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:

: This revision was by author [[User:phylingual|phylingual]] and made on <tt>2012-04-18 13:32:33 UTC</tt>.

: This revision was by author [[User:phylingual|phylingual]] and made on <tt>2012-04-18 21:28:35 UTC</tt>.

: The original revision id was <tt>~~322204226~~</tt>.

: The original revision id was <tt>322397622</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.

<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">=~~34 tone equal temperament~~=

<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 octave into 34 equal steps of approximately 35.29412 cents. 34edo contains two [[xenharmonic/17edo|17edo]]'s and the half-octave tritone of 600 cents.

34edo divides the octave into 34 equal steps of approximately 35.29412 cents. 34edo contains two [[xenharmonic/17edo|17edo]]'s and the half-octave tritone of 600 cents.

===Approximations to Just Intonation===

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|| 32 || 1129.412 || || ||

|| 33 || 1164.706 || || ||

==Commas==

34-EDO [[xenharmonic/tempering out|tempers out]] the following [[xenharmonic/comma|comma]]s. (Note: This assumes the [[xenharmonic/val|val]] < 34 54 79 95 118 126 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||

||= 134217728/129140163 ||< | 27 -17 > ||> 66.765 ||= 17-comma ||= ||= ||

||= 20000/19683 || | 5 -9 4 > ||> 27.660 ||= Minimal Diesis ||= Tetracot Comma || ||

||= 2048/2025 ||< | 11 -4 -2 > ||> 19.553 ||= Diaschisma ||= Semicomma Majeur ||= ||

||= 393216/390625 || | 17 1 -8 > ||> 11.445 ||= Würschmidt comma ||= || ||

||= 15625/15552 || | -6 -5 6 > ||> 8.11 ||= Kleisma ||= Semicomma Majeur || ||

||= 1212717/1210381 ||< | -4 4 -1 > ||> 3.338 ||= Vishnuzma ||= Semisuper ||= ||

||= 1029/1000 ||< | -3 1 -3 3 > ||> 49.492 ||= Keega ||= ||= ||

||= [[xenharmonic/49_48|49/48]] ||< | -4 -1 0 2 > ||> 35.697 ||= Slendro Diesis ||= ||= ||

||= 875/864 ||< | -5 -3 3 1 > ||> 21.902 ||= Keema ||= ||= ||

||= 126/125 ||< | 1 2 -3 1 > ||> 13.795 ||= Starling comma ||= Septimal semicomma ||= ||

||= 100/99 ||< | 2 -2 2 0 -1> ||> 17.399 ||= Ptolemisma ||= Ptolemy's comma ||= ||

||= 243/242 ||< | -1 5 0 0 -2 > ||> 7.1391 ||= Rastma ||= Neutral third comma ||= ||

||= 385/384 ||< | -7 -1 1 1 1 > ||> 4.5026 ||= Keenanisma ||= ||= ||

||= 91/90 ||< | -1 -2 -1 1 0 1 > ||> 19.120 ||= Superleap ||= ||= ||

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* [[http://www.microstick.net/34guitararticle.htm|34 Equal Guitar]] by [[xenharmonic/Larry Hanson|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"><html><head><title>34edo</title></head><body><h1 id="toc0"~~><a name="x34 tone equal temperament"></a~~>~~34 tone equal temperament~~</h1>

~~~~34edo~~~~ divides the octave into 34 equal steps of approximately 35.29412 cents. 34edo contains two <a class="wiki_link" href="http://xenharmonic.wikispaces.com/17edo">17edo</a>'s and the half-octave tritone of 600 cents.

34edo divides the octave into 34 equal steps of approximately 35.29412 cents. 34edo contains two <a class="wiki_link" href="http://xenharmonic.wikispaces.com/17edo">17edo</a>'s and the half-octave tritone of 600 cents.

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

<h3 id="toc1"><a name="x--Approximations to Just Intonation"></a>Approximations to Just Intonation</h3>

Like <a class="wiki_link" href="http://xenharmonic.wikispaces.com/17edo">17edo</a>, 34edo contains good 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 -- as well as 17 -- including 17/16, 17/18, 17/12, 17/10, 17/13, 17/15 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="http://xenharmonic.wikispaces.com/meantone">meantone </a>system.

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 [that is: 6 5 3 6 5 6 3], thus making a distinction between major tones, ratio 9/8 and minor tones, ratio 10/9. (<a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/34_equal_temperament" rel="nofollow">Wikipedia</a>)

<h3 id="toc2"><a name="~~x34 tone equal temperament~~--34edo and phi"></a>34edo and phi</h3>

<h3 id="toc2"><a name="x--34edo and phi"></a>34edo and phi</h3>

As a Fibonacci number, 34edo contains a fraction of an octave which is 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="http://xenharmonic.wikispaces.com/MOSScales">Moment of Symmetry</a> scales with near-phi relationships between the step sizes. As a 2.3.5.13 temperament, the 21\34 generator is an approximate 20/13, and the temperament tempers out 512/507 and 140625/140608. From the tempering of 512/507, two 16/13 neutral thirds are an approximate 3/2, defining an essentially tempered neutral triad with a sharp rather than a flat fifth.

<h3 id="toc3"><a name="~~x34 tone equal temperament~~--Linear temperaments"></a>Linear temperaments</h3>

<h3 id="toc3"><a name="x--Linear temperaments"></a>Linear temperaments</h3>

<a class="wiki_link" href="/List%20of%2034et%20rank%20two%20temperaments%20by%20badness">List of 34et rank two temperaments by badness</a>

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</table>

<h3 id="toc4"><a name="~~x34 tone equal temperament~~--Intervals:"></a>Intervals:</h3>

<h3 id="toc4"><a name="x--Intervals:"></a>Intervals:</h3>

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</td>

<td>

</td>

</tr>

</table>

<h2 id="toc5"><a name="x-Commas"></a>Commas</h2>

34-EDO <a class="wiki_link" href="http://xenharmonic.wikispaces.com/tempering%20out">tempers out</a> the following <a class="wiki_link" href="http://xenharmonic.wikispaces.com/comma">comma</a>s. (Note: This assumes the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/val">val</a> &lt; 34 54 79 95 118 126 |.)

<tr>

<th>Comma

</th>

<th>Monzo

</th>

<th>Value (Cents)

</th>

<th>Name 1

</th>

<th>Name 2

</th>

<th>Name 3

</th>

</tr>

<tr>

<td style="text-align: center;">134217728/129140163

</td>

<td style="text-align: left;">| 27 -17 &gt;

</td>

<td style="text-align: right;">66.765

</td>

<td style="text-align: center;">17-comma

</td>

</td>

</td>

</tr>

<tr>

<td style="text-align: center;">20000/19683

</td>

<td>| 5 -9 4 &gt;

</td>

<td style="text-align: right;">27.660

</td>

<td style="text-align: center;">Minimal Diesis

</td>

<td style="text-align: center;">Tetracot Comma

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">2048/2025

</td>

<td style="text-align: left;">| 11 -4 -2 &gt;

</td>

<td style="text-align: right;">19.553

</td>

<td style="text-align: center;">Diaschisma

</td>

<td style="text-align: center;">Semicomma Majeur

</td>

</td>

</tr>

<tr>

<td style="text-align: center;">393216/390625

</td>

<td>| 17 1 -8 &gt;

</td>

<td style="text-align: right;">11.445

</td>

<td style="text-align: center;">Würschmidt comma

</td>

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">15625/15552

</td>

<td>| -6 -5 6 &gt;

</td>

<td style="text-align: right;">8.11

</td>

<td style="text-align: center;">Kleisma

</td>

<td style="text-align: center;">Semicomma Majeur

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">1212717/1210381

</td>

<td style="text-align: left;">| -4 4 -1 &gt;

</td>

<td style="text-align: right;">3.338

</td>

<td style="text-align: center;">Vishnuzma

</td>

<td style="text-align: center;">Semisuper

</td>

</td>

</tr>

<tr>

<td style="text-align: center;">1029/1000

</td>

<td style="text-align: left;">| -3 1 -3 3 &gt;

</td>

<td style="text-align: right;">49.492

</td>

<td style="text-align: center;">Keega

</td>

</td>

</td>

</tr>

<tr>

</td>

<td style="text-align: left;">| -4 -1 0 2 &gt;

</td>

<td style="text-align: right;">35.697

</td>

<td style="text-align: center;">Slendro Diesis

</td>

</td>

</td>

</tr>

<tr>

<td style="text-align: center;">875/864

</td>

<td style="text-align: left;">| -5 -3 3 1 &gt;

</td>

<td style="text-align: right;">21.902

</td>

<td style="text-align: center;">Keema

</td>

</td>

</td>

</tr>

<tr>

<td style="text-align: center;">126/125

</td>

<td style="text-align: left;">| 1 2 -3 1 &gt;

</td>

<td style="text-align: right;">13.795

</td>

<td style="text-align: center;">Starling comma

</td>

<td style="text-align: center;">Septimal semicomma

</td>

</td>

</tr>

<tr>

<td style="text-align: center;">100/99

</td>

<td style="text-align: left;">| 2 -2 2 0 -1&gt;

</td>

<td style="text-align: right;">17.399

</td>

<td style="text-align: center;">Ptolemisma

</td>

<td style="text-align: center;">Ptolemy's comma

</td>

</td>

</tr>

<tr>

<td style="text-align: center;">243/242

</td>

<td style="text-align: left;">| -1 5 0 0 -2 &gt;

</td>

<td style="text-align: right;">7.1391

</td>

<td style="text-align: center;">Rastma

</td>

<td style="text-align: center;">Neutral third comma

</td>

</td>

</tr>

<tr>

<td style="text-align: center;">385/384

</td>

<td style="text-align: left;">| -7 -1 1 1 1 &gt;

</td>

<td style="text-align: right;">4.5026

</td>

<td style="text-align: center;">Keenanisma

</td>

</td>

</td>

</tr>

<tr>

<td style="text-align: center;">91/90

</td>

<td style="text-align: left;">| -1 -2 -1 1 0 1 &gt;

</td>

<td style="text-align: right;">19.120

</td>

<td style="text-align: center;">Superleap

</td>

</td>

</td>

</tr>

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<h2 id="~~toc5~~"><a name="~~x34 tone equal temperament~~-Listen"></a>Listen</h2>

<h2 id="toc6"><a name="x-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

<h2 id="~~toc6~~"><a name="~~x34 tone equal temperament~~-Links"></a>Links</h2>

<h2 id="toc7"><a name="x-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="http://xenharmonic.wikispaces.com/Larry%20Hanson">Larry Hanson</a></li></ul></body></html></pre></div>

@@ 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:phylingual|phylingual]] and made on <tt>2012-04-18 13:32:33 UTC</tt>.<br>
+: This revision was by author [[User:phylingual|phylingual]] and made on <tt>2012-04-18 21:28:35 UTC</tt>.<br>
-: The original revision id was <tt>322204226</tt>.<br>
+: The original revision id was <tt>322397622</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">=&lt;span style="color: #991785; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&gt;34 tone equal temperament&lt;/span&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;white-space: pre-wrap ! important" class="old-revision-html">=&lt;span style="color: #991785;"&gt;&lt;span style="font-family: 'Times New Roman',Times,serif;"&gt; &lt;/span&gt;&lt;/span&gt;=
-//**34edo**// divides the octave into 34 equal steps of approximately 35.29412 cents. 34edo contains two [[xenharmonic/17edo|17edo]]'s and the half-octave tritone of 600 cents.
+edo divides the octave into 34 equal steps of approximately 35.29412 cents. 34edo contains two [[xenharmonic/17edo|17edo]]'s and the half-octave tritone of 600 cents.
 ===Approximations to Just Intonation===
@@ Line 77: / Line 77: @@
 || 32 || 1129.412 ||   ||   ||
 || 33 || 1164.706 ||   ||   ||
+==&lt;span style="background-color: #ffffff;"&gt;Commas&lt;/span&gt;==
+&lt;span style="background-color: #ffffff;"&gt;34-EDO &lt;/span&gt;&lt;span style="background-color: #ffffff;"&gt;[[xenharmonic/tempering out|tempers out]]&lt;/span&gt;&lt;span style="background-color: #ffffff;"&gt; the following &lt;/span&gt;&lt;span style="background-color: #ffffff;"&gt;[[xenharmonic/comma|comma]]&lt;/span&gt;&lt;span style="background-color: #ffffff;"&gt;s. (Note: This assumes the &lt;/span&gt;&lt;span style="background-color: #ffffff;"&gt;[[xenharmonic/val|val]]&lt;/span&gt;&lt;span style="background-color: #ffffff;"&gt; &lt; 34 54 79 95 118 126 |.)&lt;/span&gt;
+||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
+||= &lt;span style="background-color: #ffffff; display: block; text-align: center;"&gt;134217728/129140163&lt;/span&gt; ||&lt; | 27 -17 &gt; ||&gt; 66.765 ||= 17-comma ||=   ||=   ||
+||= 20000/19683 || &lt;span style="background-color: #ffffff; display: block; text-align: left;"&gt;| 5 -9 4 &gt;&lt;/span&gt; ||&gt; &lt;span style="background-color: #ffffff; display: block; text-align: right;"&gt;27.660&lt;/span&gt; ||= Minimal Diesis ||= Tetracot Comma ||   ||
+||= &lt;span style="background-color: #ffffff; display: block; text-align: center;"&gt;2048/2025&lt;/span&gt; ||&lt; | 11 -4 -2 &gt; ||&gt; 19.553 ||= Diaschisma ||= &lt;span style="background-color: #ffffff; display: block; text-align: center;"&gt;Semicomma Majeur&lt;/span&gt; ||=   ||
+||= &lt;span style="background-color: #ffffff; display: block; text-align: center;"&gt;393216/390625&lt;/span&gt; || | &lt;span style="text-align: left;"&gt;17 1 -8 &gt;&lt;/span&gt; ||&gt; &lt;span style="background-color: #ffffff; display: block; text-align: right;"&gt;11.445&lt;/span&gt; ||= Würschmidt comma ||=   ||   ||
+||= &lt;span style="background-color: #ffffff; display: block; text-align: center;"&gt;15625/15552&lt;/span&gt; || &lt;span style="text-align: left;"&gt;| -6 -5 6 &gt;&lt;/span&gt; ||&gt; &lt;span style="text-align: right;"&gt;8.11&lt;/span&gt; ||= &lt;span style="text-align: center;"&gt;Kleisma&lt;/span&gt; ||= &lt;span style="text-align: center;"&gt;Semicomma Majeur&lt;/span&gt; ||   ||
+||= &lt;span style="background-color: #ffffff; display: block; text-align: center;"&gt;1212717/1210381&lt;/span&gt; ||&lt; | -4 4 -1 &gt; ||&gt; 3.338 ||= &lt;span style="background-color: #ffffff; display: block; text-align: center;"&gt;Vishnuzma&lt;/span&gt; ||= Semisuper ||=   ||
+||= 1029/1000 ||&lt; | -3 1 -3 3 &gt; ||&gt; 49.492 ||= &lt;span style="background-color: #ffffff;"&gt;Keega&lt;/span&gt; ||=   ||=   ||
+||= [[xenharmonic/49_48|49/48]] ||&lt; | -4 -1 0 2 &gt; ||&gt; 35.697 ||= Slendro Diesis ||=   ||=   ||
+||= 875/864 ||&lt; | -5 -3 3 1 &gt; ||&gt; 21.902 ||= Keema ||=   ||=   ||
+||= 126/125 ||&lt; | 1 2 -3 1 &gt; ||&gt; 13.795 ||= Starling comma ||= Septimal semicomma ||=   ||
+||= 100/99 ||&lt; | 2 -2 2 0 -1&gt; ||&gt; 17.399 ||= Ptolemisma ||= Ptolemy's comma ||=   ||
+||= 243/242 ||&lt; &lt;span style="background-color: #ffffff; display: block; text-align: left;"&gt;| -1 5 0 0 -2 &gt;&lt;/span&gt; ||&gt; &lt;span style="background-color: #ffffff; display: block; text-align: right;"&gt;7.1391&lt;/span&gt; ||= Rastma ||= Neutral third comma ||=   ||
+||= 385/384 ||&lt; &lt;span style="background-color: #ffffff; display: block; text-align: left;"&gt;| -7 -1 1 1 1 &gt;&lt;/span&gt; ||&gt; &lt;span style="background-color: #ffffff; display: block; text-align: right;"&gt;4.5026&lt;/span&gt; ||= Keenanisma ||=   ||=   ||
+||= 91/90 ||&lt; | -1 -2 -1 1 0 1 &gt; ||&gt; 19.120 ||= Superleap ||=   ||=   ||
@@ Line 86: / Line 105: @@
 * [[http://www.microstick.net/34guitararticle.htm|34 Equal Guitar]] by [[xenharmonic/Larry Hanson|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;&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;
+<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;!-- ws:end:WikiTextHeadingRule:0 --&gt;&lt;span style="color: #991785;"&gt;&lt;span style="font-family: 'Times New Roman',Times,serif;"&gt; &lt;/span&gt;&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="http://xenharmonic.wikispaces.com/17edo"&gt;17edo&lt;/a&gt;'s and the half-octave tritone of 600 cents.&lt;br /&gt;
+edo divides the octave into 34 equal steps of approximately 35.29412 cents. 34edo contains two &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/17edo"&gt;17edo&lt;/a&gt;'s and the half-octave tritone of 600 cents.&lt;br /&gt;
 &lt;br /&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;
+&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x--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="http://xenharmonic.wikispaces.com/17edo"&gt;17edo&lt;/a&gt;, 34edo contains good 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 -- as well as 17 -- including 17/16, 17/18, 17/12, 17/10, 17/13, 17/15 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="http://xenharmonic.wikispaces.com/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 [that is: 6 5 3 6 5 6 3], 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;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x34 tone equal temperament--34edo and phi"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;34edo and phi&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x--34edo and phi"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;34edo and phi&lt;/h3&gt;
   As a Fibonacci number, 34edo contains a fraction of an octave which is 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="http://xenharmonic.wikispaces.com/MOSScales"&gt;Moment of Symmetry&lt;/a&gt; scales with near-phi relationships between the step sizes. As a 2.3.5.13 temperament, the 21\34 generator is an approximate 20/13, and the temperament tempers out 512/507 and 140625/140608. From the tempering of 512/507, two 16/13 neutral thirds are an approximate 3/2, defining an essentially tempered neutral triad with a sharp rather than a flat fifth.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x34 tone equal temperament--Linear temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Linear temperaments&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x--Linear temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Linear temperaments&lt;/h3&gt;
   &lt;a class="wiki_link" href="/List%20of%2034et%20rank%20two%20temperaments%20by%20badness"&gt;List of 34et rank two temperaments by badness&lt;/a&gt;&lt;br /&gt;
@@ Line 286: / Line 305: @@
 &lt;/table&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="x34 tone equal temperament--Intervals:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Intervals:&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="x--Intervals:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Intervals:&lt;/h3&gt;
@@ Line 640: / Line 659: @@
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="x-Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;&lt;span style="background-color: #ffffff;"&gt;Commas&lt;/span&gt;&lt;/h2&gt;
+ &lt;span style="background-color: #ffffff;"&gt;34-EDO &lt;/span&gt;&lt;span style="background-color: #ffffff;"&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/tempering%20out"&gt;tempers out&lt;/a&gt;&lt;/span&gt;&lt;span style="background-color: #ffffff;"&gt; the following &lt;/span&gt;&lt;span style="background-color: #ffffff;"&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/comma"&gt;comma&lt;/a&gt;&lt;/span&gt;&lt;span style="background-color: #ffffff;"&gt;s. (Note: This assumes the &lt;/span&gt;&lt;span style="background-color: #ffffff;"&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/val"&gt;val&lt;/a&gt;&lt;/span&gt;&lt;span style="background-color: #ffffff;"&gt; &amp;lt; 34 54 79 95 118 126 |.)&lt;/span&gt;&lt;br /&gt;
+&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;th&gt;Comma&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;Monzo&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;Value (Cents)&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;Name 1&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;Name 2&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;Name 3&lt;br /&gt;
+&lt;/th&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;&lt;span style="background-color: #ffffff; display: block; text-align: center;"&gt;134217728/129140163&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: left;"&gt;| 27 -17 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;66.765&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;17-comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;20000/19683&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="background-color: #ffffff; display: block; text-align: left;"&gt;| 5 -9 4 &amp;gt;&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;&lt;span style="background-color: #ffffff; display: block; text-align: right;"&gt;27.660&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Minimal Diesis&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Tetracot Comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;&lt;span style="background-color: #ffffff; display: block; text-align: center;"&gt;2048/2025&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: left;"&gt;| 11 -4 -2 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;19.553&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Diaschisma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;span style="background-color: #ffffff; display: block; text-align: center;"&gt;Semicomma Majeur&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;&lt;span style="background-color: #ffffff; display: block; text-align: center;"&gt;393216/390625&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| &lt;span style="text-align: left;"&gt;17 1 -8 &amp;gt;&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;&lt;span style="background-color: #ffffff; display: block; text-align: right;"&gt;11.445&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Würschmidt comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;&lt;span style="background-color: #ffffff; display: block; text-align: center;"&gt;15625/15552&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="text-align: left;"&gt;| -6 -5 6 &amp;gt;&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;&lt;span style="text-align: right;"&gt;8.11&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;span style="text-align: center;"&gt;Kleisma&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;span style="text-align: center;"&gt;Semicomma Majeur&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;&lt;span style="background-color: #ffffff; display: block; text-align: center;"&gt;1212717/1210381&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: left;"&gt;| -4 4 -1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;3.338&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;span style="background-color: #ffffff; display: block; text-align: center;"&gt;Vishnuzma&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Semisuper&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;1029/1000&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: left;"&gt;| -3 1 -3 3 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;49.492&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;span style="background-color: #ffffff;"&gt;Keega&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/49_48"&gt;49/48&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: left;"&gt;| -4 -1 0 2 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;35.697&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Slendro Diesis&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;875/864&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: left;"&gt;| -5 -3 3 1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;21.902&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Keema&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;126/125&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: left;"&gt;| 1 2 -3 1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;13.795&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Starling comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Septimal semicomma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;100/99&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: left;"&gt;| 2 -2 2 0 -1&amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;17.399&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Ptolemisma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Ptolemy's comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;243/242&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: left;"&gt;&lt;span style="background-color: #ffffff; display: block; text-align: left;"&gt;| -1 5 0 0 -2 &amp;gt;&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;&lt;span style="background-color: #ffffff; display: block; text-align: right;"&gt;7.1391&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Rastma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Neutral third comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;385/384&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: left;"&gt;&lt;span style="background-color: #ffffff; display: block; text-align: left;"&gt;| -7 -1 1 1 1 &amp;gt;&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;&lt;span style="background-color: #ffffff; display: block; text-align: right;"&gt;4.5026&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Keenanisma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;91/90&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: left;"&gt;| -1 -2 -1 1 0 1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;19.120&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Superleap&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 646: / Line 884: @@
 &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="x34 tone equal temperament-Listen"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Listen&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="x-Listen"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Listen&lt;/h2&gt;
   &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;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="x34 tone equal temperament-Links"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Links&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="x-Links"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Links&lt;/h2&gt;
   &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="http://xenharmonic.wikispaces.com/Larry%20Hanson"&gt;Larry Hanson&lt;/a&gt;&lt;/li&gt;&lt;/ul&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>