Elf: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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-~~10 22~~:33:44 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-01-12 11:07:52 UTC</tt>.<br>

: The original revision id was <tt>~~397507214~~</tt>.<br>

: The original revision id was <tt>397858578</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>

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<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.

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. 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.

=Examples=

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[[elfmyna10]]

[[elfmyna12f]]

==13-limit sensus==

[[elfsensus7]]

[[elfsensus8d]]

[[elfsensus9]]

[[elfsensus10]]

[[elfsensus12]]

[[elfsensus12f]]

==11-limit valentine==

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[[elfvalentine9]]

[[elfvalentine10]]

[[elfvalentine12]]

[[elfvalentine12]]</pre></div>

</pre></div>

<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 />

<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. 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 />

<h1 id="toc0"><a name="Examples"></a>Examples</h1>

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<a class="wiki_link" href="/elfmyna12f">elfmyna12f</a><br />

<br />

<h2 id="toc4"><a name="Examples-11-limit valentine"></a>11-limit valentine</h2>

<h2 id="toc4"><a name="Examples-13-limit sensus"></a>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 />

<h2 id="toc5"><a name="Examples-11-limit valentine"></a>11-limit valentine</h2>

<a class="wiki_link" href="/elfvalentine7">elfvalentine7</a><br />

<a class="wiki_link" href="/elfvalentine8d">elfvalentine8d</a><br />

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2013-01-10 22:33:44 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-01-12 11:07:52 UTC</tt>.<br>
-: The original revision id was <tt>397507214</tt>.<br>
+: The original revision id was <tt>397858578</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>
@@ Line 8: / Line 8: @@
 <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.
-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. 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.
 =Examples=
@@ Line 33: / Line 33: @@
 [[elfmyna10]]
 [[elfmyna12f]]
+==13-limit sensus==
+[[elfsensus7]]
+[[elfsensus8d]]
+[[elfsensus9]]
+[[elfsensus10]]
+[[elfsensus12]]
+[[elfsensus12f]]
 ==11-limit valentine==
@@ Line 39: / Line 47: @@
 [[elfvalentine9]]
 [[elfvalentine10]]
-[[elfvalentine12]]
+[[elfvalentine12]]</pre></div>
-</pre></div>
 <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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Elves&lt;/title&gt;&lt;/head&gt;&lt;body&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. The tempering of this transversal by a tuning map for the temperament is the elf.&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:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Examples&lt;/h1&gt;
@@ Line 70: / Line 77: @@
 &lt;a class="wiki_link" href="/elfmyna12f"&gt;elfmyna12f&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="Examples-11-limit valentine"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;11-limit valentine&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Examples-13-limit sensus"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&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:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Examples-11-limit valentine"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&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;