 535.922
| 15/11
|-
| 47
| 547.573
| 11/8
|-
| 48
| 559.223
| 18/13
|-
| 49
| 570.874
| 25/18
|-
| 50
| 582.524
| 7/5
|-
| 51
| 594.175
| 24/17
|-
| …
| …
| …
|}
|}


== Regular temperament properties ==
* 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.
 
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per octave
|-
! Generator<br>(reduced)
! Periods<br>per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
Line 240: Line 105:
| 34.951
| 34.951
| 1990656/1953125
| 1990656/1953125
| [[Gammic]] (5-limit)
| [[Gammy]]
|-
|-
| 1
| 1
Line 247: Line 112:
| 27/26
| 27/26
| [[Hemisecordite]]
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
|-
| 1
| 1
Line 265: Line 136:
| 8/7
| 8/7
| [[Slendric]]
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
|-
| 1
| 1
Line 277: Line 154:
| 16/13
| 16/13
| [[Phicordial]]
| [[Phicordial]]
|-
| 1
| 37\103
| 431.06
| 77/60
| [[Lockerbie]]
|-
|-
| 1
| 1
Line 282: Line 165:
| 442.708
| 442.708
| 162/125
| 162/125
| [[Sensipent]] / [[sensei]]
| [[Sensei]]
|-
|-
| 1
| 1
Line 289: Line 172:
| 13/10
| 13/10
| [[Fibo]]
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
|-
| 1
| 1
Line 320: Line 209:
| [[Neptune]]
| [[Neptune]]
|}
|}
<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


== Scales ==
== Music ==
Since 103edo has a step of 11.6505 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
; [[Francium]]
{| class="wikitable"
* "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]
|+Circulating temperaments in 103edo
!Tones
!Pattern
!L:s
|-
|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
| 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
|}
 
=== 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"