103edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
103edo | 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 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 | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br>8ve stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
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| 2.3 | | 2.3 | ||
| {{monzo| -163 103 }} | | {{monzo| -163 103 }} | ||
| | | {{mapping| 103 166 }} | ||
| +0.923 | | +0.923 | ||
| 0.924 | | 0.924 | ||
| Line 36: | Line 55: | ||
| 2.3.5 | | 2.3.5 | ||
| 78732/78125, 34171875/33554432 | | 78732/78125, 34171875/33554432 | ||
| | | {{mapping| 103 166 239 }} | ||
| +0.881 | | +0.881 | ||
| 0.757 | | 0.757 | ||
| Line 43: | Line 62: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 225/224, 1029/1024, 78732/78125 | | 225/224, 1029/1024, 78732/78125 | ||
| | | {{mapping| 103 166 239 289 }} | ||
| +0.824 | | +0.824 | ||
| 0.663 | | 0.663 | ||
| Line 50: | Line 69: | ||
| 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 | ||
| | | {{mapping| 103 166 239 289 356 }} | ||
| +0.876 | | +0.876 | ||
| 0.602 | | 0.602 | ||
| Line 57: | Line 76: | ||
| 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 | ||
| | | {{mapping| 103 166 239 289 356 381 }} | ||
| +0.806 | | +0.806 | ||
| 0.571 | | 0.571 | ||
| Line 64: | Line 83: | ||
| 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 | ||
| | | {{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 | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br> | ! Associated<br>ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
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| 34.951 | | 34.951 | ||
| 1990656/1953125 | | 1990656/1953125 | ||
| [[ | | [[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 == | ||
; [[Francium]] | ; [[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] | * "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:Listen]] | |||
[[Category:Miracle]] | [[Category:Miracle]] | ||