34edo: 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>2011-06-~~06 14~~:06:44 UTC</tt>.<br>

: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on <tt>2011-07-12 02:23:11 UTC</tt>.<br>

: The original revision id was <tt>~~234635124~~</tt>.<br>

: The original revision id was <tt>240946537</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">34edo divides the ~~2:1~~ octave into 34 equal steps of approximately 35.~~294117~~ cents. 34edo contains two [[17edo]]'s and the half-octave tritone of 600 cents. As a Fibonacci number, 34edo contains a close approximation to the irrational interval phi -- 21 degrees of 34edo, approximately 741.2 cents. Repeated iterations of this interval generates [[MOSScales|Moment of Symmetry]] scales with near-phi relationships between the step sizes.

<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">=<span style="color: #991785; font-family: 'Times New Roman',Times,serif; font-size: 113%;">34 tone equal temperament</span>=

//**34edo**// divides the octave into 34 equal steps of approximately 35.29412 cents. 34edo contains two [[17edo]]'s and the half-octave tritone of 600 cents. As a Fibonacci number, 34edo contains a close approximation to the irrational interval phi -- 21 degrees of 34edo, approximately 741.2 cents. Repeated iterations of this interval generates [[MOSScales|Moment of Symmetry]] scales with near-phi relationships between the step sizes.

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|| degrees of 34edo || cents ||

|| 0 || 0.0 ||

|| 1 || 35.3 ||

|| 1 || 35.294 ||

|| 2 || 70.6 ||

|| 2 || 70.588 ||

|| 3 || 105.9 ||

|| 3 || 105.882 ||

|| 4 || 141.2 ||

|| 4 || 141.176 ||

|| 5 || 176.5 ||

|| 5 || 176.471 ||

|| 6 || 211.8 ||

|| 6 || 211.765 ||

|| 7 || 247.1 ||

|| 7 || 247.059 ||

|| 8 || 282.4 ||

|| 8 || 282.353 ||

|| 9 || 317.6 ||

|| 9 || 317.647 ||

|| 10 || 352.9 ||

|| 10 || 352.941 ||

|| 11 || 388.2 ||

|| 11 || 388.235 ||

|| 12 || 423.5 ||

|| 12 || 423.529 ||

|| 13 || 458.8 ||

|| 13 || 458.823 ||

|| 14 || 494.1 ||

|| 14 || 494.118 ||

|| 15 || 529.4 ||

|| 15 || 529.412 ||

|| 16 || 564.7 ||

|| 16 || 564.706 ||

|| 17 || 600.0 ||

|| 17 || 600 ||

|| 18 || 635.3 ||

|| 18 || 635.294 ||

|| 19 || 670.6 ||

|| 19 || 670.588 ||

|| 20 || 705.9 ||

|| 20 || 705.882 ||

|| 21 || 741.2 ||

|| 21 || 741.176 ||

|| 22 || 776.5 ||

|| 22 || 776.471 ||

|| 23 || 811.8 ||

|| 23 || 811.765 ||

|| 24 || 847.1 ||

|| 24 || 847.059 ||

|| 25 || 882.4 ||

|| 25 || 882.353 ||

|| 26 || 917.6 ||

|| 26 || 917.647 ||

|| 27 || 952.9 ||

|| 27 || 952.941 ||

|| 28 || 988.2 ||

|| 28 || 988.235 ||

|| 29 || 1023.5 ||

|| 29 || 1023.529 ||

|| 30 || 1058.8 ||

|| 30 || 1058.823 ||

|| 31 || 1094.1 ||

|| 31 || 1094.118 ||

|| 32 || 1129.4 ||

|| 32 || 1129.412 ||

|| 33 || 1164.7 ||

|| 33 || 1164.706 ||

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Drums Bass and 34-tone guitar

==Links==

* [[http://www.microstick.net/34guitararticle.htm|34 Equal Guitar]] by [[Larry Hanson]]</pre></div>

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<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>34edo</title></head><body>34edo divides the ~~2:1~~ octave into 34 equal steps of approximately 35.~~294117~~ cents. 34edo contains two <a class="wiki_link" href="/17edo">17edo</a>'s and the half-octave tritone of 600 cents. As a Fibonacci number, 34edo contains a close approximation to the irrational interval phi -- 21 degrees of 34edo, approximately 741.2 cents. Repeated iterations of this interval generates <a class="wiki_link" href="/MOSScales">Moment of Symmetry</a> scales with near-phi relationships between the step sizes.<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>34edo</title></head><body><h1 id="toc0"><a name="x34 tone equal temperament"></a><span style="color: #991785; font-family: 'Times New Roman',Times,serif; font-size: 113%;">34 tone equal temperament</span></h1>

<br />

<em><strong>34edo</strong></em> divides the octave into 34 equal steps of approximately 35.29412 cents. 34edo contains two <a class="wiki_link" href="/17edo">17edo</a>'s and the half-octave tritone of 600 cents. As a Fibonacci number, 34edo contains a close approximation to the irrational interval phi -- 21 degrees of 34edo, approximately 741.2 cents. Repeated iterations of this interval generates <a class="wiki_link" href="/MOSScales">Moment of Symmetry</a> scales with near-phi relationships between the step sizes.<br />

<br />

<h3 id="~~toc0~~"><a name="x--Approximations to Just Intonation"></a>Approximations to Just Intonation</h3>

<h3 id="toc1"><a name="x34 tone equal temperament--Approximations to Just Intonation"></a>Approximations to Just Intonation</h3>

Like <a class="wiki_link" href="/17edo">17edo</a>, 34edo contains workable approximations of just intervals involving 13 and 3 -- specifically, 13/8, 13/12, 13/9 and their inversions -- while failing to closely approximate ratios of 7 or 11. 34edo adds ratios of 5 into the mix, including: 5/4, 6/5, 9/5, 15/8, 13/10, 15/13, and their inversions. Since it distinguishes between 9/8 and 10/9 (exaggerating the difference between them, the syntonic comma of 81/80, from 21.5 cents to 35.3 cents), it is suitable for 5-limit JI. It is not a <a class="wiki_link" href="/meantone">meantone </a>system.<br />

<br />

<em>Viewed in light of Western diatonic theory, the three extra steps (of 34-et compared to 31-et) in effect widen the intervals between C and D, F and G, and A and B, thus making a distinction between major tones, ratio 9/8 and minor tones, ratio 10/9.</em> (<a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/34_equal_temperament" rel="nofollow">Wikipedia</a>)<br />

<br />

<h3 id="~~toc1~~"><a name="x--Intervals:"></a>Intervals:</h3>

<h3 id="toc2"><a name="x34 tone equal temperament--Intervals:"></a>Intervals:</h3>

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<br />

<h2 id="~~toc2~~"><a name="x-Listen"></a>Listen</h2>

<h2 id="toc3"><a name="x34 tone equal temperament-Listen"></a>Listen</h2>

<ul><li><a class="wiki_link_ext" href="http://www.archive.org/details/Ascension_105" rel="nofollow" target="_blank">Ascension</a></li></ul>Drums Bass and 34-tone guitar<br />

<br />

<h2 id="~~toc3~~"><a name="x-Links"></a>Links</h2>

<h2 id="toc4"><a name="x34 tone equal temperament-Links"></a>Links</h2>

<ul><li><a class="wiki_link_ext" href="http://www.microstick.net/34guitararticle.htm" rel="nofollow">34 Equal Guitar</a> by <a class="wiki_link" href="/Larry%20Hanson">Larry Hanson</a></li></ul></body></html></pre></div>

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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>2011-06-06 14:06:44 UTC</tt>.<br>
+: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on <tt>2011-07-12 02:23:11 UTC</tt>.<br>
-: The original revision id was <tt>234635124</tt>.<br>
+: The original revision id was <tt>240946537</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">34edo divides the 2:1 octave into 34 equal steps of approximately 35.294117 cents. 34edo contains two [[17edo]]'s and the half-octave tritone of 600 cents. As a Fibonacci number, 34edo contains a close approximation to the irrational interval phi -- 21 degrees of 34edo, approximately 741.2 cents. Repeated iterations of this interval generates [[MOSScales|Moment of Symmetry]] scales with near-phi relationships between the step sizes.
+<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">=&lt;span style="color: #991785; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&gt;34 tone equal temperament&lt;/span&gt;=
+//**34edo**// divides the octave into 34 equal steps of approximately 35.29412 cents. 34edo contains two [[17edo]]'s and the half-octave tritone of 600 cents. As a Fibonacci number, 34edo contains a close approximation to the irrational interval phi -- 21 degrees of 34edo, approximately 741.2 cents. Repeated iterations of this interval generates [[MOSScales|Moment of Symmetry]] scales with near-phi relationships between the step sizes.
@@ Line 17: / Line 19: @@
 || degrees of 34edo || cents ||
 || 0 || 0.0 ||
-|| 1 || 35.3 ||
+|| 1 || 35.294 ||
-|| 2 || 70.6 ||
+|| 2 || 70.588 ||
-|| 3 || 105.9 ||
+|| 3 || 105.882 ||
-|| 4 || 141.2 ||
+|| 4 || 141.176 ||
-|| 5 || 176.5 ||
+|| 5 || 176.471 ||
-|| 6 || 211.8 ||
+|| 6 || 211.765 ||
-|| 7 || 247.1 ||
+|| 7 || 247.059 ||
-|| 8 || 282.4 ||
+|| 8 || 282.353 ||
-|| 9 || 317.6 ||
+|| 9 || 317.647 ||
-|| 10 || 352.9 ||
+|| 10 || 352.941 ||
-|| 11 || 388.2 ||
+|| 11 || 388.235 ||
-|| 12 || 423.5 ||
+|| 12 || 423.529 ||
-|| 13 || 458.8 ||
+|| 13 || 458.823 ||
-|| 14 || 494.1 ||
+|| 14 || 494.118 ||
-|| 15 || 529.4 ||
+|| 15 || 529.412 ||
-|| 16 || 564.7 ||
+|| 16 || 564.706 ||
-|| 17 || 600.0 ||
+|| 17 || 600 ||
-|| 18 || 635.3 ||
+|| 18 || 635.294 ||
-|| 19 || 670.6 ||
+|| 19 || 670.588 ||
-|| 20 || 705.9 ||
+|| 20 || 705.882 ||
-|| 21 || 741.2 ||
+|| 21 || 741.176 ||
-|| 22 || 776.5 ||
+|| 22 || 776.471 ||
-|| 23 || 811.8 ||
+|| 23 || 811.765 ||
-|| 24 || 847.1 ||
+|| 24 || 847.059 ||
-|| 25 || 882.4 ||
+|| 25 || 882.353 ||
-|| 26 || 917.6 ||
+|| 26 || 917.647 ||
-|| 27 || 952.9 ||
+|| 27 || 952.941 ||
-|| 28 || 988.2 ||
+|| 28 || 988.235 ||
-|| 29 || 1023.5 ||
+|| 29 || 1023.529 ||
-|| 30 || 1058.8 ||
+|| 30 || 1058.823 ||
-|| 31 || 1094.1 ||
+|| 31 || 1094.118 ||
-|| 32 || 1129.4 ||
+|| 32 || 1129.412 ||
-|| 33 || 1164.7 ||
+|| 33 || 1164.706 ||
@@ Line 56: / Line 58: @@
 Drums Bass and 34-tone guitar
 ==Links==
 * [[http://www.microstick.net/34guitararticle.htm|34 Equal Guitar]] by [[Larry Hanson]]</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;34edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;34edo divides the 2:1 octave into 34 equal steps of approximately 35.294117 cents. 34edo contains two &lt;a class="wiki_link" href="/17edo"&gt;17edo&lt;/a&gt;'s and the half-octave tritone of 600 cents. As a Fibonacci number, 34edo contains a close approximation to the irrational interval phi -- 21 degrees of 34edo, approximately 741.2 cents. Repeated iterations of this interval generates &lt;a class="wiki_link" href="/MOSScales"&gt;Moment of Symmetry&lt;/a&gt; scales with near-phi relationships between the step sizes.&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;34edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x34 tone equal temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;&lt;span style="color: #991785; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&gt;34 tone equal temperament&lt;/span&gt;&lt;/h1&gt;
+ &lt;br /&gt;
+&lt;em&gt;&lt;strong&gt;34edo&lt;/strong&gt;&lt;/em&gt; divides the octave into 34 equal steps of approximately 35.29412 cents. 34edo contains two &lt;a class="wiki_link" href="/17edo"&gt;17edo&lt;/a&gt;'s and the half-octave tritone of 600 cents. As a Fibonacci number, 34edo contains a close approximation to the irrational interval phi -- 21 degrees of 34edo, approximately 741.2 cents. Repeated iterations of this interval generates &lt;a class="wiki_link" href="/MOSScales"&gt;Moment of Symmetry&lt;/a&gt; scales with near-phi relationships between the step sizes.&lt;br /&gt;
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc0"&gt;&lt;a name="x--Approximations to Just Intonation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Approximations to Just Intonation&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x34 tone equal temperament--Approximations to Just Intonation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Approximations to Just Intonation&lt;/h3&gt;
   Like &lt;a class="wiki_link" href="/17edo"&gt;17edo&lt;/a&gt;, 34edo contains workable approximations of just intervals involving 13 and 3 -- specifically, 13/8, 13/12, 13/9 and their inversions -- while failing to closely approximate ratios of 7 or 11. 34edo adds ratios of 5 into the mix, including: 5/4, 6/5, 9/5, 15/8, 13/10, 15/13, and their inversions. Since it distinguishes between 9/8 and 10/9 (exaggerating the difference between them, the syntonic comma of 81/80, from 21.5 cents to 35.3 cents), it is suitable for 5-limit JI. It is not a &lt;a class="wiki_link" href="/meantone"&gt;meantone &lt;/a&gt;system.&lt;br /&gt;
 &lt;br /&gt;
 &lt;em&gt;Viewed in light of Western diatonic theory, the three extra steps (of 34-et compared to 31-et) in effect widen the intervals between C and D, F and G, and A and B, thus making a distinction between major tones, ratio 9/8 and minor tones, ratio 10/9.&lt;/em&gt; (&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/34_equal_temperament" rel="nofollow"&gt;Wikipedia&lt;/a&gt;)&lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x34 tone equal temperament--Intervals:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Intervals:&lt;/h3&gt;
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-         &lt;td&gt;35.3&lt;br /&gt;
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   &lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.archive.org/details/Ascension_105" rel="nofollow" target="_blank"&gt;Ascension&lt;/a&gt;&lt;/li&gt;&lt;/ul&gt;Drums Bass and 34-tone guitar&lt;br /&gt;
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-&lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.microstick.net/34guitararticle.htm" rel="nofollow"&gt;34 Equal Guitar&lt;/a&gt; by &lt;a class="wiki_link" href="/Larry%20Hanson"&gt;Larry Hanson&lt;/a&gt;&lt;/li&gt;&lt;/ul&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
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