5edo: 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:~~xenwolf~~|~~xenwolf~~]] and made on <tt>~~2010~~-09-~~23 16~~:38:31 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-22 18:32:03 UTC</tt>.<br>

: The original revision id was <tt>~~164989553~~</tt>.<br>

: The original revision id was <tt>212989204</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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==As a temperament==

If 5-edo is regarded as a temperament, which is to say as 5-et, then the most salient fact is that 16/15 is tempered out. This means in 5-et the major third and the fourth, and the minor sixth and the fifth, are not distinguished. This is at the very edge what can sensibly be called temperament, but it does make sense and can be used.

If 5-edo is regarded as a temperament, which is to say as 5-et, then the most salient fact is that 16/15 is tempered out. This means in 5-et the major third and the fourth, and the minor sixth and the fifth, are not distinguished; this is 5-limit [[Trienstonic clan|father temperament]]. This is at the very edge what can sensibly be called temperament, but it does make sense and can be used.

Also tempered out is 27/25, ~~leading~~ to [[~~beep~~ temperament]], which equates 10/9 with 6/5: it is a little more perverse even than [[father ~~temperament|father]]~~. Because these intervals are so large, this sort of analysis is less significant with 5 than it becomes with larger and more accurate divisions, but it still plays a role. For example, I-IV-V-I is the same as 1-III-V-I and involves triads with common intervals because of fourth-thirds equivalence.

Also tempered out is 27/25, if we temper this out in preference to 16/15 we obtain [[Bug family|bug temperament]], which equates 10/9 with 6/5: it is a little more perverse even than father. Because these intervals are so large, this sort of analysis is less significant with 5 than it becomes with larger and more accurate divisions, but it still plays a role. For example, I-IV-V-I is the same as 1-III-V-I and involves triads with common intervals because of fourth-thirds equivalence.

Despite its lack of accuracy, 5EDO is the second Zeta function integral tuning, after 2EDO. See http://www.research.att.com/~njas/sequences/A117538

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<ul><li>By its cardinality, 5-edo is related to other <a class="wiki_link" href="/pentatonic">pentatonic</a> scales, and it is especially close in sound to many Indonesian <a class="wiki_link" href="/slendro">slendros</a>.</li><li>Due to the interest around the &quot;fifth&quot; interval size, there are many <a class="wiki_link" href="/nonoctave">nonoctave</a> &quot;stretch sisters&quot; to 5-edo: square root of 4/3, cube root of 3/2, 8th root of 3, etc.</li><li>For the same reason there are many &quot;circle sisters&quot;:<ul><li>Make a chain of five &quot;bigger fifths&quot; (50/33), which makes three octaves 3.227¢ flat. (50/33)^5=7.985099.</li></ul></li></ul><br />

<h2 id="toc4"><a name="x5 Equal Divisions of the Octave: Theory-As a temperament"></a>As a temperament</h2>

If 5-edo is regarded as a temperament, which is to say as 5-et, then the most salient fact is that 16/15 is tempered out. This means in 5-et the major third and the fourth, and the minor sixth and the fifth, are not distinguished. This is at the very edge what can sensibly be called temperament, but it does make sense and can be used.<br />

If 5-edo is regarded as a temperament, which is to say as 5-et, then the most salient fact is that 16/15 is tempered out. This means in 5-et the major third and the fourth, and the minor sixth and the fifth, are not distinguished; this is 5-limit <a class="wiki_link" href="/Trienstonic%20clan">father temperament</a>. This is at the very edge what can sensibly be called temperament, but it does make sense and can be used.<br />

<br />

Also tempered out is 27/25, ~~leading~~ to <a class="wiki_link" href="/~~beep~~%~~20temperament~~">~~beep~~ temperament</a>, which equates 10/9 with 6/5: it is a little more perverse even than ~~<a class="wiki_link" href="/~~father~~%20temperament">father</a>~~. Because these intervals are so large, this sort of analysis is less significant with 5 than it becomes with larger and more accurate divisions, but it still plays a role. For example, I-IV-V-I is the same as 1-III-V-I and involves triads with common intervals because of fourth-thirds equivalence.<br />

Also tempered out is 27/25, if we temper this out in preference to 16/15 we obtain <a class="wiki_link" href="/Bug%20family">bug temperament</a>, which equates 10/9 with 6/5: it is a little more perverse even than father. Because these intervals are so large, this sort of analysis is less significant with 5 than it becomes with larger and more accurate divisions, but it still plays a role. For example, I-IV-V-I is the same as 1-III-V-I and involves triads with common intervals because of fourth-thirds equivalence.<br />

<br />

Despite its lack of accuracy, 5EDO is the second Zeta function integral tuning, after 2EDO. See <a class="wiki_link_ext" href="http://www.research.att.com/~njas/sequences/A117538" rel="nofollow">http://www.research.att.com/~njas/sequences/A117538</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:xenwolf|xenwolf]] and made on <tt>2010-09-23 16:38:31 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-22 18:32:03 UTC</tt>.<br>
-: The original revision id was <tt>164989553</tt>.<br>
+: The original revision id was <tt>212989204</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 49: / Line 49: @@
 ==As a temperament==
-If 5-edo is regarded as a temperament, which is to say as 5-et, then the most salient fact is that 16/15 is tempered out. This means in 5-et the major third and the fourth, and the minor sixth and the fifth, are not distinguished. This is at the very edge what can sensibly be called temperament, but it does make sense and can be used.
+If 5-edo is regarded as a temperament, which is to say as 5-et, then the most salient fact is that 16/15 is tempered out. This means in 5-et the major third and the fourth, and the minor sixth and the fifth, are not distinguished; this is 5-limit [[Trienstonic clan|father temperament]]. This is at the very edge what can sensibly be called temperament, but it does make sense and can be used.
-Also tempered out is 27/25, leading to [[beep temperament]], which equates 10/9 with 6/5: it is a little more perverse even than [[father temperament|father]]. Because these intervals are so large, this sort of analysis is less significant with 5 than it becomes with larger and more accurate divisions, but it still plays a role. For example, I-IV-V-I is the same as 1-III-V-I and involves triads with common intervals because of fourth-thirds equivalence.
+Also tempered out is 27/25, if we temper this out in preference to 16/15 we obtain [[Bug family|bug temperament]], which equates 10/9 with 6/5: it is a little more perverse even than father. Because these intervals are so large, this sort of analysis is less significant with 5 than it becomes with larger and more accurate divisions, but it still plays a role. For example, I-IV-V-I is the same as 1-III-V-I and involves triads with common intervals because of fourth-thirds equivalence.
 Despite its lack of accuracy, 5EDO is the second Zeta function integral tuning, after 2EDO. See http://www.research.att.com/~njas/sequences/A117538
@@ Line 203: / Line 203: @@
   &lt;ul&gt;&lt;li&gt;By its cardinality, 5-edo is related to other &lt;a class="wiki_link" href="/pentatonic"&gt;pentatonic&lt;/a&gt; scales, and it is especially close in sound to many Indonesian &lt;a class="wiki_link" href="/slendro"&gt;slendros&lt;/a&gt;.&lt;/li&gt;&lt;li&gt;Due to the interest around the &amp;quot;fifth&amp;quot; interval size, there are many &lt;a class="wiki_link" href="/nonoctave"&gt;nonoctave&lt;/a&gt; &amp;quot;stretch sisters&amp;quot; to 5-edo: square root of 4/3, cube root of 3/2, 8th root of 3, etc.&lt;/li&gt;&lt;li&gt;For the same reason there are many &amp;quot;circle sisters&amp;quot;:&lt;ul&gt;&lt;li&gt;Make a chain of five &amp;quot;bigger fifths&amp;quot; (50/33), which makes three octaves 3.227¢ flat. (50/33)^5=7.985099.&lt;/li&gt;&lt;/ul&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="x5 Equal Divisions of the Octave: Theory-As a temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;As a temperament&lt;/h2&gt;
-  If 5-edo is regarded as a temperament, which is to say as 5-et, then the most salient fact is that 16/15 is tempered out. This means in 5-et the major third and the fourth, and the minor sixth and the fifth, are not distinguished. This is at the very edge what can sensibly be called temperament, but it does make sense and can be used.&lt;br /&gt;
+  If 5-edo is regarded as a temperament, which is to say as 5-et, then the most salient fact is that 16/15 is tempered out. This means in 5-et the major third and the fourth, and the minor sixth and the fifth, are not distinguished; this is 5-limit &lt;a class="wiki_link" href="/Trienstonic%20clan"&gt;father temperament&lt;/a&gt;. This is at the very edge what can sensibly be called temperament, but it does make sense and can be used.&lt;br /&gt;
 &lt;br /&gt;
-Also tempered out is 27/25, leading to &lt;a class="wiki_link" href="/beep%20temperament"&gt;beep temperament&lt;/a&gt;, which equates 10/9 with 6/5: it is a little more perverse even than &lt;a class="wiki_link" href="/father%20temperament"&gt;father&lt;/a&gt;. Because these intervals are so large, this sort of analysis is less significant with 5 than it becomes with larger and more accurate divisions, but it still plays a role. For example, I-IV-V-I is the same as 1-III-V-I and involves triads with common intervals because of fourth-thirds equivalence.&lt;br /&gt;
+Also tempered out is 27/25, if we temper this out in preference to 16/15 we obtain &lt;a class="wiki_link" href="/Bug%20family"&gt;bug temperament&lt;/a&gt;, which equates 10/9 with 6/5: it is a little more perverse even than father. Because these intervals are so large, this sort of analysis is less significant with 5 than it becomes with larger and more accurate divisions, but it still plays a role. For example, I-IV-V-I is the same as 1-III-V-I and involves triads with common intervals because of fourth-thirds equivalence.&lt;br /&gt;
 &lt;br /&gt;
 Despite its lack of accuracy, 5EDO is the second Zeta function integral tuning, after 2EDO. See &lt;!-- ws:start:WikiTextUrlRule:248:http://www.research.att.com/~njas/sequences/A117538 --&gt;&lt;a class="wiki_link_ext" href="http://www.research.att.com/~njas/sequences/A117538" rel="nofollow"&gt;http://www.research.att.com/~njas/sequences/A117538&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:248 --&gt;&lt;br /&gt;