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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|103}}
{{ED intro}}


== Theory ==
== Theory ==
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.
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.


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%).
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 ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
103edo is the 27th [[prime edo]].
103edo is the 27th [[prime edo]], following [[101edo]] and before [[107edo]].


== Intervals ==
== Intervals ==
{{Main|Table of 103edo 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 ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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| 2.3
| 2.3
| {{monzo| -163 103 }}
| {{monzo| -163 103 }}
| [{{val| 103 166 }}]
| {{mapping| 103 166 }}
| +0.923
| +0.923
| 0.924
| 0.924
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| 2.3.5
| 2.3.5
| 78732/78125, 34171875/33554432
| 78732/78125, 34171875/33554432
| [{{val| 103 166 239 }}]
| {{mapping| 103 166 239 }}
| +0.881
| +0.881
| 0.757
| 0.757
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| 2.3.5.7
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| 225/224, 1029/1024, 78732/78125
| [{{val| 103 166 239 289 }}]
| {{mapping| 103 166 239 289 }}
| +0.824
| +0.824
| 0.663
| 0.663
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| 2.3.5.7.11
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| 225/224, 243/242, 385/384, 43923/43750
| [{{val| 103 166 239 289 356 }}]
| {{mapping| 103 166 239 289 356 }}
| +0.876
| +0.876
| 0.602
| 0.602
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| 225/224, 243/242, 351/350, 385/384, 847/845
| [{{val| 103 166 239 289 356 381 }}]
| {{mapping| 103 166 239 289 356 381 }}
| +0.806
| +0.806
| 0.571
| 0.571
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| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| [{{val| 103 166 239 289 356 381 421 }}]
| {{mapping| 103 166 239 289 356 381 421 }}
| +0.694
| +0.694
| 0.595
| 0.595
| 5.10
| 5.10
|}
|}
* 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 8ve
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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| 34.951
| 34.951
| 1990656/1953125
| 1990656/1953125
| [[Gammic]] (5-limit)
| [[Gammy]]
|-
|-
| 1
| 1
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| [[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


== Music ==
== Music ==
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[[Category:Benediction]]
[[Category:Benediction]]
[[Category:Listen]]
[[Category:Miracle]]
[[Category:Miracle]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/103edo"