Hobbit: 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>2011-03-10 14:47:50 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-10 15:01:25 UTC</tt>.<br>

: The original revision id was <tt>~~209342662~~</tt>.<br>

: The original revision id was <tt>209350434</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">A //hobbit scale// is a generalization of [[MOSScales|MOS]] for arbitrary regular temperaments which is a sort of cousin to [[Dwarves|dwarf scales]]; examples may be found on the [[Scalesmith]] page. The idea is that MOS scales give us a means of contructing scales for a [[~~http://xenharmonic.wikispaces.com/~~Regular+Temperaments#Rank 2 (including "linear") temperaments|rank two regular temperament]] which gives priority to the intervals of least complexity in that temperament, and so makes efficient use of it; a hobbit does the same in higher ranks, and so using them is one way to make higher ranks, including especially the interesting rank three case, accessible for musical purposes.

<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">A //hobbit scale// is a generalization of [[MOSScales|MOS]] for arbitrary regular temperaments which is a sort of cousin to [[Dwarves|dwarf scales]]; examples may be found on the [[Scalesmith]] page. The idea is that MOS scales give us a means of contructing scales for a [[Regular+Temperaments#Rank 2 (including "linear") temperaments|rank two regular temperament]] which gives priority to the intervals of least complexity in that temperament, and so makes efficient use of it; a hobbit does the same in higher ranks, and so using them is one way to make higher ranks, including especially the interesting rank three case, accessible for musical purposes.

Given a regular temperament and an equal temperament val v which supports (or belongs to) the temperament, there is a unique scale for the temperament, which can be tuned to any tuning of the temperament, containing v[1] notes to the octave.

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After applying such a tuning, we discover than there seems to be a certain irregularity or inconsistency in action, in that some of the 11-limit intervals do not stem from the mapping for minerva, but represent additional temperings by 243/242 or 4000/3993. By adding one of these, we can flatten out the irregularity to a corresponding rank two temperament; by adding both, we obtain the rank one temperament with val <65 103 151 183 225|, giving a scale with steps 2433333242432424233333. Examples of this sort inconsistency seem to increase with increasing rank.</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>Hobbits</title></head><body>A <em>hobbit scale</em> is a generalization of <a class="wiki_link" href="/MOSScales">MOS</a> for arbitrary regular temperaments which is a sort of cousin to <a class="wiki_link" href="/Dwarves">dwarf scales</a>; examples may be found on the <a class="wiki_link" href="/Scalesmith">Scalesmith</a> page. The idea is that MOS scales give us a means of contructing scales for a [[~~<a href="http://xenharmonic.wikispaces.com/Regular+Temperaments#Rank">http://xenharmonic.wikispaces.com/Regular+Temperaments#Rank</a>~~ 2 (including &quot;linear&quot;) temperaments|rank two regular temperament]] which gives priority to the intervals of least complexity in that temperament, and so makes efficient use of it; a hobbit does the same in higher ranks, and so using them is one way to make higher ranks, including especially the interesting rank three case, accessible for musical purposes. <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>Hobbits</title></head><body>A <em>hobbit scale</em> is a generalization of <a class="wiki_link" href="/MOSScales">MOS</a> for arbitrary regular temperaments which is a sort of cousin to <a class="wiki_link" href="/Dwarves">dwarf scales</a>; examples may be found on the <a class="wiki_link" href="/Scalesmith">Scalesmith</a> page. The idea is that MOS scales give us a means of contructing scales for a [[Regular+Temperaments#Rank 2 (including &quot;linear&quot;) temperaments|rank two regular temperament]] which gives priority to the intervals of least complexity in that temperament, and so makes efficient use of it; a hobbit does the same in higher ranks, and so using them is one way to make higher ranks, including especially the interesting rank three case, accessible for musical purposes. <br />

<br />

Given a regular temperament and an equal temperament val v which supports (or belongs to) the temperament, there is a unique scale for the temperament, which can be tuned to any tuning of the temperament, containing v[1] notes to the octave.<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>2011-03-10 14:47:50 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-10 15:01:25 UTC</tt>.<br>
-: The original revision id was <tt>209342662</tt>.<br>
+: The original revision id was <tt>209350434</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">A //hobbit scale// is a generalization of [[MOSScales|MOS]] for arbitrary regular temperaments which is a sort of cousin to [[Dwarves|dwarf scales]]; examples may be found on the [[Scalesmith]] page. The idea is that MOS scales give us a means of contructing scales for a [[http://xenharmonic.wikispaces.com/Regular+Temperaments#Rank 2 (including "linear") temperaments|rank two regular temperament]] which gives priority to the intervals of least complexity in that temperament, and so makes efficient use of it; a hobbit does the same in higher ranks, and so using them is one way to make higher ranks, including especially the interesting rank three case, accessible for musical purposes.
+<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">A //hobbit scale// is a generalization of [[MOSScales|MOS]] for arbitrary regular temperaments which is a sort of cousin to [[Dwarves|dwarf scales]]; examples may be found on the [[Scalesmith]] page. The idea is that MOS scales give us a means of contructing scales for a [[Regular+Temperaments#Rank 2 (including "linear") temperaments|rank two regular temperament]] which gives priority to the intervals of least complexity in that temperament, and so makes efficient use of it; a hobbit does the same in higher ranks, and so using them is one way to make higher ranks, including especially the interesting rank three case, accessible for musical purposes.
 Given a regular temperament and an equal temperament val v which supports (or belongs to) the temperament, there is a unique scale for the temperament, which can be tuned to any tuning of the temperament, containing v[1] notes to the octave.
@@ Line 26: / Line 26: @@
 After applying such a tuning, we discover than there seems to be a certain irregularity or inconsistency in action, in that some of the 11-limit intervals do not stem from the mapping for minerva, but represent additional temperings by 243/242 or 4000/3993. By adding one of these, we can flatten out the irregularity to a corresponding rank two temperament; by adding both, we obtain the rank one temperament with val &lt;65 103 151 183 225|, giving a scale with steps 2433333242432424233333. Examples of this sort inconsistency seem to increase with increasing rank.</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;Hobbits&lt;/title&gt;&lt;/head&gt;&lt;body&gt;A &lt;em&gt;hobbit scale&lt;/em&gt; is a generalization of &lt;a class="wiki_link" href="/MOSScales"&gt;MOS&lt;/a&gt; for arbitrary regular temperaments which is a sort of cousin to &lt;a class="wiki_link" href="/Dwarves"&gt;dwarf scales&lt;/a&gt;; examples may be found on the &lt;a class="wiki_link" href="/Scalesmith"&gt;Scalesmith&lt;/a&gt; page. The idea is that MOS scales give us a means of contructing scales for a [[&lt;!-- ws:start:WikiTextUrlRule:29:http://xenharmonic.wikispaces.com/Regular+Temperaments#Rank --&gt;&lt;a href="http://xenharmonic.wikispaces.com/Regular+Temperaments#Rank"&gt;http://xenharmonic.wikispaces.com/Regular+Temperaments#Rank&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:29 --&gt; 2 (including &amp;quot;linear&amp;quot;) temperaments|rank two regular temperament]] which gives priority to the intervals of least complexity in that temperament, and so makes efficient use of it; a hobbit does the same in higher ranks, and so using them is one way to make higher ranks, including especially the interesting rank three case, accessible for musical purposes. &lt;br /&gt;
+<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;Hobbits&lt;/title&gt;&lt;/head&gt;&lt;body&gt;A &lt;em&gt;hobbit scale&lt;/em&gt; is a generalization of &lt;a class="wiki_link" href="/MOSScales"&gt;MOS&lt;/a&gt; for arbitrary regular temperaments which is a sort of cousin to &lt;a class="wiki_link" href="/Dwarves"&gt;dwarf scales&lt;/a&gt;; examples may be found on the &lt;a class="wiki_link" href="/Scalesmith"&gt;Scalesmith&lt;/a&gt; page. The idea is that MOS scales give us a means of contructing scales for a [[Regular+Temperaments#Rank 2 (including &amp;quot;linear&amp;quot;) temperaments|rank two regular temperament]] which gives priority to the intervals of least complexity in that temperament, and so makes efficient use of it; a hobbit does the same in higher ranks, and so using them is one way to make higher ranks, including especially the interesting rank three case, accessible for musical purposes. &lt;br /&gt;
 &lt;br /&gt;
 Given a regular temperament and an equal temperament val v which supports (or belongs to) the temperament, there is a unique scale for the temperament, which can be tuned to any tuning of the temperament, containing v[1] notes to the octave.&lt;br /&gt;