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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
An '''elf''' is a scale in a [[regular temperament]] which is tempered from a [[just intonation]] (JI) scale in the group of the temperament which is a one-to-one [[detempering]] of ''n''-edo via a [[val]] ''V'' which may not, and characteristically does not, support the temperament. This allows the elf to have more freedom in scale size and structure but still to possess the coherence induced by the one-to-one detempering.
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>2013-01-09 17:55:29 UTC</tt>.<br>
: The original revision id was <tt>397124154</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">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 given an ''n''-edo val ''V'' and a temperament, the following steps are used:
# Take all intervals in the JI group of the temperament which lie within an octave.
# For each interval of the temperament, keep only the least complex (in terms of [[Benedetti height]]) JI interpretation of that interval.
# Construct a detempering of ''n''-edo as follows: For each integer value {{nowrap|1 &le; ''i'' &le; ''n''}}, select the ''i''th degree of the scale from the JI preimage of ''i''\''n'' under ''V'' to be the least temperamentally-complex interval (with ties broken by Benedetti height) among the JI intervals that remain after step 2.
# Temper this detempering of ''n''-edo using a [[tuning map]] for the temperament. This step is done so that it becomes a scale of the given temperament. The result is an elf.


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


==11-limit magic==
=== 11-limit magic ===
[[elfmagic7]]
* [[elfmagic7]]
[[elfmagic8]]
* [[elfmagic8]]
[[elfmagic8d]]
* [[elfmagic8d]]
[[elfmagic9]]
* [[elfmagic9]]
[[elfmagic10]]
* [[elfmagic10]]
</pre></div>
* [[elfmagic12]]
<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;
=== 11-limit miracle ===
&lt;br /&gt;
* [[elfmiracle7]]
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;
* [[elfmiracle8d]]
&lt;br /&gt;
* [[elfmiracle9]]
&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;
* [[elfmiracle10]]
&lt;br /&gt;
* [[elfmiracle12]]
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Examples-11-limit magic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11-limit magic&lt;/h2&gt;
 
&lt;a class="wiki_link" href="/elfmagic7"&gt;elfmagic7&lt;/a&gt;&lt;br /&gt;
=== 13-limit myna ===
&lt;a class="wiki_link" href="/elfmagic8"&gt;elfmagic8&lt;/a&gt;&lt;br /&gt;
* [[elfmyna7]]
&lt;a class="wiki_link" href="/elfmagic8d"&gt;elfmagic8d&lt;/a&gt;&lt;br /&gt;
* [[elfmyna8d]]
&lt;a class="wiki_link" href="/elfmagic9"&gt;elfmagic9&lt;/a&gt;&lt;br /&gt;
* [[elfmyna9]]
&lt;a class="wiki_link" href="/elfmagic10"&gt;elfmagic10&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
* [[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]]
 
[[Category:Elves]] <!-- main article -->

Latest revision as of 19:40, 17 May 2026

An elf is a scale in a regular temperament which is tempered from a just intonation (JI) scale in the group of the temperament which is a one-to-one detempering of n-edo via a val V which may not, and characteristically does not, support the temperament. This allows the elf to have more freedom in scale size and structure but still to possess the coherence induced by the one-to-one detempering.

To construct an elf given an n-edo val V and a temperament, the following steps are used:

  1. Take all intervals in the JI group of the temperament which lie within an octave.
  2. For each interval of the temperament, keep only the least complex (in terms of Benedetti height) JI interpretation of that interval.
  3. Construct a detempering of n-edo as follows: For each integer value 1 ≤ i ≤ n, select the ith degree of the scale from the JI preimage of i\n under V to be the least temperamentally-complex interval (with ties broken by Benedetti height) among the JI intervals that remain after step 2.
  4. Temper this detempering of n-edo using a tuning map for the temperament. This step is done so that it becomes a scale of the given temperament. The result is an elf.

Rank two examples

13-limit leapday

11-limit magic

11-limit miracle

13-limit myna

13-limit octacot

13-limit qilin

13-limit sensus

11-limit valentine

Rank three examples

11-limit jove

13-limit madagascar

11-limit portent

11-limit thrush

11-limit zeus

Rank four examples

Keenanismic

Swetismic

Valinorsmic

Retrieved from "https://en.xen.wiki/index.php?title=Elf&oldid=230500"