2L 8s: 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:

: This revision was by author [[User:~~JosephRuhf~~|~~JosephRuhf~~]] and made on <tt>~~2012~~-11-~~25 23~~:39:58 UTC</tt>.

: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2013-01-20 16:29:34 UTC</tt>.

: The original revision id was <tt>~~385933550~~</tt>.

: The original revision id was <tt>399961292</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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|| || || || || || 1\20 || || || || 60 ||= ||

|| || || || || 1\18 || || || || || 66.67 ||= ||

|| || || || 1\16 || || || || || || 75 ||= L/s=4:

|| || || || 1\16 || || || || || || 75 ||= L/s = 4 ||

~~generators smaller than this are "impractical"~~ ||

|| || || 1\14 || || || || || || || 85.71 ||= L/s = 3 ||

|| || || 1\14 || || || || || || || 85.71 ||= L/s=3:

~~generators smaller than this may be considered "impractical"~~ ||

|| || || || 2\26 || || || || || || 92.31 ||= ||

|| || || || || 3\38 || || || || || 94.74 ||= ||

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The only significant harmonic entropy minimum that is <a class="wiki_link" href="/Rothenberg%20propriety">proper</a> is the decatonic scale itself (<a class="wiki_link" href="/Diaschismic%20family">pajara</a>[10]), in which the period is 7/5~10/7 (tempered to be the same interval), one generator down from that makes <a class="wiki_link" href="/4_3">4/3</a>, and another generator down makes <a class="wiki_link" href="/5_4">5/4</a>. More than a few people think this is a beautiful scale that deserves a lot of investigation and use, with some going so far as to say it's the next step up from the diatonic scale that preserves the most desirable features of diatonic melody and harmony. Paul Erlich's original paper on this scale can be found at either of these links:

<a class="wiki_link_ext" href="http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf" rel="nofollow">http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf</a>

<a class="wiki_link_ext" href="http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf" rel="nofollow">http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf</a>

<a class="wiki_link_ext" href="http://www.lumma.org/tuning/erlich/" rel="nofollow">http://www.lumma.org/tuning/erlich/</a>

<a class="wiki_link_ext" href="http://www.lumma.org/tuning/erlich/" rel="nofollow">http://www.lumma.org/tuning/erlich/</a>

Improper harmonic entropy minima include <a class="wiki_link" href="/Meantone%20family#Injera">injera</a> (which is similar to pajara except that 5/4 is now four generators up and no periods) and <a class="wiki_link" href="/Diaschismic%20family#Shrutar">shrutar</a> (which is basically pajara with the generator divided in two).

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<td>75

</td>

<td style="text-align: center;">L/s=4~~: ~~

<td style="text-align: center;">L/s = 4

~~generators smaller than this are &quot;impractical&quot;~~

</td>

</tr>

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<td>85.71

</td>

<td style="text-align: center;">L/s=3~~: ~~

<td style="text-align: center;">L/s = 3

~~generators smaller than this may be considered &quot;impractical&quot;~~

</td>

</tr>

@@ 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>2012-11-25 23:39:58 UTC</tt>.<br>
+: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2013-01-20 16:29:34 UTC</tt>.<br>
-: The original revision id was <tt>385933550</tt>.<br>
+: The original revision id was <tt>399961292</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 23: / Line 23: @@
 ||   ||   ||   ||   ||   || 1\20 ||   ||   ||   || 60 ||=   ||
 ||   ||   ||   ||   || 1\18 ||   ||   ||   ||   || 66.67 ||=   ||
-||   ||   ||   || 1\16 ||   ||   ||   ||   ||   || 75 ||= L/s=4:
+||   ||   ||   || 1\16 ||   ||   ||   ||   ||   || 75 ||= L/s = 4 ||
-generators smaller than this are "impractical" ||
+||   ||   || 1\14 ||   ||   ||   ||   ||   ||   || 85.71 ||= L/s = 3 ||
-||   ||   || 1\14 ||   ||   ||   ||   ||   ||   || 85.71 ||= L/s=3:
-generators smaller than this may be considered "impractical" ||
 ||   ||   ||   || 2\26 ||   ||   ||   ||   ||   || 92.31 ||=   ||
 ||   ||   ||   ||   || 3\38 ||   ||   ||   ||   || 94.74 ||=   ||
@@ Line 48: / Line 46: @@
 &lt;br /&gt;
 The only significant harmonic entropy minimum that is &lt;a class="wiki_link" href="/Rothenberg%20propriety"&gt;proper&lt;/a&gt; is the decatonic scale itself (&lt;a class="wiki_link" href="/Diaschismic%20family"&gt;pajara&lt;/a&gt;[10]), in which the period is 7/5~10/7 (tempered to be the same interval), one generator down from that makes &lt;a class="wiki_link" href="/4_3"&gt;4/3&lt;/a&gt;, and another generator down makes &lt;a class="wiki_link" href="/5_4"&gt;5/4&lt;/a&gt;. More than a few people think this is a beautiful scale that deserves a lot of investigation and use, with some going so far as to say it's the next step up from the diatonic scale that preserves the most desirable features of diatonic melody and harmony. Paul Erlich's original paper on this scale can be found at either of these links:&lt;br /&gt;
-&lt;!-- ws:start:WikiTextUrlRule:908:http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf --&gt;&lt;a class="wiki_link_ext" href="http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf" rel="nofollow"&gt;http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:908 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:906:http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf --&gt;&lt;a class="wiki_link_ext" href="http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf" rel="nofollow"&gt;http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:906 --&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextUrlRule:909:http://www.lumma.org/tuning/erlich/ --&gt;&lt;a class="wiki_link_ext" href="http://www.lumma.org/tuning/erlich/" rel="nofollow"&gt;http://www.lumma.org/tuning/erlich/&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:909 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:907:http://www.lumma.org/tuning/erlich/ --&gt;&lt;a class="wiki_link_ext" href="http://www.lumma.org/tuning/erlich/" rel="nofollow"&gt;http://www.lumma.org/tuning/erlich/&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:907 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 Improper harmonic entropy minima include &lt;a class="wiki_link" href="/Meantone%20family#Injera"&gt;injera&lt;/a&gt; (which is similar to pajara except that 5/4 is now four generators &lt;strong&gt;up&lt;/strong&gt; and no periods) and &lt;a class="wiki_link" href="/Diaschismic%20family#Shrutar"&gt;shrutar&lt;/a&gt; (which is basically pajara with the generator divided in two).&lt;br /&gt;
@@ Line 254: / Line 252: @@
          &lt;td&gt;75&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;L/s=4:&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;L/s = 4&lt;br /&gt;
-generators smaller than this are &amp;quot;impractical&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 279: / Line 276: @@
          &lt;td&gt;85.71&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;L/s=3:&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;L/s = 3&lt;br /&gt;
-generators smaller than this may be considered &amp;quot;impractical&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;