34edo: Difference between revisions
Wikispaces>JosephRuhf **Imported revision 600630016 - Original comment: ** |
Wikispaces>JosephRuhf **Imported revision 601502432 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | 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>2016- | : This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-12-06 08:24:18 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>601502432</tt>.<br> | ||
: The revision comment was: <tt></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> | 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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=34edo and phi= | =34edo and phi= | ||
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 [[xenharmonic/MOSScales|Moment of Symmetry]] 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 | 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 [[xenharmonic/MOSScales|Moment of Symmetry]] 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 | -6 2 6 0 0 -13 >. 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. Yes. But, to be clear the harmonic ratio of phi is ~ 833 cents, and the equal divisions of octave approximating this interval closely are 13edo and [[36edo]]. | ||
=Rank two temperaments= | =Rank two temperaments= | ||
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|| degrees of 34edo || solfege || cents | || degrees of 34edo || solfege || cents | ||
DMS || approx. ratios of | DMS || approx. ratios of | ||
2.3.5.13.17/1 [[xenharmonic/subgroup|subgroup]] || additional ratios | |||
of 7 and 11 || pseudo-traditional | of 7 and 11 || pseudo-traditional | ||
notation || | notation || | ||
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158°49'15" || || 15/11 || F^ = E#v = Gbv || | 158°49'15" || || 15/11 || F^ = E#v = Gbv || | ||
|| 16 || fu || 564.706 | || 16 || fu || 564.706 | ||
169°24'42" || 18/13 || 11/8 || E# = F^^ = Gb || | 169°24'42" || 18/13 || 11/8 || E# | ||
=F^^= | |||
Gb || | |||
|| 17 || fi/se || 600 | || 17 || fi/se || 600 | ||
180° || 17/12, 24/17 || 7/5, 10/7 || Gb^ || | 180° || 17/12, 24/17 || 7/5, 10/7 || Gb^ || | ||
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=Commas= | =Commas= | ||
34-EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] < | 34-EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] < 34 54 79 95 118 126/1 |.) | ||
||= **Comma** ||= **Monzo** ||= **Value (Cents)** ||= **Names** || | ||= **Comma** ||= **Monzo** ||= **Value (Cents)** ||= **Names** || | ||
||= | ||= || | 27 -17 > ||> 66.765 ||= 17-comma || | ||
||= 20000/19683 || | 5 -9 4 > ||> 27.660 ||= Minimal Diesis, Tetracot Comma || | ||= 20000/19683 || | 5 -9 4 > ||> 27.660 ||= Minimal Diesis, Tetracot Comma || | ||
||= 2048/2025 || | 11 -4 -2 > ||> 19.553 ||= Diaschisma || | ||= 2048/2025 || | 11 -4 -2 > ||> 19.553 ||= Diaschisma || | ||
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* [[https://myspace.com/microstick]]</pre></div> | * [[https://myspace.com/microstick]]</pre></div> | ||
<h4>Original HTML content:</h4> | <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><!-- ws:start:WikiTextTocRule:20:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --><a href="#Approximations to Just Intonation">Approximations to Just Intonation</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | <a href="#x34edo and phi">34edo and phi</a><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --> | <a href="#Rank two temperaments">Rank two temperaments</a><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --> | <a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --> | <a href="#F^^"> F^^ </a><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --> | <a href="#Notations">Notations</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --> | <a href="#Commas">Commas</a><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --> | <a href="#Listen">Listen</a><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --> | <a href="#Links">Links</a><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --> | <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><!-- ws:start:WikiTextTocRule:20:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --><a href="#Approximations to Just Intonation">Approximations to Just Intonation</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | <a href="#x34edo and phi">34edo and phi</a><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --> | <a href="#Rank two temperaments">Rank two temperaments</a><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --> | <a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --> | <a href="#F^^">F^^</a><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --> | <a href="#Notations">Notations</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --> | <a href="#Commas">Commas</a><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --> | <a href="#Listen">Listen</a><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --> | <a href="#Links">Links</a><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --> | ||
<!-- ws:end:WikiTextTocRule:31 --><hr /> | <!-- ws:end:WikiTextTocRule:31 --><hr /> | ||
<br /> | <br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="x34edo and phi"></a><!-- ws:end:WikiTextHeadingRule:2 -->34edo and phi</h1> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="x34edo and phi"></a><!-- ws:end:WikiTextHeadingRule:2 -->34edo and phi</h1> | ||
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 | 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 | -6 2 6 0 0 -13 &gt;. 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. Yes. But, to be clear the harmonic ratio of phi is ~ 833 cents, and the equal divisions of octave approximating this interval closely are 13edo and <a class="wiki_link" href="/36edo">36edo</a>.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Rank two temperaments"></a><!-- ws:end:WikiTextHeadingRule:4 -->Rank two temperaments</h1> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Rank two temperaments"></a><!-- ws:end:WikiTextHeadingRule:4 -->Rank two temperaments</h1> | ||
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</td> | </td> | ||
<td>approx. ratios of<br /> | <td>approx. ratios of<br /> | ||
2.3.5.13.17/1 <a class="wiki_link" href="http://xenharmonic.wikispaces.com/subgroup">subgroup</a><br /> | |||
</td> | </td> | ||
<td>additional ratios<br /> | <td>additional ratios<br /> | ||
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<td>11/8<br /> | <td>11/8<br /> | ||
</td> | </td> | ||
<td>E# | <td>E#<br /> | ||
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="F^^"></a><!-- ws:end:WikiTextHeadingRule:8 -->F^^</h1> | |||
Gb<br /> | Gb<br /> | ||
</td> | </td> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:14 -->Commas</h1> | <!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:14 -->Commas</h1> | ||
34-EDO <a class="wiki_link" href="/tempering%20out">tempers out</a> the following <a class="wiki_link" href="/comma">comma</a>s. (Note: This assumes the <a class="wiki_link" href="/val">val</a> &lt | 34-EDO <a class="wiki_link" href="/tempering%20out">tempers out</a> the following <a class="wiki_link" href="/comma">comma</a>s. (Note: This assumes the <a class="wiki_link" href="/val">val</a> &lt; 34 54 79 95 118 126/1 |.)<br /> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td>| 27 -17 &gt;<br /> | <td>| 27 -17 &gt;<br /> | ||