Elf
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[[toc]] =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. 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 zeus== [[zeus7tri]] =Rank four examples= ==Keenanismic== [[elfkeenanismic7]] [[elfkeenanismic8d]] [[elfkeenanismic9]] [[elfkeenanismic10]] [[elfkeenanismic11c]] [[elfkeenanismic12]] ==Swetismic== [[elfswetismic8d]] [[elfswetismic9]] [[elfswetismic10]] [[elfswetismic12]] ==Valinorsmic== [[elfvalinorsmic7]] [[elfvalinorsmic8d]] [[elfvalinorsmic9]] [[elfvalinorsmic10]] [[elfvalinorsmic11c]] [[elfvalinorsmic12]]
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<html><head><title>Elves</title></head><body><!-- ws:start:WikiTextTocRule:38:<img id="wikitext@@toc@@normal" class="WikiMedia WikiMediaToc" title="Table of Contents" src="/site/embedthumbnail/toc/normal?w=225&h=100"/> --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><div style="margin-left: 1em;"><a href="#Definition of elves">Definition of elves</a></div> <!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --><div style="margin-left: 1em;"><a href="#Rank two examples">Rank two examples</a></div> <!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><div style="margin-left: 2em;"><a href="#Rank two examples-13-limit leapday">13-limit leapday</a></div> <!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><div style="margin-left: 2em;"><a href="#Rank two examples-11-limit magic">11-limit magic</a></div> <!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><div style="margin-left: 2em;"><a href="#Rank two examples-11-limit miracle">11-limit miracle</a></div> <!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --><div style="margin-left: 2em;"><a href="#Rank two examples-13-limit myna">13-limit myna</a></div> <!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --><div style="margin-left: 2em;"><a href="#Rank two examples-13-limit octacot">13-limit octacot</a></div> <!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --><div style="margin-left: 2em;"><a href="#Rank two examples-13-limit qilin">13-limit qilin</a></div> <!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --><div style="margin-left: 2em;"><a href="#Rank two examples-13-limit sensus">13-limit sensus</a></div> <!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --><div style="margin-left: 2em;"><a href="#Rank two examples-11-limit valentine">11-limit valentine</a></div> <!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --><div style="margin-left: 1em;"><a href="#Rank three examples">Rank three examples</a></div> <!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --><div style="margin-left: 2em;"><a href="#Rank three examples-11-limit jove">11-limit jove</a></div> <!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --><div style="margin-left: 2em;"><a href="#Rank three examples-13-limit madagascar">13-limit madagascar</a></div> <!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --><div style="margin-left: 2em;"><a href="#Rank three examples-11-limit portent">11-limit portent</a></div> <!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --><div style="margin-left: 2em;"><a href="#Rank three examples-11-limit zeus">11-limit zeus</a></div> <!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --><div style="margin-left: 1em;"><a href="#Rank four examples">Rank four examples</a></div> <!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --><div style="margin-left: 2em;"><a href="#Rank four examples-Keenanismic">Keenanismic</a></div> <!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextTocRule:56: --><div style="margin-left: 2em;"><a href="#Rank four examples-Swetismic">Swetismic</a></div> <!-- ws:end:WikiTextTocRule:56 --><!-- ws:start:WikiTextTocRule:57: --><div style="margin-left: 2em;"><a href="#Rank four examples-Valinorsmic">Valinorsmic</a></div> <!-- ws:end:WikiTextTocRule:57 --><!-- ws:start:WikiTextTocRule:58: --></div> <!-- ws:end:WikiTextTocRule:58 --><br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Definition of elves"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition of elves</h1> An <em>elf</em> is a scale in a <a class="wiki_link" href="/regular%20temperament">regular temperament</a> which is tempered from a JI scale in the group of the temperament which is <a class="wiki_link" href="/Periodic%20scale">epimorphic</a> 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.<br /> <br /> 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 <a class="wiki_link" href="/Benedetti%20height">Benedetti height</a>) 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 <a class="wiki_link" href="/transversal">transversal</a> 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.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Rank two examples"></a><!-- ws:end:WikiTextHeadingRule:2 -->Rank two examples</h1> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="Rank two examples-13-limit leapday"></a><!-- ws:end:WikiTextHeadingRule:4 -->13-limit leapday</h2> <a class="wiki_link" href="/elfleapday7">elfleapday7</a><br /> <a class="wiki_link" href="/elfleapday8d">elfleapday8d</a><br /> <a class="wiki_link" href="/elfleapday9">elfleapday9</a><br /> <a class="wiki_link" href="/elfleapday10">elfleapday10</a><br /> <a class="wiki_link" href="/elfleapday12f">elfleapday12f</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="Rank two examples-11-limit magic"></a><!-- ws:end:WikiTextHeadingRule:6 -->11-limit magic</h2> <a class="wiki_link" href="/elfmagic7">elfmagic7</a><br /> <a class="wiki_link" href="/elfmagic8">elfmagic8</a><br /> <a class="wiki_link" href="/elfmagic8d">elfmagic8d</a><br /> <a class="wiki_link" href="/elfmagic9">elfmagic9</a><br /> <a class="wiki_link" href="/elfmagic10">elfmagic10</a><br /> <a class="wiki_link" href="/elfmagic12">elfmagic12</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h2> --><h2 id="toc4"><a name="Rank two examples-11-limit miracle"></a><!-- ws:end:WikiTextHeadingRule:8 -->11-limit miracle</h2> <a class="wiki_link" href="/elfmiracle7">elfmiracle7</a><br /> <a class="wiki_link" href="/elfmiracle8d">elfmiracle8d</a><br /> <a class="wiki_link" href="/elfmiracle9">elfmiracle9</a><br /> <a class="wiki_link" href="/elfmiracle10">elfmiracle10</a><br /> <a class="wiki_link" href="/elfmiracle12">elfmiracle12</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h2> --><h2 id="toc5"><a name="Rank two examples-13-limit myna"></a><!-- ws:end:WikiTextHeadingRule:10 -->13-limit myna</h2> <a class="wiki_link" href="/elfmyna7">elfmyna7</a><br /> <a class="wiki_link" href="/elfmyna8d">elfmyna8d</a><br /> <a class="wiki_link" href="/elfmyna9">elfmyna9</a><br /> <a class="wiki_link" href="/elfmyna10">elfmyna10</a><br /> <a class="wiki_link" href="/elfmyna12f">elfmyna12f</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h2> --><h2 id="toc6"><a name="Rank two examples-13-limit octacot"></a><!-- ws:end:WikiTextHeadingRule:12 -->13-limit octacot</h2> <a class="wiki_link" href="/elfoctacot7">elfoctacot7</a><br /> <a class="wiki_link" href="/elfoctacot8d">elfoctacot8d</a><br /> <a class="wiki_link" href="/elfoctacot9">elfoctacot9</a><br /> <a class="wiki_link" href="/elfoctacot10">elfoctacot10</a><br /> <a class="wiki_link" href="/elfoctacot12f">elfoctacot12f</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h2> --><h2 id="toc7"><a name="Rank two examples-13-limit qilin"></a><!-- ws:end:WikiTextHeadingRule:14 -->13-limit qilin</h2> <a class="wiki_link" href="/elfqilin7">elfqilin7</a><br /> <a class="wiki_link" href="/elfqilin8d">elfqilin8d</a><br /> <a class="wiki_link" href="/elfqilin9">elfqilin9</a><br /> <a class="wiki_link" href="/elfqilin10">elfqilin10</a><br /> <a class="wiki_link" href="/elfqilin12f">elfqilin12f</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:16:<h2> --><h2 id="toc8"><a name="Rank two examples-13-limit sensus"></a><!-- ws:end:WikiTextHeadingRule:16 -->13-limit sensus</h2> <a class="wiki_link" href="/elfsensus7">elfsensus7</a><br /> <a class="wiki_link" href="/elfsensus8d">elfsensus8d</a><br /> <a class="wiki_link" href="/elfsensus9">elfsensus9</a><br /> <a class="wiki_link" href="/elfsensus10">elfsensus10</a><br /> <a class="wiki_link" href="/elfsensus12">elfsensus12</a><br /> <a class="wiki_link" href="/elfsensus12f">elfsensus12f</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:18:<h2> --><h2 id="toc9"><a name="Rank two examples-11-limit valentine"></a><!-- ws:end:WikiTextHeadingRule:18 -->11-limit valentine</h2> <a class="wiki_link" href="/elfvalentine7">elfvalentine7</a><br /> <a class="wiki_link" href="/elfvalentine8d">elfvalentine8d</a><br /> <a class="wiki_link" href="/elfvalentine9">elfvalentine9</a><br /> <a class="wiki_link" href="/elfvalentine10">elfvalentine10</a><br /> <a class="wiki_link" href="/elfvalentine12">elfvalentine12</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:20:<h1> --><h1 id="toc10"><a name="Rank three examples"></a><!-- ws:end:WikiTextHeadingRule:20 -->Rank three examples</h1> <!-- ws:start:WikiTextHeadingRule:22:<h2> --><h2 id="toc11"><a name="Rank three examples-11-limit jove"></a><!-- ws:end:WikiTextHeadingRule:22 -->11-limit jove</h2> <a class="wiki_link" href="/elfjove7">elfjove7</a><br /> <a class="wiki_link" href="/elfjove8d">elfjove8d</a><br /> <a class="wiki_link" href="/elfjove9">elfjove9</a><br /> <a class="wiki_link" href="/elfjove10">elfjove10</a><br /> <a class="wiki_link" href="/elfjove11c">elfjove11c</a><br /> <a class="wiki_link" href="/elfjove12">elfjove12</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:24:<h2> --><h2 id="toc12"><a name="Rank three examples-13-limit madagascar"></a><!-- ws:end:WikiTextHeadingRule:24 -->13-limit madagascar</h2> <a class="wiki_link" href="/elfmadagascar7">elfmadagascar7</a><br /> <a class="wiki_link" href="/elfmadagascar8d">elfmadagascar8d</a><br /> <a class="wiki_link" href="/elfmadagascar9">elfmadagascar9</a><br /> <a class="wiki_link" href="/elfmadagascar10">elfmadagascar10</a><br /> <a class="wiki_link" href="/elfmadagascar12f">elfmadagascar12f</a><br /> <a class="wiki_link" href="/elfmadagascar14c">elfmadagascar14c</a><br /> <a class="wiki_link" href="/elfmadagascar15">elfmadagascar15</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:26:<h2> --><h2 id="toc13"><a name="Rank three examples-11-limit portent"></a><!-- ws:end:WikiTextHeadingRule:26 -->11-limit portent</h2> <a class="wiki_link" href="/elfportent9">elfportent9</a><br /> <a class="wiki_link" href="/elfportent10">elfportent10</a><br /> <a class="wiki_link" href="/elfportent11c">elfportent11c</a><br /> <a class="wiki_link" href="/elfportent12">elfportent12</a><br /> <a class="wiki_link" href="/elfportent15">elfportent15</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:28:<h2> --><h2 id="toc14"><a name="Rank three examples-11-limit zeus"></a><!-- ws:end:WikiTextHeadingRule:28 -->11-limit zeus</h2> <a class="wiki_link" href="/zeus7tri">zeus7tri</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:30:<h1> --><h1 id="toc15"><a name="Rank four examples"></a><!-- ws:end:WikiTextHeadingRule:30 -->Rank four examples</h1> <!-- ws:start:WikiTextHeadingRule:32:<h2> --><h2 id="toc16"><a name="Rank four examples-Keenanismic"></a><!-- ws:end:WikiTextHeadingRule:32 -->Keenanismic</h2> <a class="wiki_link" href="/elfkeenanismic7">elfkeenanismic7</a><br /> <a class="wiki_link" href="/elfkeenanismic8d">elfkeenanismic8d</a><br /> <a class="wiki_link" href="/elfkeenanismic9">elfkeenanismic9</a><br /> <a class="wiki_link" href="/elfkeenanismic10">elfkeenanismic10</a><br /> <a class="wiki_link" href="/elfkeenanismic11c">elfkeenanismic11c</a><br /> <a class="wiki_link" href="/elfkeenanismic12">elfkeenanismic12</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:34:<h2> --><h2 id="toc17"><a name="Rank four examples-Swetismic"></a><!-- ws:end:WikiTextHeadingRule:34 -->Swetismic</h2> <a class="wiki_link" href="/elfswetismic8d">elfswetismic8d</a><br /> <a class="wiki_link" href="/elfswetismic9">elfswetismic9</a><br /> <a class="wiki_link" href="/elfswetismic10">elfswetismic10</a><br /> <a class="wiki_link" href="/elfswetismic12">elfswetismic12</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:36:<h2> --><h2 id="toc18"><a name="Rank four examples-Valinorsmic"></a><!-- ws:end:WikiTextHeadingRule:36 -->Valinorsmic</h2> <a class="wiki_link" href="/elfvalinorsmic7">elfvalinorsmic7</a><br /> <a class="wiki_link" href="/elfvalinorsmic8d">elfvalinorsmic8d</a><br /> <a class="wiki_link" href="/elfvalinorsmic9">elfvalinorsmic9</a><br /> <a class="wiki_link" href="/elfvalinorsmic10">elfvalinorsmic10</a><br /> <a class="wiki_link" href="/elfvalinorsmic11c">elfvalinorsmic11c</a><br /> <a class="wiki_link" href="/elfvalinorsmic12">elfvalinorsmic12</a></body></html>