Elf: Difference between revisions
Wikispaces>genewardsmith **Imported revision 398004978 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 398584508 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | 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>2013-01- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-01-15 10:16:21 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>398584508</tt>.<br> | ||
: The revision comment was: <tt></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> | 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> | <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">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. | <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= | |||
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. | 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== | ==13-limit leapday== | ||
[[elfleapday7]] | [[elfleapday7]] | ||
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[[elfvalentine9]] | [[elfvalentine9]] | ||
[[elfvalentine10]] | [[elfvalentine10]] | ||
[[elfvalentine12]]</pre></div> | [[elfvalentine12]] | ||
=Rank three examples= | |||
==11-limit portent== | |||
[[elfportent9]] | |||
[[elfportent10]] | |||
[[elfportent11c]] | |||
[[elfportent12]] | |||
[[elfportent15]]</pre></div> | |||
<h4>Original HTML content:</h4> | <h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Elves</title></head><body>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 /> | <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"><html><head><title>Elves</title></head><body><!-- ws:start:WikiTextTocRule:24:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --><div style="margin-left: 1em;"><a href="#Definition of elves">Definition of elves</a></div> | ||
<!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --><div style="margin-left: 1em;"><a href="#Rank two examples">Rank two examples</a></div> | |||
<!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><div style="margin-left: 2em;"><a href="#Rank two examples-13-limit leapday">13-limit leapday</a></div> | |||
<!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --><div style="margin-left: 2em;"><a href="#Rank two examples-11-limit magic">11-limit magic</a></div> | |||
<!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><div style="margin-left: 2em;"><a href="#Rank two examples-11-limit miracle">11-limit miracle</a></div> | |||
<!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><div style="margin-left: 2em;"><a href="#Rank two examples-13-limit myna">13-limit myna</a></div> | |||
<!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><div style="margin-left: 2em;"><a href="#Rank two examples-13-limit octacot">13-limit octacot</a></div> | |||
<!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><div style="margin-left: 2em;"><a href="#Rank two examples-13-limit qilin">13-limit qilin</a></div> | |||
<!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><div style="margin-left: 2em;"><a href="#Rank two examples-13-limit sensus">13-limit sensus</a></div> | |||
<!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><div style="margin-left: 2em;"><a href="#Rank two examples-11-limit valentine">11-limit valentine</a></div> | |||
<!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><div style="margin-left: 1em;"><a href="#Rank three examples">Rank three examples</a></div> | |||
<!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><div style="margin-left: 2em;"><a href="#Rank three examples-11-limit portent">11-limit portent</a></div> | |||
<!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --></div> | |||
<!-- ws:end:WikiTextTocRule:37 --><br /> | |||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><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 /> | <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 /> | 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 /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Rank two examples"></a><!-- ws:end:WikiTextHeadingRule:2 -->Rank two examples</h1> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Rank two examples-13-limit leapday"></a><!-- ws:end:WikiTextHeadingRule:4 -->13-limit leapday</h2> | |||
<!-- ws:start:WikiTextHeadingRule: | |||
<a class="wiki_link" href="/elfleapday7">elfleapday7</a><br /> | <a class="wiki_link" href="/elfleapday7">elfleapday7</a><br /> | ||
<a class="wiki_link" href="/elfleapday8d">elfleapday8d</a><br /> | <a class="wiki_link" href="/elfleapday8d">elfleapday8d</a><br /> | ||
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<a class="wiki_link" href="/elfleapday12">elfleapday12</a><br /> | <a class="wiki_link" href="/elfleapday12">elfleapday12</a><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><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="/elfmagic7">elfmagic7</a><br /> | ||
<a class="wiki_link" href="/elfmagic8">elfmagic8</a><br /> | <a class="wiki_link" href="/elfmagic8">elfmagic8</a><br /> | ||
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<a class="wiki_link" href="/elfmagic12">elfmagic12</a><br /> | <a class="wiki_link" href="/elfmagic12">elfmagic12</a><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><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="/elfmiracle7">elfmiracle7</a><br /> | ||
<a class="wiki_link" href="/elfmiracle8d">elfmiracle8d</a><br /> | <a class="wiki_link" href="/elfmiracle8d">elfmiracle8d</a><br /> | ||
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<a class="wiki_link" href="/elfmiracle12">elfmiracle12</a><br /> | <a class="wiki_link" href="/elfmiracle12">elfmiracle12</a><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><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="/elfmyna7">elfmyna7</a><br /> | ||
<a class="wiki_link" href="/elfmyna8d">elfmyna8d</a><br /> | <a class="wiki_link" href="/elfmyna8d">elfmyna8d</a><br /> | ||
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<a class="wiki_link" href="/elfmyna12f">elfmyna12f</a><br /> | <a class="wiki_link" href="/elfmyna12f">elfmyna12f</a><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><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="/elfoctacot7">elfoctacot7</a><br /> | ||
<a class="wiki_link" href="/elfoctacot8d">elfoctacot8d</a><br /> | <a class="wiki_link" href="/elfoctacot8d">elfoctacot8d</a><br /> | ||
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<a class="wiki_link" href="/elfoctacot12f">elfoctacot12f</a><br /> | <a class="wiki_link" href="/elfoctacot12f">elfoctacot12f</a><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><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="/elfqilin7">elfqilin7</a><br /> | ||
<a class="wiki_link" href="/elfqilin8d">elfqilin8d</a><br /> | <a class="wiki_link" href="/elfqilin8d">elfqilin8d</a><br /> | ||
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<a class="wiki_link" href="/elfqilin12f">elfqilin12f</a><br /> | <a class="wiki_link" href="/elfqilin12f">elfqilin12f</a><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><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="/elfsensus7">elfsensus7</a><br /> | ||
<a class="wiki_link" href="/elfsensus8d">elfsensus8d</a><br /> | <a class="wiki_link" href="/elfsensus8d">elfsensus8d</a><br /> | ||
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<a class="wiki_link" href="/elfsensus12f">elfsensus12f</a><br /> | <a class="wiki_link" href="/elfsensus12f">elfsensus12f</a><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><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="/elfvalentine7">elfvalentine7</a><br /> | ||
<a class="wiki_link" href="/elfvalentine8d">elfvalentine8d</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="/elfvalentine9">elfvalentine9</a><br /> | ||
<a class="wiki_link" href="/elfvalentine10">elfvalentine10</a><br /> | <a class="wiki_link" href="/elfvalentine10">elfvalentine10</a><br /> | ||
<a class="wiki_link" href="/elfvalentine12">elfvalentine12</a></body></html></pre></div> | <a class="wiki_link" href="/elfvalentine12">elfvalentine12</a><br /> | ||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:20:&lt;h1&gt; --><h1 id="toc10"><a name="Rank three examples"></a><!-- ws:end:WikiTextHeadingRule:20 -->Rank three examples</h1> | |||
<!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="Rank three examples-11-limit portent"></a><!-- ws:end:WikiTextHeadingRule:22 -->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></body></html></pre></div> | |||