Xenharmonic Wiki

103edo: Difference between revisions

VisualWikitext
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**Imported revision 214054878 - Original comment: **
 
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
 
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{ED intro}}
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-25 14:55:42 UTC</tt>.<br>
 
: The original revision id was <tt>214054878</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
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.
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 
<h4>Original Wikitext content:</h4>
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.
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The //103 equal division// divides the octave into 103 equal parts of 11.650 cents each. It is a good [[Gamelismic clan|miracle]] tuning, especially for the 7-limit and [[Gamelismic clan|benediction]], one of the 13-limit extensions of miracle. It 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 in the 13-limit.</pre></div>
 
<h4>Original HTML content:</h4>
=== Prime harmonics ===
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;103edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;103 equal division&lt;/em&gt; divides the octave into 103 equal parts of 11.650 cents each. It is a good &lt;a class="wiki_link" href="/Gamelismic%20clan"&gt;miracle&lt;/a&gt; tuning, especially for the 7-limit and &lt;a class="wiki_link" href="/Gamelismic%20clan"&gt;benediction&lt;/a&gt;, one of the 13-limit extensions of miracle. It 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 in the 13-limit.&lt;/body&gt;&lt;/html&gt;</pre></div>
{{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"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| {{mapping| 103 166 }}
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| {{mapping| 103 166 239 }}
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| {{mapping| 103 166 239 289 }}
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| {{mapping| 103 166 239 289 356 }}
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| {{mapping| 103 166 239 289 356 381 }}
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| {{mapping| 103 166 239 289 356 381 421 }}
| +0.694
| 0.595
| 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 ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammy]]
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 37\103
| 431.06
| 77/60
| [[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[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 ==
; [[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]
 
[[Category:Benediction]]
[[Category:Listen]]
[[Category:Miracle]]
Retrieved from "https://en.xen.wiki/w/103edo"