EDT: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-10-16 22:19:18 UTC</tt>.<br>

: The original revision id was <tt>~~595502644~~</tt>.<br>

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The other no twos rank two temperament which 13edt "supports" is [[Arcturus]], which takes an ~5:3 as a generator. I speak advisedly of 13edt supporting this temperament because the lowest-error proper MOS of it is a 2L 11s triskaidecatonic scale. However, if you do not mind having a smeary 5, you will need only a 2L 7s (nonatonic) scale to make an understandable rendition of it.

~~A final~~ rank two temperament which 13edt "supports" is [[Sirius]], which takes a generator between ~7:6 and -6:5. Like Arcturus, I speak advisedly of 13edt supporting it because the most proper small MOS of it is triskaidecatonic. Unlike Arcturus, there is a smaller MOS of it than this which is technically proper. However, this MOS is the Grumpy heptatonic scale the use of which is made problematic by the uniqueness of the step of the second size. It is problematic to have the step of the second size be unique in a subscale of an edX because it creates a strong sense of a second equal division of a Y strictly less than X, and this sense of two different equal divisions a trying to happen in the same scale causes ordinary concepts of equivalence to break down in spectacular ways.

The named but not necessarily no twos rank two temperament which 13edt "supports" is [[Sirius]], which takes a generator between ~7:6 and -6:5. Like Arcturus, I speak advisedly of 13edt supporting it because the most proper small MOS of it is triskaidecatonic. Unlike Arcturus, there is a smaller MOS of it than this which is technically proper. However, this MOS is the Grumpy heptatonic scale the use of which is made problematic by the uniqueness of the step of the second size. It is problematic to have the step of the second size be unique in a subscale of an edX because it creates a strong sense of a second equal division of a Y strictly less than X, and this sense of two different equal divisions a trying to happen in the same scale causes ordinary concepts of equivalence to break down in spectacular ways.

The final interval which 13edt can reasonably use to generate a rank two temperament is its false 3\2 of 5 degrees. By a weird coincidence, it will generate the [[5L 3s]] unfair father octatonic scale just as if it were an interval of an edo, except that the scale will not always contain a false 4\3 as it must in an edo. This means, most importantly, that 16\15 cannot be assumed to be a "comma" tempered out by this false Father temperament when it is taken as a temperament of full just intonation. By a second, and totally separate, weird coincidence, the well-known Bohlen-Pierce temperament is its index-2 subtemperament.

Among the EDTs tempering out 245/243, 13EDT stands out. An apt analogy can be drawn with EDOs supporting meantone: 4EDT and 9EDT are to BP what 5EDO and 7EDO are to meantone. However, in contrast to meantone, the simplest EDO supporting the BP nonatonic scale - 13EDT, the traditional tempered BP scale - is the most accurate and lowest in tuning error until 56EDT. However, there are many EDTs supporting BP temperament which also support extensions to higher-limit temperaments; in particular 2, 3, and 4 times 13 in the form of 26EDT, 39EDT and 52EDT as well as 56EDT. For tempering out 16875/16807, 13EDT again stands out, though much better accuracy can be found in more complex divisions such as 114EDT or 127EDT. All of this explains the focus on 13EDT to the exclusion of other EDTs among practitioners of the art of nonoctave composition, but it must be noted that the analysis is only valid if consideration is confined to the 7-limit, which is exactly analogous to confining it to the 5-limit with EDOs. There's a whole other world out there which has not been much explored.

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<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>edt</title></head><body>See also Heinz Bohlen's work:<br />

<a class="wiki_link_ext" href="http://www.huygens-fokker.org/bpsite/otherscales.html" rel="nofollow">http://www.huygens-fokker.org/bpsite/otherscales.html</a><br />

<a class="wiki_link_ext" href="http://www.huygens-fokker.org/bpsite/otherscales.html" rel="nofollow">http://www.huygens-fokker.org/bpsite/otherscales.html</a><br />

<a href="#Division of the tritave (3/1) into n equal parts">Division of the tritave (3/1) into n equal parts</a> | <a href="#Rank two temperaments">Rank two temperaments</a> | <a href="#Individual pages for EDT's">Individual pages for EDT's</a> | <a href="#Multiples of 13EDT which approximate EDO">Multiples of 13EDT which approximate EDO</a>

<hr />

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The other no twos rank two temperament which 13edt &quot;supports&quot; is <a class="wiki_link" href="/Arcturus">Arcturus</a>, which takes an ~5:3 as a generator. I speak advisedly of 13edt supporting this temperament because the lowest-error proper MOS of it is a 2L 11s triskaidecatonic scale. However, if you do not mind having a smeary 5, you will need only a 2L 7s (nonatonic) scale to make an understandable rendition of it.<br />

<br />

~~A final~~ rank two temperament which 13edt &quot;supports&quot; is <a class="wiki_link" href="/Sirius">Sirius</a>, which takes a generator between ~7:6 and -6:5. Like Arcturus, I speak advisedly of 13edt supporting it because the most proper small MOS of it is triskaidecatonic. Unlike Arcturus, there is a smaller MOS of it than this which is technically proper. However, this MOS is the Grumpy heptatonic scale the use of which is made problematic by the uniqueness of the step of the second size. It is problematic to have the step of the second size be unique in a subscale of an edX because it creates a strong sense of a second equal division of a Y strictly less than X, and this sense of two different equal divisions a trying to happen in the same scale causes ordinary concepts of equivalence to break down in spectacular ways.<br />

The named but not necessarily no twos rank two temperament which 13edt &quot;supports&quot; is <a class="wiki_link" href="/Sirius">Sirius</a>, which takes a generator between ~7:6 and -6:5. Like Arcturus, I speak advisedly of 13edt supporting it because the most proper small MOS of it is triskaidecatonic. Unlike Arcturus, there is a smaller MOS of it than this which is technically proper. However, this MOS is the Grumpy heptatonic scale the use of which is made problematic by the uniqueness of the step of the second size. It is problematic to have the step of the second size be unique in a subscale of an edX because it creates a strong sense of a second equal division of a Y strictly less than X, and this sense of two different equal divisions a trying to happen in the same scale causes ordinary concepts of equivalence to break down in spectacular ways.<br />

<br />

The final interval which 13edt can reasonably use to generate a rank two temperament is its false 3\2 of 5 degrees. By a weird coincidence, it will generate the <a class="wiki_link" href="/5L%203s">5L 3s</a> unfair father octatonic scale just as if it were an interval of an edo, except that the scale will not always contain a false 4\3 as it must in an edo. This means, most importantly, that 16\15 cannot be assumed to be a &quot;comma&quot; tempered out by this false Father temperament when it is taken as a temperament of full just intonation. By a second, and totally separate, weird coincidence, the well-known Bohlen-Pierce temperament is its index-2 subtemperament. <br />

<br />

Among the EDTs tempering out 245/243, 13EDT stands out. An apt analogy can be drawn with EDOs supporting meantone: 4EDT and 9EDT are to BP what 5EDO and 7EDO are to meantone. However, in contrast to meantone, the simplest EDO supporting the BP nonatonic scale - 13EDT, the traditional tempered BP scale - is the most accurate and lowest in tuning error until 56EDT. However, there are many EDTs supporting BP temperament which also support extensions to higher-limit temperaments; in particular 2, 3, and 4 times 13 in the form of 26EDT, 39EDT and 52EDT as well as 56EDT. For tempering out 16875/16807, 13EDT again stands out, though much better accuracy can be found in more complex divisions such as 114EDT or 127EDT. All of this explains the focus on 13EDT to the exclusion of other EDTs among practitioners of the art of nonoctave composition, but it must be noted that the analysis is only valid if consideration is confined to the 7-limit, which is exactly analogous to confining it to the 5-limit with EDOs. There's a whole other world out there which has not been much explored.<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:JosephRuhf|JosephRuhf]] and made on <tt>2016-10-15 10:00:17 UTC</tt>.<br>
+: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-10-16 22:19:18 UTC</tt>.<br>
-: The original revision id was <tt>595502644</tt>.<br>
+: The original revision id was <tt>595575930</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 25: / Line 25: @@
 The other no twos rank two temperament which 13edt "supports" is [[Arcturus]], which takes an ~5:3 as a generator. I speak advisedly of 13edt supporting this temperament because the lowest-error proper MOS of it is a 2L 11s triskaidecatonic scale. However, if you do not mind having a smeary 5, you will need only a 2L 7s (nonatonic) scale to make an understandable rendition of it.
-A final rank two temperament which 13edt "supports" is [[Sirius]], which takes a generator between ~7:6 and -6:5. Like Arcturus, I speak advisedly of 13edt supporting it because the most proper small MOS of it is triskaidecatonic. Unlike Arcturus, there is a smaller MOS of it than this which is technically proper. However, this MOS is the Grumpy heptatonic scale the use of which is made problematic by the uniqueness of the step of the second size. It is problematic to have the step of the second size be unique in a subscale of an edX because it creates a strong sense of a second equal division of a Y strictly less than X, and this sense of two different equal divisions a trying to happen in the same scale causes ordinary concepts of equivalence to break down in spectacular ways.
+The named but not necessarily no twos rank two temperament which 13edt "supports" is [[Sirius]], which takes a generator between ~7:6 and -6:5. Like Arcturus, I speak advisedly of 13edt supporting it because the most proper small MOS of it is triskaidecatonic. Unlike Arcturus, there is a smaller MOS of it than this which is technically proper. However, this MOS is the Grumpy heptatonic scale the use of which is made problematic by the uniqueness of the step of the second size. It is problematic to have the step of the second size be unique in a subscale of an edX because it creates a strong sense of a second equal division of a Y strictly less than X, and this sense of two different equal divisions a trying to happen in the same scale causes ordinary concepts of equivalence to break down in spectacular ways.
+The final interval which 13edt can reasonably use to generate a rank two temperament is its false 3\2 of 5 degrees. By a weird coincidence, it will generate the [[5L 3s]] unfair father octatonic scale just as if it were an interval of an edo, except that the scale will not always contain a false 4\3 as it must in an edo. This means, most importantly, that 16\15 cannot be assumed to be a "comma" tempered out by this false Father temperament when it is taken as a temperament of full just intonation. By a second, and totally separate, weird coincidence, the well-known Bohlen-Pierce temperament is its index-2 subtemperament.
 Among the EDTs tempering out 245/243, 13EDT stands out. An apt analogy can be drawn with EDOs supporting meantone: 4EDT and 9EDT are to BP what 5EDO and 7EDO are to meantone. However, in contrast to meantone, the simplest EDO supporting the BP nonatonic scale - 13EDT, the traditional tempered BP scale - is the most accurate and lowest in tuning error until 56EDT. However, there are many EDTs supporting BP temperament which also support extensions to higher-limit temperaments; in particular 2, 3, and 4 times 13 in the form of 26EDT, 39EDT and 52EDT as well as 56EDT. For tempering out 16875/16807, 13EDT again stands out, though much better accuracy can be found in more complex divisions such as 114EDT or 127EDT. All of this explains the focus on 13EDT to the exclusion of other EDTs among practitioners of the art of nonoctave composition, but it must be noted that the analysis is only valid if consideration is confined to the 7-limit, which is exactly analogous to confining it to the 5-limit with EDOs. There's a whole other world out there which has not been much explored.
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 <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;edt&lt;/title&gt;&lt;/head&gt;&lt;body&gt;See also Heinz Bohlen's work:&lt;br /&gt;
-&lt;!-- ws:start:WikiTextUrlRule:938:http://www.huygens-fokker.org/bpsite/otherscales.html --&gt;&lt;a class="wiki_link_ext" href="http://www.huygens-fokker.org/bpsite/otherscales.html" rel="nofollow"&gt;http://www.huygens-fokker.org/bpsite/otherscales.html&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:938 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:941:http://www.huygens-fokker.org/bpsite/otherscales.html --&gt;&lt;a class="wiki_link_ext" href="http://www.huygens-fokker.org/bpsite/otherscales.html" rel="nofollow"&gt;http://www.huygens-fokker.org/bpsite/otherscales.html&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:941 --&gt;&lt;br /&gt;
 &lt;!-- ws:start:WikiTextTocRule:8:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt;&lt;a href="#Division of the tritave (3/1) into n equal parts"&gt;Division of the tritave (3/1) into n equal parts&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;!-- ws:start:WikiTextTocRule:10: --&gt; | &lt;a href="#Rank two temperaments"&gt;Rank two temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt; | &lt;a href="#Individual pages for EDT's"&gt;Individual pages for EDT's&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextTocRule:12: --&gt; | &lt;a href="#Multiples of 13EDT which approximate EDO"&gt;Multiples of 13EDT which approximate EDO&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;
 &lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;hr /&gt;
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 The other no twos rank two temperament which 13edt &amp;quot;supports&amp;quot; is &lt;a class="wiki_link" href="/Arcturus"&gt;Arcturus&lt;/a&gt;, which takes an ~5:3 as a generator. I speak advisedly of 13edt supporting this temperament because the lowest-error proper MOS of it is a 2L 11s triskaidecatonic scale. However, if you do not mind having a smeary 5, you will need only a 2L 7s (nonatonic) scale to make an understandable rendition of it.&lt;br /&gt;
 &lt;br /&gt;
-A final rank two temperament which 13edt &amp;quot;supports&amp;quot; is &lt;a class="wiki_link" href="/Sirius"&gt;Sirius&lt;/a&gt;, which takes a generator between ~7:6 and -6:5. Like Arcturus, I speak advisedly of 13edt supporting it because the most proper small MOS of it is triskaidecatonic. Unlike Arcturus, there is a smaller MOS of it than this which is technically proper. However, this MOS is the Grumpy heptatonic scale the use of which is made problematic by the uniqueness of the step of the second size. It is problematic to have the step of the second size be unique in a subscale of an edX because it creates a strong sense of a second equal division of a Y strictly less than X, and this sense of two different equal divisions a trying to happen in the same scale causes ordinary concepts of equivalence to break down in spectacular ways.&lt;br /&gt;
+The named but not necessarily no twos rank two temperament which 13edt &amp;quot;supports&amp;quot; is &lt;a class="wiki_link" href="/Sirius"&gt;Sirius&lt;/a&gt;, which takes a generator between ~7:6 and -6:5. Like Arcturus, I speak advisedly of 13edt supporting it because the most proper small MOS of it is triskaidecatonic. Unlike Arcturus, there is a smaller MOS of it than this which is technically proper. However, this MOS is the Grumpy heptatonic scale the use of which is made problematic by the uniqueness of the step of the second size. It is problematic to have the step of the second size be unique in a subscale of an edX because it creates a strong sense of a second equal division of a Y strictly less than X, and this sense of two different equal divisions a trying to happen in the same scale causes ordinary concepts of equivalence to break down in spectacular ways.&lt;br /&gt;
+&lt;br /&gt;
+The final interval which 13edt can reasonably use to generate a rank two temperament is its false 3\2 of 5 degrees. By a weird coincidence, it will generate the &lt;a class="wiki_link" href="/5L%203s"&gt;5L 3s&lt;/a&gt; unfair father octatonic scale just as if it were an interval of an edo, except that the scale will not always contain a false 4\3 as it must in an edo. This means, most importantly, that 16\15 cannot be assumed to be a &amp;quot;comma&amp;quot; tempered out by this false Father temperament when it is taken as a temperament of full just intonation. By a second, and totally separate, weird coincidence, the well-known Bohlen-Pierce temperament is its index-2 subtemperament. &lt;br /&gt;
 &lt;br /&gt;
 Among the EDTs tempering out 245/243, 13EDT stands out. An apt analogy can be drawn with EDOs supporting meantone: 4EDT and 9EDT are to BP what 5EDO and 7EDO are to meantone. However, in contrast to meantone, the simplest EDO supporting the BP nonatonic scale - 13EDT, the traditional tempered BP scale - is the most accurate and lowest in tuning error until 56EDT. However, there are many EDTs supporting BP temperament which also support extensions to higher-limit temperaments; in particular 2, 3, and 4 times 13 in the form of 26EDT, 39EDT and 52EDT as well as 56EDT. For tempering out 16875/16807, 13EDT again stands out, though much better accuracy can be found in more complex divisions such as 114EDT or 127EDT. All of this explains the focus on 13EDT to the exclusion of other EDTs among practitioners of the art of nonoctave composition, but it must be noted that the analysis is only valid if consideration is confined to the 7-limit, which is exactly analogous to confining it to the 5-limit with EDOs. There's a whole other world out there which has not been much explored.&lt;br /&gt;