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>2011-08-~~22 03~~:02:43 UTC</tt>.<br>

: This revision was by author [[User:Natebedell|Natebedell]] and made on <tt>2011-09-05 19:18:42 UTC</tt>.<br>

: The original revision id was <tt>~~247583949~~</tt>.<br>

: The original revision id was <tt>250973102</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 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~~.

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]].

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 [[The Riemann Zeta Function and Tuning#Zeta ~~EDO lists~~|zeta integral edo]], after 2EDO. It also is the smallest equal division representing the [[9-limit]] [[consistent]]ly, giving a distinct value modulo five to 2, 3, 5, 7 and 9. Hence in a way similar to how [[4edo]] can be used, and which is discussed in that article, it can be used to represent [[7-limit]] intervals in terms of their position in a pentad, by giving a triple of integers representing a pentad in the [[The Seven Limit Symmetrical Lattices|lattice]] of tetrads/pentads together with the number of scale steps in 5EDO. However, while [[2edo]] represents the [[3-limit]] consistently, [[3edo]] the [[5-limit]], [[4edo]] the [[7-limit]] and [[5edo]] the [[9-limit]], to represent the [[11-limit]] consistently with a [[patent val]] requires going all the way to [[22edo]].

Despite its lack of accuracy, 5EDO is the second [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]], after 2EDO. It also is the smallest equal division representing the [[9-limit]] [[consistent]]ly, giving a distinct value modulo five to 2, 3, 5, 7 and 9. Hence in a way similar to how [[4edo]] can be used, and which is discussed in that article, it can be used to represent [[7-limit]] intervals in terms of their position in a pentad, by giving a triple of integers representing a pentad in the [[The Seven Limit Symmetrical Lattices|lattice]] of tetrads/pentads together with the number of scale steps in 5EDO. However, while [[2edo]] represents the [[3-limit]] consistently, [[3edo]] the [[5-limit]], [[4edo]] the [[7-limit]] and [[5edo]] the [[9-limit]], to represent the [[11-limit]] consistently with a [[patent val]] requires going all the way to [[22edo]].

==Cycles, Divisions==

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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 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 />

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>.<br />

<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 <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta ~~EDO lists~~">zeta integral edo</a>, after 2EDO. It also is the smallest equal division representing the <a class="wiki_link" href="/9-limit">9-limit</a> <a class="wiki_link" href="/consistent">consistent</a>ly, giving a distinct value modulo five to 2, 3, 5, 7 and 9. Hence in a way similar to how <a class="wiki_link" href="/4edo">4edo</a> can be used, and which is discussed in that article, it can be used to represent <a class="wiki_link" href="/7-limit">7-limit</a> intervals in terms of their position in a pentad, by giving a triple of integers representing a pentad in the <a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices">lattice</a> of tetrads/pentads together with the number of scale steps in 5EDO. However, while <a class="wiki_link" href="/2edo">2edo</a> represents the <a class="wiki_link" href="/3-limit">3-limit</a> consistently, <a class="wiki_link" href="/3edo">3edo</a> the <a class="wiki_link" href="/5-limit">5-limit</a>, <a class="wiki_link" href="/4edo">4edo</a> the <a class="wiki_link" href="/7-limit">7-limit</a> and <a class="wiki_link" href="/5edo">5edo</a> the <a class="wiki_link" href="/9-limit">9-limit</a>, to represent the <a class="wiki_link" href="/11-limit">11-limit</a> consistently with a <a class="wiki_link" href="/patent%20val">patent val</a> requires going all the way to <a class="wiki_link" href="/22edo">22edo</a>.<br />

Despite its lack of accuracy, 5EDO is the second <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists">zeta integral edo</a>, after 2EDO. It also is the smallest equal division representing the <a class="wiki_link" href="/9-limit">9-limit</a> <a class="wiki_link" href="/consistent">consistent</a>ly, giving a distinct value modulo five to 2, 3, 5, 7 and 9. Hence in a way similar to how <a class="wiki_link" href="/4edo">4edo</a> can be used, and which is discussed in that article, it can be used to represent <a class="wiki_link" href="/7-limit">7-limit</a> intervals in terms of their position in a pentad, by giving a triple of integers representing a pentad in the <a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices">lattice</a> of tetrads/pentads together with the number of scale steps in 5EDO. However, while <a class="wiki_link" href="/2edo">2edo</a> represents the <a class="wiki_link" href="/3-limit">3-limit</a> consistently, <a class="wiki_link" href="/3edo">3edo</a> the <a class="wiki_link" href="/5-limit">5-limit</a>, <a class="wiki_link" href="/4edo">4edo</a> the <a class="wiki_link" href="/7-limit">7-limit</a> and <a class="wiki_link" href="/5edo">5edo</a> the <a class="wiki_link" href="/9-limit">9-limit</a>, to represent the <a class="wiki_link" href="/11-limit">11-limit</a> consistently with a <a class="wiki_link" href="/patent%20val">patent val</a> requires going all the way to <a class="wiki_link" href="/22edo">22edo</a>.<br />

<br />

<h2 id="toc5"><a name="x5 Equal Divisions of the Octave: Theory-Cycles, Divisions"></a>Cycles, Divisions</h2>

@@ 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>2011-08-22 03:02:43 UTC</tt>.<br>
+: This revision was by author [[User:Natebedell|Natebedell]] and made on <tt>2011-09-05 19:18:42 UTC</tt>.<br>
-: The original revision id was <tt>247583949</tt>.<br>
+: The original revision id was <tt>250973102</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 50: / Line 50: @@
 ==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 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.
+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]].
 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 [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]], after 2EDO. It also is the smallest equal division representing the [[9-limit]] [[consistent]]ly, giving a distinct value modulo five to 2, 3, 5, 7 and 9. Hence in a way similar to how [[4edo]] can be used, and which is discussed in that article, it can be used to represent [[7-limit]] intervals in terms of their position in a pentad, by giving a triple of integers representing a pentad in the [[The Seven Limit Symmetrical Lattices|lattice]] of tetrads/pentads together with the number of scale steps in 5EDO. However, while [[2edo]] represents the [[3-limit]] consistently, [[3edo]] the [[5-limit]], [[4edo]] the [[7-limit]] and [[5edo]] the [[9-limit]], to represent the [[11-limit]] consistently with a [[patent val]] requires going all the way to [[22edo]].
+Despite its lack of accuracy, 5EDO is the second [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]], after 2EDO. It also is the smallest equal division representing the [[9-limit]] [[consistent]]ly, giving a distinct value modulo five to 2, 3, 5, 7 and 9. Hence in a way similar to how [[4edo]] can be used, and which is discussed in that article, it can be used to represent [[7-limit]] intervals in terms of their position in a pentad, by giving a triple of integers representing a pentad in the [[The Seven Limit Symmetrical Lattices|lattice]] of tetrads/pentads together with the number of scale steps in 5EDO. However, while [[2edo]] represents the [[3-limit]] consistently, [[3edo]] the [[5-limit]], [[4edo]] the [[7-limit]] and [[5edo]] the [[9-limit]], to represent the [[11-limit]] consistently with a [[patent val]] requires going all the way to [[22edo]].
 ==Cycles, Divisions==
@@ Line 229: / Line 229: @@
   &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 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;
+  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;.&lt;br /&gt;
 &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 &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists"&gt;zeta integral edo&lt;/a&gt;, after 2EDO. It also is the smallest equal division representing the &lt;a class="wiki_link" href="/9-limit"&gt;9-limit&lt;/a&gt; &lt;a class="wiki_link" href="/consistent"&gt;consistent&lt;/a&gt;ly, giving a distinct value modulo five to 2, 3, 5, 7 and 9. Hence in a way similar to how &lt;a class="wiki_link" href="/4edo"&gt;4edo&lt;/a&gt; can be used, and which is discussed in that article, it can be used to represent &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; intervals in terms of their position in a pentad, by giving a triple of integers representing a pentad in the &lt;a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices"&gt;lattice&lt;/a&gt; of tetrads/pentads together with the number of scale steps in 5EDO. However, while &lt;a class="wiki_link" href="/2edo"&gt;2edo&lt;/a&gt; represents the &lt;a class="wiki_link" href="/3-limit"&gt;3-limit&lt;/a&gt; consistently, &lt;a class="wiki_link" href="/3edo"&gt;3edo&lt;/a&gt; the &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt;, &lt;a class="wiki_link" href="/4edo"&gt;4edo&lt;/a&gt; the &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; and &lt;a class="wiki_link" href="/5edo"&gt;5edo&lt;/a&gt; the &lt;a class="wiki_link" href="/9-limit"&gt;9-limit&lt;/a&gt;, to represent the &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt; consistently with a &lt;a class="wiki_link" href="/patent%20val"&gt;patent val&lt;/a&gt; requires going all the way to &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt;.&lt;br /&gt;
+Despite its lack of accuracy, 5EDO is the second &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&gt;zeta integral edo&lt;/a&gt;, after 2EDO. It also is the smallest equal division representing the &lt;a class="wiki_link" href="/9-limit"&gt;9-limit&lt;/a&gt; &lt;a class="wiki_link" href="/consistent"&gt;consistent&lt;/a&gt;ly, giving a distinct value modulo five to 2, 3, 5, 7 and 9. Hence in a way similar to how &lt;a class="wiki_link" href="/4edo"&gt;4edo&lt;/a&gt; can be used, and which is discussed in that article, it can be used to represent &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; intervals in terms of their position in a pentad, by giving a triple of integers representing a pentad in the &lt;a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices"&gt;lattice&lt;/a&gt; of tetrads/pentads together with the number of scale steps in 5EDO. However, while &lt;a class="wiki_link" href="/2edo"&gt;2edo&lt;/a&gt; represents the &lt;a class="wiki_link" href="/3-limit"&gt;3-limit&lt;/a&gt; consistently, &lt;a class="wiki_link" href="/3edo"&gt;3edo&lt;/a&gt; the &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt;, &lt;a class="wiki_link" href="/4edo"&gt;4edo&lt;/a&gt; the &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; and &lt;a class="wiki_link" href="/5edo"&gt;5edo&lt;/a&gt; the &lt;a class="wiki_link" href="/9-limit"&gt;9-limit&lt;/a&gt;, to represent the &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt; consistently with a &lt;a class="wiki_link" href="/patent%20val"&gt;patent val&lt;/a&gt; requires going all the way to &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="x5 Equal Divisions of the Octave: Theory-Cycles, Divisions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Cycles, Divisions&lt;/h2&gt;