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Escapade family: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-02-07 09:25:24 UTC</tt>.<br>
: The original revision id was <tt>299261208</tt>.<br>
: The revision comment was: <tt></tt><br>
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>
<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">[[toc|flat]]


=Escapade=
<div style="float: right;">
The **escapade family** tempers out the [[escapade comma]], |32 -7 -9&gt;, of size 9.492 [[cent]]s.
[[File:Escapade.png|alt=Escapade.png|thumb|600x560px|An image of the tuning spectrum of 2.3.5.11 escapade, in terms of the generator; [[Edo]] [[patent val]] tunings are marked with vertical lines whose length indicates the edo's tolerance, i.e. half of its step size in either direction of just, and some small edos supporting the temperament are labeled.]]
</div>


[[POTE tuning|POTE generator]]: (~16875/16384) 55.293
The '''escapade family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[escapade comma]], {{monzo| 32 -7 -9 }}, of size 9.492 [[cent]]s. The defining feature of this comma is splitting [[5/3]] into sixteen quartertones of which [[5/4]] makes up seven and [[4/3]] makes up nine; therefore [[16/15]] is two generator steps. It most naturally manifests as a [[2.3.5.11 subgroup|2.3.5.11-subgroup]] temperament, tempering out [[4000/3993]] and [[5632/5625]].


Map: [&lt;1 2 2|, &lt;0 -9 7|]
Extensions of escapade to incorporate prime 7 (and therefore the full [[11-limit]]) include escapist ({{nowrap| 21 & 22 }}), tempering out [[225/224]] and mapping 7 to −4 generators; escaped ({{nowrap| 22 & 87 }}), tempering out [[245/243]] and mapping 7 to −26 generators; alphaquarter ({{nowrap| 65d & 87 }}), tempering out [[5120/5103]] and mapping 7 to 61 generators; septisuperfourth (a.k.a. biscapade) ({{nowrap| 22 & 86 }}), tempering out [[6144/6125]], splitting the octave in half and mapping 7 to −15 generators; and arch ({{nowrap| 43 & 87 }}), tempering out [[3136/3125]] and splitting the generator into two [[64/63]] intervals; all are considered below.
EDOs: [[21edo|21]], [[22edo|22]], [[43edo|43]], [[65edo|65]], 152, 217, 586, 803


==7-limit==
== Escapade ==
Commas: 225/224, 12288/12005
For intervals along the chain of generators in the 2.3.5.11.31 subgroup temperament, out to 22 generators up, see the third column of [[16ed5/3#Intervals]].


[[POTE tuning|POTE temperament]]: (~49/48) 55.327
=== 5-limit ===
[[Subgroup]]: 2.3.5


Map: [&lt;1 2 2 3|, &lt;0 -9 7 -4|]
[[Comma list]]: 4294967296/4271484375 ({{monzo|32 -7 -9}})
Wedgie: &lt;&lt;9 -7 4 -32 -19 29||
EDOs: 21, 22, 43, 65


==11-limit==
{{Mapping|legend=1| 1 2 2 | 0 -9 7 }}
Commas: 99/98, 176/175, 2560/2541


[[POTE tuning|POTE temperament]]: 55.354
: mapping generators: ~2, ~16875/16384


Map: [&lt;1 2 2 3 3|, &lt;0 -9 7 -4 10|]
[[Optimal tuning]]s:
EDOs: 21, 22, 43, 65
* [[WE]]: ~2 = 1199.8082{{c}}, ~16875/16384 = 55.2840{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16875/16384 = 55.2961{{c}}


=Escaped=
{{Optimal ET sequence|legend=1| 21, 22, 43, 65, 152, 217, 586, 803 }}
Commas: 245/243, 65625/65536


[[POTE tuning|POTE generator]]: 55.122
[[Badness]] (Sintel): 1.965


Map: [&lt;1 2 2 4|, &lt;0 -9 7 -26|]
{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
Wedgie: &lt;&lt;9 -7 26 -32 16 80||
|+ style="font-size: 105%;" | Harmonics
EDOs: 22, 65, [[87edo|87]], 196, 283
|-
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation from just
|-
| 3/2 || 702.253 || +0.298
|-
| 5/4 || 387.136 || +0.823
|}


==11-limit==
=== 2.3.5.11 subgroup ===
Commas: 245/243, 385/384, 4000/3993
Since (an ideally slightly flat) 4/3 is split into 9 generators, it makes sense to equate the 3-generator interval to [[11/10]] by tempering out 4000/3993, and therefore the generator to {{nowrap|(11/10)/(16/15) {{=}} [[33/32]]}}; this does minimal damage to the temperament. This structure in 2.3.5.11 occurs in all extensions of escapade to include prime 7, and therefore will be considered the fount of all further extensions.


[[POTE tuning|POTE generator]]: 55.126
Subgroup: 2.3.5.11


Map: [&lt;1 2 2 4 3|, &lt;0 -9 7 -26 10|]
Comma list: 4000/3993 ({{monzo|5 -1 3 -3}}), 5632/5625 ({{monzo|9 -2 -4 1}})
EDOs: 22, 65, 87, 196, 283


=Biscapade=
Mapping: {{Mapping| 1 2 2 3 | 0 -9 7 10 }}
Commas: 6144/6125, 118098/117649


[[POTE tuning|POTE generator]]: 55.230
Optimal tunings:
* WE: ~2 = 1199.7406{{c}}, ~33/32 = 55.2653{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 55.2770{{c}}


Map: [&lt;2 4 4 7|, &lt;0 -9 7 -15|]
{{Optimal ET sequence|legend=0| 21, 22, 43, 65, 87, 152, 369, 521e }}
Wedgie: &lt;&lt;18 -14 30 -64 -3 109||
EDOs: 22, [[86edo|86]], 108, 130, 152, 282


==11-limit==
Badness (Sintel): 0.335
Commas: 540/539, 4000/3993, 5632/5625


[[POTE tuning|POTE generator]]: 55.304
{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
|+ style="font-size: 105%;" | Harmonics
|-
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation from just
|-
| 3/2 || 702.516 || +0.561
|-
| 5/4 || 386.932 || +0.618
|-
| 11/8 || 552.760 || +1.442
|}


Map: [&lt;2 4 4 7 6|, &lt;0 -9 7 -15 10|]
=== 2.3.5.11.31 subgroup ===
EDOs: 22, 86, 108, 130, 152, 282
One may note that the generator represents the square root of [[16/15]] and therefore it would be logical to also temper out {{nowrap| S31 {{=}} [[961/960]] }} so that the generator is equated to {{nowrap| [[32/31]] ~ [[31/30]] }} in addition to 33/32.


==13-limit==
Subgroup: 2.3.5.11.31
Commas: 351/350, 364/363, 540/539, 4096/4095


[[POTE tuning|POTE generator]]: 55.359
Comma list: 496/495 ({{monzo| 4 -2 -1 -1 1 }}), 961/960 ({{monzo| -6 -1 -1 0 2 }}), 4000/3993 ({{monzo| 5 -1 3 -3 0 }})


Map: [&lt;2 4 4 7 6 5|, &lt;0 -9 7 -15 10 26|]
Mapping: {{mapping| 1 2 2 3 5 | 0 -9 7 10 -1 }}
EDOs: 22, 108, 130</pre></div>
 
<h4>Original HTML content:</h4>
Optimal tunings:
<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;Escapade family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:16:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt;&lt;a href="#Escapade"&gt;Escapade&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt; | &lt;a href="#Escaped"&gt;Escaped&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt; | &lt;a href="#Biscapade"&gt;Biscapade&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt;
* WE: ~2 = 1199.8050{{c}}, ~32/31 = 55.2669{{c}}
&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~32/31 = 55.2759{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Escapade"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Escapade&lt;/h1&gt;
 
The &lt;strong&gt;escapade family&lt;/strong&gt; tempers out the &lt;a class="wiki_link" href="/escapade%20comma"&gt;escapade comma&lt;/a&gt;, |32 -7 -9&amp;gt;, of size 9.492 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s.&lt;br /&gt;
{{Optimal ET sequence|legend=0| 21, 22, 43, 65, 87, 152, 369, 521e, 673e }}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: (~16875/16384) 55.293&lt;br /&gt;
Badness (Sintel): 0.251
&lt;br /&gt;
 
Map: [&amp;lt;1 2 2|, &amp;lt;0 -9 7|]&lt;br /&gt;
{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
EDOs: &lt;a class="wiki_link" href="/21edo"&gt;21&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22&lt;/a&gt;, &lt;a class="wiki_link" href="/43edo"&gt;43&lt;/a&gt;, &lt;a class="wiki_link" href="/65edo"&gt;65&lt;/a&gt;, 152, 217, 586, 803&lt;br /&gt;
|+ style="font-size: 105%;" | Harmonics
&lt;br /&gt;
|-
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Escapade-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;7-limit&lt;/h2&gt;
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
Commas: 225/224, 12288/12005&lt;br /&gt;
|-
&lt;br /&gt;
! CTE tuning !! Deviation from just
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: (~49/48) 55.327&lt;br /&gt;
|-
&lt;br /&gt;
| 3/2 || 702.518 || +0.563
Map: [&amp;lt;1 2 2 3|, &amp;lt;0 -9 7 -4|]&lt;br /&gt;
|-
Wedgie: &amp;lt;&amp;lt;9 -7 4 -32 -19 29||&lt;br /&gt;
| 5/4 || 386.931 || +0.617
EDOs: 21, 22, 43, 65&lt;br /&gt;
|-
&lt;br /&gt;
| 11/8 || 552.758 || +1.440
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Escapade-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;11-limit&lt;/h2&gt;
|-
Commas: 99/98, 176/175, 2560/2541&lt;br /&gt;
| 31/16 || 1144.724 || -0.311
&lt;br /&gt;
|}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: 55.354&lt;br /&gt;
 
&lt;br /&gt;
= Strong extensions =
Map: [&amp;lt;1 2 2 3 3|, &amp;lt;0 -9 7 -4 10|]&lt;br /&gt;
{| class="wikitable center-all"
EDOs: 21, 22, 43, 65&lt;br /&gt;
|+ style="font-size: 105%;" | Map to strong full 7- and 11-limit extensions
&lt;br /&gt;
|-
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Escaped"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Escaped&lt;/h1&gt;
! rowspan="1" | Extension !! rowspan="1" | Mapping of 7 !! rowspan="1" | Tuning range*
Commas: 245/243, 65625/65536&lt;br /&gt;
|-
&lt;br /&gt;
| [[#Escapist|Escapist]] || -4 || ↓ [[65edo|65]]
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 55.122&lt;br /&gt;
|-
&lt;br /&gt;
| [[#Alphaquarter|Alphaquarter]] || +61 || ↑ 65 <br> ↓ [[87edo|87]]
Map: [&amp;lt;1 2 2 4|, &amp;lt;0 -9 7 -26|]&lt;br /&gt;
|-
Wedgie: &amp;lt;&amp;lt;9 -7 26 -32 16 80||&lt;br /&gt;
| [[#Escaped|Escaped]] || -26 || ↑ 87
EDOs: 22, 65, &lt;a class="wiki_link" href="/87edo"&gt;87&lt;/a&gt;, 196, 283&lt;br /&gt;
|}
&lt;br /&gt;
<nowiki/>* Defined as the range in which the extension specified has a better mapping of 7 compared to its neighboring extensions
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Escaped-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;11-limit&lt;/h2&gt;
 
Commas: 245/243, 385/384, 4000/3993&lt;br /&gt;
== Escaped ==
&lt;br /&gt;
''[[#Strong extensions|Return to the map]]''
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 55.126&lt;br /&gt;
 
&lt;br /&gt;
This temperament was also known as "sensa" in earlier materials because it tempers out 245/243, 352/351, and 385/384 as a sensamagic temperament. ''Not to be confused with the {{nowrap| 19e & 27 }} temperament (sensi extension).''
Map: [&amp;lt;1 2 2 4 3|, &amp;lt;0 -9 7 -26 10|]&lt;br /&gt;
 
EDOs: 22, 65, 87, 196, 283&lt;br /&gt;
Here, [[245/243]] is tempered out so that [[9/7]] is equated to the square root of 5/3 (at 8 generators) present in the temperament. This works best where 5/3 is slightly flat, therefore on the end of the spectrum approaching [[22edo]].
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Biscapade"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Biscapade&lt;/h1&gt;
=== 7-limit ===
Commas: 6144/6125, 118098/117649&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 55.230&lt;br /&gt;
[[Comma list]]: 245/243, 65625/65536
&lt;br /&gt;
 
Map: [&amp;lt;2 4 4 7|, &amp;lt;0 -9 7 -15|]&lt;br /&gt;
{{Mapping|legend=1| 1 2 2 4 | 0 -9 7 -26 }}
Wedgie: &amp;lt;&amp;lt;18 -14 30 -64 -3 109||&lt;br /&gt;
 
EDOs: 22, &lt;a class="wiki_link" href="/86edo"&gt;86&lt;/a&gt;, 108, 130, 152, 282&lt;br /&gt;
[[Optimal tuning]]s:
&lt;br /&gt;
* [[WE]]: ~2 = 1199.9190{{c}}, ~28/27 = 55.1186{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Biscapade-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;11-limit&lt;/h2&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/27 = 55.1242{{c}}
Commas: 540/539, 4000/3993, 5632/5625&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 22, 65, 87, 196, 283 }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 55.304&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Sintel): 2.246
Map: [&amp;lt;2 4 4 7 6|, &amp;lt;0 -9 7 -15 10|]&lt;br /&gt;
 
EDOs: 22, 86, 108, 130, 152, 282&lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Biscapade-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;13-limit&lt;/h2&gt;
 
Commas: 351/350, 364/363, 540/539, 4096/4095&lt;br /&gt;
Comma list: 245/243, 385/384, 4000/3993
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 55.359&lt;br /&gt;
Mapping: {{mapping| 1 2 2 4 3 | 0 -9 7 -26 10 }}
&lt;br /&gt;
 
Map: [&amp;lt;2 4 4 7 6 5|, &amp;lt;0 -9 7 -15 10 26|]&lt;br /&gt;
Optimal tunings:
EDOs: 22, 108, 130&lt;/body&gt;&lt;/html&gt;</pre></div>
* WE: ~2 = 1199.9480{{c}}, ~28/27 = 55.1241{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 55.1271{{c}}
 
{{Optimal ET sequence|legend=0| 22, 65, 87, 196, 283 }}
 
Badness (Sintel): 1.185
 
{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
|+ style="font-size: 105%;" | Harmonics
|-
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation from just
|-
| 3/2 || 703.831 || +1.876
|-
| 5/4 || 385.909 || -0.405
|-
| 7/4 || 966.624 || -2.202
|-
| 11/8 || 551.299 || -0.019
|}
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 245/243, 352/351, 385/384, 625/624
 
Mapping: {{mapping| 1 2 2 4 3 2 | 0 -9 7 -26 10 37 }}
 
Optimal tunings:
* WE: ~2 = 1199.9926{{c}}, ~28/27 = 55.1378{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 55.1382{{c}}
 
{{Optimal ET sequence|legend=0| 22, 65, 87, 283 }}
 
Badness (Sintel): 1.296
 
== Alphaquarter ==
''[[#Strong extensions|Return to the map]]''
 
Given the slightly sharp ~[[3/2]] in ideal tunings of escapade (between [[65edo]] and [[87edo]]), it does very little damage to temper out [[5120/5103]] to extend it to prime 7; the cost is that the harmonic 7 is exceedingly complex, located all the way at 61 generators up.
 
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 5120/5103, 29360128/29296875
 
{{Mapping|legend=1| 1 2 2 0 | 0 -9 7 61 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7349{{c}}, ~16128/15625 = 55.2306{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16128/15625 = 55.2405{{c}}
 
{{Optimal ET sequence|legend=1| 65d, 87, 152, 239, 391 }}
 
[[Badness]] (Sintel): 2.951
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4000/3993, 5120/5103
 
Mapping: {{mapping| 1 2 2 0 3 | 0 -9 7 61 10 }}
 
Optimal tunings:
* WE: ~2 = 1199.7229{{c}}, ~33/32 = 55.2303{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 55.2407{{c}}
 
{{Optimal ET sequence|legend=0| 65d, 87, 152, 239, 391 }}
 
Badness (Sintel): 0.980
 
{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
|+ style="font-size: 105%;" | Harmonics
|-
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation from just
|-
| 3/2 || 702.918 || +0.963
|-
| 5/4 || 386.620 || +0.306
|-
| 7/4 || 969.113 || +0.287
|-
| 11/8 || 552.314 || +0.996
|}
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 625/624, 847/845, 1575/1573
 
Mapping: {{mapping| 1 2 2 0 3 2 | 0 -9 7 61 10 37 }}
 
Optimal tunings:
* WE: ~2 = 1199.6491{{c}}, ~33/32 = 55.2200{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 55.2328{{c}}
 
{{Optimal ET sequence|legend=0| 65d, 87, 152f, 239f }}
 
Badness (Sintel): 1.047
 
== Escapist ==
''[[#Strong extensions|Return to the map]]''
 
This temperament makes the identification of the 4-generator interval, representing (16/15)<sup>2</sup>, with [[8/7]] by tempering out [[225/224]] (along with [[12288/12005]]); however, this is somewhat inaccurate as the ~16/15 in escapade is slightly flat, while for a good marvel tuning it needs to be tempered sharpward to equate it with [[15/14]].
 
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 12288/12005
 
{{Mapping|legend=1| 1 2 2 3 | 0 -9 7 -4 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.9926{{c}}, ~49/48 = 55.2809{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 55.3479{{c}}
 
{{Optimal ET sequence|legend=1| 21, 22, 43, 65d }}
 
[[Badness]] (Sintel): 1.973
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 99/98, 176/175, 2560/2541
 
Mapping: {{mapping| 1 2 2 3 3 | 0 -9 7 -4 10 }}
 
Optimal tunings:
* WE: ~2 = 1199.0859{{c}}, ~33/32 = 55.3117{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 55.3574{{c}}
 
{{Optimal ET sequence|legend=0| 21, 22, 43, 65d }}
 
Badness (Sintel): 1.213
 
{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
|+ style="font-size: 105%;" | Harmonics
|-
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation from just
|-
| 3/2 || 701.626 || -0.329
|-
| 5/4 || 387.624 || +1.310
|-
| 7/4 || 978.501 || +9.675
|-
| 11/8 || 553.749 || +2.431
|}
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 78/77, 99/98, 176/175, 507/500
 
Mapping: {{mapping| 1 2 2 3 3 3 | 0 -9 7 -4 10 15 }}
 
Optimal tunings:
* WE: ~2 = 1199.5949{{c}}, ~33/32 = 55.5317{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 55.5480{{c}}
 
{{Optimal ET sequence|legend=0| 21, 22, 43 }}
 
Badness (Sintel): 1.457
 
= Weak extensions =
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Map to weak extensions
|-
! rowspan="2" | Extensions !! rowspan="2" | Periods per octave !! colspan="2" | Position of original generator
|-
! Number of generators !! Number of periods
|-
| [[#Septisuperfourth|Septisuperfourth]] || period = 1/2 octave || 1 generator || + 0 periods
|-
| [[#Arch|Arch]] || period = octave || 2 generators || + 0 periods
|}
 
== Septisuperfourth ==
''[[#Weak extensions|Return to map]]''
 
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 6144/6125, 118098/117649
 
{{Mapping|legend=1| 2 4 4 7 | 0 -9 7 -15 }}
 
: mapping generators: ~343/243, ~405/392
 
[[Optimal tuning]]s:
* [[WE]]: ~343/243 = 599.8762{{c}}, ~405/392 = 55.3089{{c}}
* [[CWE]]: ~343/243 = 600.0000{{c}}, ~405/392 = 55.3273{{c}}
 
{{Optimal ET sequence|legend=1| 22, 86, 108, 130, 152, 282 }}
 
[[Badness]] (Sintel): 1.499
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 4000/3993, 5632/5625
 
Mapping: {{mapping| 2 4 4 7 6 | 0 -9 7 -15 10 }}
 
Optimal tunings:
* WE: ~99/70 = 599.8383{{c}}, ~33/32 = 55.2895{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~33/32 = 55.3081{{c}}
 
{{Optimal ET sequence|legend=0| 22, 86, 108, 130, 152, 282 }}
 
Badness (Sintel): 0.814
 
{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
|+ style="font-size: 105%;" | Harmonics
|-
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation from just
|-
| 3/2 || 702.070 || +0.115
|-
| 5/4 || 387.279 || +0.965
|-
| 7/4 || 970.117 || +1.291
|-
| 11/8 || 553.255 || +1.937
|}
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 729/728, 1575/1573, 3584/3575
 
Mapping: {{mapping| 2 4 4 7 6 11 | 0 -9 7 -15 10 -39 }}
 
Optimal tunings:
* WE: ~99/70 = 599.8331{{c}}, ~33/32 = 55.3093{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~33/32 = 55.3295{{c}}
 
{{Optimal ET sequence|legend=0| 22f, 108f, 130, 282 }}
 
Badness (Sintel): 0.946
 
==== Septisuperquad ====
This temperament is also known as "biscapade".
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 364/363, 540/539, 4096/4095
 
Mapping: {{mapping| 2 4 4 7 6 5 | 0 -9 7 -15 10 26 }}
 
Optimal tunings:
* WE: ~55/39 = 599.9152{{c}}, ~33/32 = 55.3509{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~33/32 = 55.3584{{c}}
 
{{Optimal ET sequence|legend=0| 22, 108, 130 }}
 
Badness (Sintel): 1.365
 
== Arch ==
''[[#Weak extensions|Return to map]]''
 
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 3136/3125, 5250987/5242880
 
{{Mapping|legend=1| 1 2 2 2 | 0 -18 14 35 }}
 
: mapping generators: ~2, ~64/63
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9246{{c}}, ~64/63 = 27.6662{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~64/63 = 27.6676{{c}}
 
{{Optimal ET sequence|legend=1| 43, 87, 130, 217, 347, 824c, 1171c, 1518cd }}
 
[[Badness]] (Sintel): 2.388
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 3136/3125, 4000/3993
 
Mapping: {{mapping| 1 2 2 2 3 | 0 -18 14 35 20 }}
 
Optimal tunings:
* WE: ~2 = 1199.8347{{c}}, ~64/63 = 27.6590{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~64/63 = 27.6617{{c}}
 
{{Optimal ET sequence|legend=0| 43, 87, 130, 217, 347e, 911cde }}
 
Badness (Sintel): 1.208
 
{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
|+ style="font-size: 105%;" | Harmonics
|-
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation from just
|-
| 3/2 || 702.178 || +0.223
|-
| 5/4 || 387.195 || +0.881
|-
| 7/4 || 967.987 || -0.839
|-
| 11/8 || 553.135 || +1.817
|}
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 364/363, 441/440, 676/675, 3136/3125
 
Mapping: {{mapping| 1 2 2 2 3 4 | 0 -18 14 35 20 -13 }}
 
Optimal tunings:
* WE: ~2 = 1199.8733{{c}}, ~64/63 = 27.6569{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~64/63 = 27.6594{{c}}
 
{{Optimal ET sequence|legend=0| 43, 87, 130, 217, 347e, 564e }}
 
Badness (Sintel): 0.806
 
[[Category:Escapade family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Escapade_family"