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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
__FORCETOC__
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>2014-05-01 17:11:39 UTC</tt>.<br>
: The original revision id was <tt>505890632</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]]
 
=Definition of elves=
=Definition of elves=
An //elf// is a scale in a [[regular temperament]] which is tempered from a JI scale in the group of the temperament which is [[Periodic scale|epimorphic]] via a val V which may not be, and characteristically is not, a val supporting the temperament. This allows the elf to have more freedom in scale size and structure but still to possess the coherence induced by the epimorphic mapping.
An ''elf'' is a scale in a [[Regular_temperament|regular temperament]] which is tempered from a JI scale in the group of the temperament which is [[Periodic_scale|epimorphic]] via a val V which may not be, and characteristically is not, a val supporting the temperament. This allows the elf to have more freedom in scale size and structure but still to possess the coherence induced by the epimorphic mapping.


To construct an elf, take the intervals in the JI group of the temperament which lie within an octave and keep only the least complex (in terms of [[Benedetti height]]) representative for each corresponding interval of the temperament. Order the remaining JI intervals by increasing temperamental complexity, breaking ties by increasing Benedetti complexity. For each integer value 1 ≤ i ≤ V(2), set the ith element of a [[transversal]] for the scale to be the first interval c in the listing such that V(c) = i; which is to say, the interval of least temperamental complexity with ties broken by Benedetti height. The tempering of this transversal by a tuning map for the temperament is the elf.
To construct an elf, take the intervals in the JI group of the temperament which lie within an octave and keep only the least complex (in terms of [[Benedetti_height|Benedetti height]]) representative for each corresponding interval of the temperament. Order the remaining JI intervals by increasing temperamental complexity, breaking ties by increasing Benedetti complexity. For each integer value 1 ≤ i ≤ V(2), set the ith element of a [[Transversal|transversal]] for the scale to be the first interval c in the listing such that V(c) = i; which is to say, the interval of least temperamental complexity with ties broken by Benedetti height. The tempering of this transversal by a tuning map for the temperament is the elf.


=Rank two examples=
=Rank two examples=
==13-limit leapday==
==13-limit leapday==
[[elfleapday7]]
[[elfleapday7|elfleapday7]]
[[elfleapday8d]]
 
[[elfleapday9]]
[[elfleapday8d|elfleapday8d]]
[[elfleapday10]]
 
[[elfleapday12f]]
[[elfleapday9|elfleapday9]]
 
[[elfleapday10|elfleapday10]]
 
[[elfleapday12f|elfleapday12f]]


==11-limit magic==
==11-limit magic==
[[elfmagic7]]
[[elfmagic7|elfmagic7]]
[[elfmagic8]]
 
[[elfmagic8d]]
[[elfmagic8|elfmagic8]]
[[elfmagic9]]
 
[[elfmagic10]]
[[elfmagic8d|elfmagic8d]]
[[elfmagic12]]
 
[[elfmagic9|elfmagic9]]
 
[[elfmagic10|elfmagic10]]
 
[[elfmagic12|elfmagic12]]


==11-limit miracle==
==11-limit miracle==
[[elfmiracle7]]
[[elfmiracle7|elfmiracle7]]
[[elfmiracle8d]]
 
[[elfmiracle9]]
[[elfmiracle8d|elfmiracle8d]]
[[elfmiracle10]]
 
[[elfmiracle12]]
[[elfmiracle9|elfmiracle9]]
 
[[elfmiracle10|elfmiracle10]]
 
[[elfmiracle12|elfmiracle12]]


==13-limit myna==
==13-limit myna==
[[elfmyna7]]
[[elfmyna7|elfmyna7]]
[[elfmyna8d]]
 
[[elfmyna9]]
[[elfmyna8d|elfmyna8d]]
[[elfmyna10]]
 
[[elfmyna12f]]
[[elfmyna9|elfmyna9]]
 
[[elfmyna10|elfmyna10]]
 
[[elfmyna12f|elfmyna12f]]


==13-limit octacot==
==13-limit octacot==
[[elfoctacot7]]
[[elfoctacot7|elfoctacot7]]
[[elfoctacot8d]]
 
[[elfoctacot9]]
[[elfoctacot8d|elfoctacot8d]]
[[elfoctacot10]]
 
[[elfoctacot12f]]
[[elfoctacot9|elfoctacot9]]
 
[[elfoctacot10|elfoctacot10]]
 
[[elfoctacot12f|elfoctacot12f]]


==13-limit qilin==
==13-limit qilin==
[[elfqilin7]]
[[elfqilin7|elfqilin7]]
[[elfqilin8d]]
 
[[elfqilin9]]
[[elfqilin8d|elfqilin8d]]
[[elfqilin10]]
 
[[elfqilin12f]]
[[elfqilin9|elfqilin9]]
 
[[elfqilin10|elfqilin10]]
 
[[elfqilin12f|elfqilin12f]]


==13-limit sensus==
==13-limit sensus==
[[elfsensus7]]
[[elfsensus7|elfsensus7]]
[[elfsensus8d]]
 
[[elfsensus9]]
[[elfsensus8d|elfsensus8d]]
[[elfsensus10]]
 
[[elfsensus12]]
[[elfsensus9|elfsensus9]]
[[elfsensus12f]]
 
[[elfsensus10|elfsensus10]]
 
[[elfsensus12|elfsensus12]]
 
[[elfsensus12f|elfsensus12f]]


==11-limit valentine==
==11-limit valentine==
[[elfvalentine7]]
[[elfvalentine7|elfvalentine7]]
[[elfvalentine8d]]
 
[[elfvalentine9]]
[[elfvalentine8d|elfvalentine8d]]
[[elfvalentine10]]
 
[[elfvalentine12]]
[[elfvalentine9|elfvalentine9]]
 
[[elfvalentine10|elfvalentine10]]
 
[[elfvalentine12|elfvalentine12]]


=Rank three examples=
=Rank three examples=
==11-limit jove==
==11-limit jove==
[[elfjove7]]
[[elfjove7|elfjove7]]
[[elfjove8d]]
 
[[elfjove9]]
[[elfjove8d|elfjove8d]]
[[elfjove10]]
 
[[elfjove11c]]
[[elfjove9|elfjove9]]
[[elfjove12]]
 
[[elfjove10|elfjove10]]
 
[[elfjove11c|elfjove11c]]
 
[[elfjove12|elfjove12]]


==13-limit madagascar==
==13-limit madagascar==
[[elfmadagascar7]]
[[elfmadagascar7|elfmadagascar7]]
[[elfmadagascar8d]]
 
[[elfmadagascar9]]
[[elfmadagascar8d|elfmadagascar8d]]
[[elfmadagascar10]]
 
[[elfmadagascar12f]]
[[elfmadagascar9|elfmadagascar9]]
[[elfmadagascar14c]]
 
[[elfmadagascar15]]
[[elfmadagascar10|elfmadagascar10]]
 
[[elfmadagascar12f|elfmadagascar12f]]
 
[[elfmadagascar14c|elfmadagascar14c]]
 
[[elfmadagascar15|elfmadagascar15]]


==11-limit portent==
==11-limit portent==
[[elfportent9]]
[[elfportent9|elfportent9]]
[[elfportent10]]
 
[[elfportent11c]]
[[elfportent10|elfportent10]]
[[elfportent12]]
 
[[elfportent15]]
[[elfportent11c|elfportent11c]]
 
[[elfportent12|elfportent12]]
 
[[elfportent15|elfportent15]]


==11-limit thrush==
==11-limit thrush==
[[elfthrush7]]
[[elfthrush7|elfthrush7]]
[[elfthrush8d]]
 
[[elfthrush9]]
[[elfthrush8d|elfthrush8d]]
[[elfthrush10]]
 
[[elfthrush12]]
[[elfthrush9|elfthrush9]]
 
[[elfthrush10|elfthrush10]]
 
[[elfthrush12|elfthrush12]]


==11-limit zeus==
==11-limit zeus==
[[zeus7tri]]
[[zeus7tri|zeus7tri]]
[[elfzeus8]]
 
[[elfzeus9]]
[[elfzeus8|elfzeus8]]
[[elfzeus10]]
 
[[elfzeus12]]
[[elfzeus9|elfzeus9]]
 
[[elfzeus10|elfzeus10]]
 
[[elfzeus12|elfzeus12]]


=Rank four examples=
=Rank four examples=
==Keenanismic==
==Keenanismic==
[[elfkeenanismic7]]
[[elfkeenanismic7|elfkeenanismic7]]
[[elfkeenanismic8d]]
 
[[elfkeenanismic9]]
[[elfkeenanismic8d|elfkeenanismic8d]]
[[elfkeenanismic10]]
 
[[elfkeenanismic11c]]
[[elfkeenanismic9|elfkeenanismic9]]
[[elfkeenanismic12]]
 
[[elfkeenanismic19]]
[[elfkeenanismic10|elfkeenanismic10]]
 
[[elfkeenanismic11c|elfkeenanismic11c]]
 
[[elfkeenanismic12|elfkeenanismic12]]
 
[[elfkeenanismic19|elfkeenanismic19]]


==Swetismic==
==Swetismic==
[[elfswetismic8d]]
[[elfswetismic8d|elfswetismic8d]]
[[elfswetismic9]]
 
[[elfswetismic10]]
[[elfswetismic9|elfswetismic9]]
[[elfswetismic12]]
 
[[elfswetismic10|elfswetismic10]]
 
[[elfswetismic12|elfswetismic12]]


==Valinorsmic==
==Valinorsmic==
[[elfvalinorsmic7]]
[[elfvalinorsmic7|elfvalinorsmic7]]
[[elfvalinorsmic8d]]
 
[[elfvalinorsmic9]]
[[elfvalinorsmic8d|elfvalinorsmic8d]]
[[elfvalinorsmic10]]
 
[[elfvalinorsmic11c]]
[[elfvalinorsmic9|elfvalinorsmic9]]
[[elfvalinorsmic12]]</pre></div>
 
<h4>Original HTML content:</h4>
[[elfvalinorsmic10|elfvalinorsmic10]]
<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;Elves&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:40:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:40 --&gt;&lt;!-- ws:start:WikiTextTocRule:41: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Definition of elves"&gt;Definition of elves&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:41 --&gt;&lt;!-- ws:start:WikiTextTocRule:42: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Rank two examples"&gt;Rank two examples&lt;/a&gt;&lt;/div&gt;
[[elfvalinorsmic11c|elfvalinorsmic11c]]
&lt;!-- ws:end:WikiTextTocRule:42 --&gt;&lt;!-- ws:start:WikiTextTocRule:43: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Rank two examples-13-limit leapday"&gt;13-limit leapday&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:43 --&gt;&lt;!-- ws:start:WikiTextTocRule:44: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Rank two examples-11-limit magic"&gt;11-limit magic&lt;/a&gt;&lt;/div&gt;
[[elfvalinorsmic12|elfvalinorsmic12]]     [[Category:elf]]
&lt;!-- ws:end:WikiTextTocRule:44 --&gt;&lt;!-- ws:start:WikiTextTocRule:45: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Rank two examples-11-limit miracle"&gt;11-limit miracle&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:45 --&gt;&lt;!-- ws:start:WikiTextTocRule:46: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Rank two examples-13-limit myna"&gt;13-limit myna&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:46 --&gt;&lt;!-- ws:start:WikiTextTocRule:47: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Rank two examples-13-limit octacot"&gt;13-limit octacot&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:47 --&gt;&lt;!-- ws:start:WikiTextTocRule:48: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Rank two examples-13-limit qilin"&gt;13-limit qilin&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:48 --&gt;&lt;!-- ws:start:WikiTextTocRule:49: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Rank two examples-13-limit sensus"&gt;13-limit sensus&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:49 --&gt;&lt;!-- ws:start:WikiTextTocRule:50: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Rank two examples-11-limit valentine"&gt;11-limit valentine&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:50 --&gt;&lt;!-- ws:start:WikiTextTocRule:51: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Rank three examples"&gt;Rank three examples&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:51 --&gt;&lt;!-- ws:start:WikiTextTocRule:52: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Rank three examples-11-limit jove"&gt;11-limit jove&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:52 --&gt;&lt;!-- ws:start:WikiTextTocRule:53: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Rank three examples-13-limit madagascar"&gt;13-limit madagascar&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:53 --&gt;&lt;!-- ws:start:WikiTextTocRule:54: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Rank three examples-11-limit portent"&gt;11-limit portent&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:54 --&gt;&lt;!-- ws:start:WikiTextTocRule:55: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Rank three examples-11-limit thrush"&gt;11-limit thrush&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:55 --&gt;&lt;!-- ws:start:WikiTextTocRule:56: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Rank three examples-11-limit zeus"&gt;11-limit zeus&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:56 --&gt;&lt;!-- ws:start:WikiTextTocRule:57: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Rank four examples"&gt;Rank four examples&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:57 --&gt;&lt;!-- ws:start:WikiTextTocRule:58: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Rank four examples-Keenanismic"&gt;Keenanismic&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:58 --&gt;&lt;!-- ws:start:WikiTextTocRule:59: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Rank four examples-Swetismic"&gt;Swetismic&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:59 --&gt;&lt;!-- ws:start:WikiTextTocRule:60: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Rank four examples-Valinorsmic"&gt;Valinorsmic&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:60 --&gt;&lt;!-- ws:start:WikiTextTocRule:61: --&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:61 --&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Definition of elves"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Definition of elves&lt;/h1&gt;
An &lt;em&gt;elf&lt;/em&gt; is a scale in a &lt;a class="wiki_link" href="/regular%20temperament"&gt;regular temperament&lt;/a&gt; which is tempered from a JI scale in the group of the temperament which is &lt;a class="wiki_link" href="/Periodic%20scale"&gt;epimorphic&lt;/a&gt; via a val V which may not be, and characteristically is not, a val supporting the temperament. This allows the elf to have more freedom in scale size and structure but still to possess the coherence induced by the epimorphic mapping.&lt;br /&gt;
&lt;br /&gt;
To construct an elf, take the intervals in the JI group of the temperament which lie within an octave and keep only the least complex (in terms of &lt;a class="wiki_link" href="/Benedetti%20height"&gt;Benedetti height&lt;/a&gt;) representative for each corresponding interval of the temperament. Order the remaining JI intervals by increasing temperamental complexity, breaking ties by increasing Benedetti complexity. For each integer value 1 ≤ i ≤ V(2), set the ith element of a &lt;a class="wiki_link" href="/transversal"&gt;transversal&lt;/a&gt; for the scale to be the first interval c in the listing such that V(c) = i; which is to say, the interval of least temperamental complexity with ties broken by Benedetti height. The tempering of this transversal by a tuning map for the temperament is the elf.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Rank two examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Rank two examples&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Rank two examples-13-limit leapday"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;13-limit leapday&lt;/h2&gt;
&lt;a class="wiki_link" href="/elfleapday7"&gt;elfleapday7&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfleapday8d"&gt;elfleapday8d&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfleapday9"&gt;elfleapday9&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfleapday10"&gt;elfleapday10&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfleapday12f"&gt;elfleapday12f&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Rank two examples-11-limit magic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;11-limit magic&lt;/h2&gt;
&lt;a class="wiki_link" href="/elfmagic7"&gt;elfmagic7&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfmagic8"&gt;elfmagic8&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfmagic8d"&gt;elfmagic8d&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfmagic9"&gt;elfmagic9&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfmagic10"&gt;elfmagic10&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfmagic12"&gt;elfmagic12&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Rank two examples-11-limit miracle"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;11-limit miracle&lt;/h2&gt;
&lt;a class="wiki_link" href="/elfmiracle7"&gt;elfmiracle7&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfmiracle8d"&gt;elfmiracle8d&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfmiracle9"&gt;elfmiracle9&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfmiracle10"&gt;elfmiracle10&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfmiracle12"&gt;elfmiracle12&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Rank two examples-13-limit myna"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;13-limit myna&lt;/h2&gt;
&lt;a class="wiki_link" href="/elfmyna7"&gt;elfmyna7&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfmyna8d"&gt;elfmyna8d&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfmyna9"&gt;elfmyna9&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfmyna10"&gt;elfmyna10&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfmyna12f"&gt;elfmyna12f&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Rank two examples-13-limit octacot"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;13-limit octacot&lt;/h2&gt;
&lt;a class="wiki_link" href="/elfoctacot7"&gt;elfoctacot7&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfoctacot8d"&gt;elfoctacot8d&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfoctacot9"&gt;elfoctacot9&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfoctacot10"&gt;elfoctacot10&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfoctacot12f"&gt;elfoctacot12f&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Rank two examples-13-limit qilin"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;13-limit qilin&lt;/h2&gt;
&lt;a class="wiki_link" href="/elfqilin7"&gt;elfqilin7&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfqilin8d"&gt;elfqilin8d&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfqilin9"&gt;elfqilin9&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfqilin10"&gt;elfqilin10&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfqilin12f"&gt;elfqilin12f&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Rank two examples-13-limit sensus"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;13-limit sensus&lt;/h2&gt;
&lt;a class="wiki_link" href="/elfsensus7"&gt;elfsensus7&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfsensus8d"&gt;elfsensus8d&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfsensus9"&gt;elfsensus9&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfsensus10"&gt;elfsensus10&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfsensus12"&gt;elfsensus12&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfsensus12f"&gt;elfsensus12f&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Rank two examples-11-limit valentine"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;11-limit valentine&lt;/h2&gt;
&lt;a class="wiki_link" href="/elfvalentine7"&gt;elfvalentine7&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfvalentine8d"&gt;elfvalentine8d&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfvalentine9"&gt;elfvalentine9&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfvalentine10"&gt;elfvalentine10&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfvalentine12"&gt;elfvalentine12&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc10"&gt;&lt;a name="Rank three examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Rank three examples&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Rank three examples-11-limit jove"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;11-limit jove&lt;/h2&gt;
&lt;a class="wiki_link" href="/elfjove7"&gt;elfjove7&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfjove8d"&gt;elfjove8d&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfjove9"&gt;elfjove9&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfjove10"&gt;elfjove10&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfjove11c"&gt;elfjove11c&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfjove12"&gt;elfjove12&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Rank three examples-13-limit madagascar"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;13-limit madagascar&lt;/h2&gt;
&lt;a class="wiki_link" href="/elfmadagascar7"&gt;elfmadagascar7&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfmadagascar8d"&gt;elfmadagascar8d&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfmadagascar9"&gt;elfmadagascar9&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfmadagascar10"&gt;elfmadagascar10&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfmadagascar12f"&gt;elfmadagascar12f&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfmadagascar14c"&gt;elfmadagascar14c&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfmadagascar15"&gt;elfmadagascar15&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="Rank three examples-11-limit portent"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;11-limit portent&lt;/h2&gt;
&lt;a class="wiki_link" href="/elfportent9"&gt;elfportent9&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfportent10"&gt;elfportent10&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfportent11c"&gt;elfportent11c&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfportent12"&gt;elfportent12&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfportent15"&gt;elfportent15&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc14"&gt;&lt;a name="Rank three examples-11-limit thrush"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;11-limit thrush&lt;/h2&gt;
&lt;a class="wiki_link" href="/elfthrush7"&gt;elfthrush7&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfthrush8d"&gt;elfthrush8d&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfthrush9"&gt;elfthrush9&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfthrush10"&gt;elfthrush10&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfthrush12"&gt;elfthrush12&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc15"&gt;&lt;a name="Rank three examples-11-limit zeus"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;11-limit zeus&lt;/h2&gt;
&lt;a class="wiki_link" href="/zeus7tri"&gt;zeus7tri&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfzeus8"&gt;elfzeus8&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfzeus9"&gt;elfzeus9&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfzeus10"&gt;elfzeus10&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfzeus12"&gt;elfzeus12&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc16"&gt;&lt;a name="Rank four examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;Rank four examples&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc17"&gt;&lt;a name="Rank four examples-Keenanismic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;Keenanismic&lt;/h2&gt;
&lt;a class="wiki_link" href="/elfkeenanismic7"&gt;elfkeenanismic7&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfkeenanismic8d"&gt;elfkeenanismic8d&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfkeenanismic9"&gt;elfkeenanismic9&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfkeenanismic10"&gt;elfkeenanismic10&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfkeenanismic11c"&gt;elfkeenanismic11c&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfkeenanismic12"&gt;elfkeenanismic12&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfkeenanismic19"&gt;elfkeenanismic19&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="Rank four examples-Swetismic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;Swetismic&lt;/h2&gt;
&lt;a class="wiki_link" href="/elfswetismic8d"&gt;elfswetismic8d&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfswetismic9"&gt;elfswetismic9&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfswetismic10"&gt;elfswetismic10&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfswetismic12"&gt;elfswetismic12&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc19"&gt;&lt;a name="Rank four examples-Valinorsmic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt;Valinorsmic&lt;/h2&gt;
&lt;a class="wiki_link" href="/elfvalinorsmic7"&gt;elfvalinorsmic7&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfvalinorsmic8d"&gt;elfvalinorsmic8d&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfvalinorsmic9"&gt;elfvalinorsmic9&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfvalinorsmic10"&gt;elfvalinorsmic10&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfvalinorsmic11c"&gt;elfvalinorsmic11c&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/elfvalinorsmic12"&gt;elfvalinorsmic12&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Revision as of 00:00, 17 July 2018

Definition of elves

An elf is a scale in a regular temperament which is tempered from a JI scale in the group of the temperament which is epimorphic via a val V which may not be, and characteristically is not, a val supporting the temperament. This allows the elf to have more freedom in scale size and structure but still to possess the coherence induced by the epimorphic mapping.

To construct an elf, take the intervals in the JI group of the temperament which lie within an octave and keep only the least complex (in terms of Benedetti height) representative for each corresponding interval of the temperament. Order the remaining JI intervals by increasing temperamental complexity, breaking ties by increasing Benedetti complexity. For each integer value 1 ≤ i ≤ V(2), set the ith element of a transversal for the scale to be the first interval c in the listing such that V(c) = i; which is to say, the interval of least temperamental complexity with ties broken by Benedetti height. The tempering of this transversal by a tuning map for the temperament is the elf.

Rank two examples

13-limit leapday

elfleapday7

elfleapday8d

elfleapday9

elfleapday10

elfleapday12f

11-limit magic

elfmagic7

elfmagic8

elfmagic8d

elfmagic9

elfmagic10

elfmagic12

11-limit miracle

elfmiracle7

elfmiracle8d

elfmiracle9

elfmiracle10

elfmiracle12

13-limit myna

elfmyna7

elfmyna8d

elfmyna9

elfmyna10

elfmyna12f

13-limit octacot

elfoctacot7

elfoctacot8d

elfoctacot9

elfoctacot10

elfoctacot12f

13-limit qilin

elfqilin7

elfqilin8d

elfqilin9

elfqilin10

elfqilin12f

13-limit sensus

elfsensus7

elfsensus8d

elfsensus9

elfsensus10

elfsensus12

elfsensus12f

11-limit valentine

elfvalentine7

elfvalentine8d

elfvalentine9

elfvalentine10

elfvalentine12

Rank three examples

11-limit jove

elfjove7

elfjove8d

elfjove9

elfjove10

elfjove11c

elfjove12

13-limit madagascar

elfmadagascar7

elfmadagascar8d

elfmadagascar9

elfmadagascar10

elfmadagascar12f

elfmadagascar14c

elfmadagascar15

11-limit portent

elfportent9

elfportent10

elfportent11c

elfportent12

elfportent15

11-limit thrush

elfthrush7

elfthrush8d

elfthrush9

elfthrush10

elfthrush12

11-limit zeus

zeus7tri

elfzeus8

elfzeus9

elfzeus10

elfzeus12

Rank four examples

Keenanismic

elfkeenanismic7

elfkeenanismic8d

elfkeenanismic9

elfkeenanismic10

elfkeenanismic11c

elfkeenanismic12

elfkeenanismic19

Swetismic

elfswetismic8d

elfswetismic9

elfswetismic10

elfswetismic12

Valinorsmic

elfvalinorsmic7

elfvalinorsmic8d

elfvalinorsmic9

elfvalinorsmic10

elfvalinorsmic11c

elfvalinorsmic12

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