26edo: Difference between revisions
Wikispaces>genewardsmith **Imported revision 491236292 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 491250148 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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:genewardsmith|genewardsmith]] and made on <tt>2014-02-22 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-02-22 17:02:34 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>491250148</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> | ||
| Line 90: | Line 90: | ||
|| 2 || 2\26 || [[Injera]] || | || 2 || 2\26 || [[Injera]] || | ||
|| 2 || 3\26 || [[Fifive]]/Crepuscular || | || 2 || 3\26 || [[Fifive]]/Crepuscular || | ||
|| 2 || 4\26 || [[Unidec]]/Dubbla || | || 2 || 4\26 || [[Unidec]]/[[Gamelismic clan#Unidec-Hendec|Hendec]]/Dubbla || | ||
|| 2 || 5\26 || [[Lemba]] || | || 2 || 5\26 || [[Lemba]] || | ||
|| 2 || 6\26 || [[Doublewide]]/Cavalier || | || 2 || 6\26 || [[Doublewide]]/Cavalier || | ||
|| 13 || 1\26 || Triskaidekic || | || 13 || 1\26 || Triskaidekic || | ||
==Hendec in 26et== | |||
[[Gamelismic clan#Unidec-Hendec|Hendec]], the 13-limit 26&46 temperament with generator ~10/9, concentrates the intervals of greatest accuracy in 26et into the lower ranges of complexity. It has a period of half an octave, with 13/12 reachable by four generators, 8/7 by two, 14/11 by one, 10/9 by one, and 11/8 by three. All of these are tuned to within 2.5 cents of accuracy. | |||
=Commas= | =Commas= | ||
| Line 177: | Line 180: | ||
////[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20Guitar%20Serenade%20in%20Q%20Major.mp3|Guitar Serenade in Q Major]]//// by [[Zach Curley]]</pre></div> | ////[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20Guitar%20Serenade%20in%20Q%20Major.mp3|Guitar Serenade in Q Major]]//// by [[Zach Curley]]</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>26edo</title></head><body><!-- ws:start:WikiTextTocRule: | <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>26edo</title></head><body><!-- ws:start:WikiTextTocRule:23:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --><div style="margin-left: 1em;"><a href="#Structure">Structure</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --><div style="margin-left: 1em;"><a href="#Intervals">Intervals</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --><div style="margin-left: 2em;"><a href="#Intervals-Intervals by patent val error">Intervals by patent val error</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><div style="margin-left: 1em;"><a href="#Rank two temperaments">Rank two temperaments</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --><div style="margin-left: 2em;"><a href="#Rank two temperaments-Hendec in 26et">Hendec in 26et</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><div style="margin-left: 1em;"><a href="#Commas">Commas</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><div style="margin-left: 1em;"><a href="#Orgone Temperament">Orgone Temperament</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><div style="margin-left: 1em;"><a href="#toc7"> </a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><div style="margin-left: 1em;"><a href="#Additional Scalar Bases Available in 26-EDO:">Additional Scalar Bases Available in 26-EDO:</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><div style="margin-left: 1em;"><a href="#Literature">Literature</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><div style="margin-left: 1em;"><a href="#Compositions">Compositions</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --></div> | ||
<!-- ws:end:WikiTextTocRule:35 --><em>26edo</em> divides the <a class="wiki_link" href="/octave">octave</a> into 26 equal parts of 46.154 <a class="wiki_link" href="/cent">cent</a>s each. It tempers out 81/80 in the <a class="wiki_link" href="/5-limit">5-limit</a>, making it a meantone tuning with a very flat fifth. In the <a class="wiki_link" href="/7-limit">7-limit</a>, it tempers out 50/49, 525/512 and 875/864, and supports <a class="wiki_link" href="/Meantone%20family">injera</a>, <a class="wiki_link" href="/Meantone%20family">flattone</a>, <a class="wiki_link" href="/Jubilismic%20clan#Lemba">lemba</a> and <a class="wiki_link" href="/Jubilismic%20clan#Doublewide">doublewide</a> temperaments. It really comes into its own as a higher-limit temperament, being the smallest equal division which represents the <a class="wiki_link" href="/13-limit">13 odd limit</a> <a class="wiki_link" href="/consistent">consistently</a>. 26edo has a very good approximation of the harmonic seventh (<a class="wiki_link" href="/7_4">7/4</a>).<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:1:&lt;h1&gt; --><h1 id="toc0"><a name="Structure"></a><!-- ws:end:WikiTextHeadingRule:1 --><strong>Structure</strong></h1> | <!-- ws:start:WikiTextHeadingRule:1:&lt;h1&gt; --><h1 id="toc0"><a name="Structure"></a><!-- ws:end:WikiTextHeadingRule:1 --><strong>Structure</strong></h1> | ||
| Line 736: | Line 740: | ||
<td>4\26<br /> | <td>4\26<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href="/Unidec">Unidec</a>/Dubbla<br /> | <td><a class="wiki_link" href="/Unidec">Unidec</a>/<a class="wiki_link" href="/Gamelismic%20clan#Unidec-Hendec">Hendec</a>/Dubbla<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 766: | Line 770: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:9:&lt;h1&gt; --><h1 id=" | <!-- ws:start:WikiTextHeadingRule:9:&lt;h2&gt; --><h2 id="toc4"><a name="Rank two temperaments-Hendec in 26et"></a><!-- ws:end:WikiTextHeadingRule:9 -->Hendec in 26et</h2> | ||
<a class="wiki_link" href="/Gamelismic%20clan#Unidec-Hendec">Hendec</a>, the 13-limit 26&amp;46 temperament with generator ~10/9, concentrates the intervals of greatest accuracy in 26et into the lower ranges of complexity. It has a period of half an octave, with 13/12 reachable by four generators, 8/7 by two, 14/11 by one, 10/9 by one, and 11/8 by three. All of these are tuned to within 2.5 cents of accuracy.<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:11:&lt;h1&gt; --><h1 id="toc5"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:11 -->Commas</h1> | |||
26et tempers out the following commas. (Note: This assumes the val &lt; 26 41 60 73 90 96 |.)<br /> | 26et tempers out the following commas. (Note: This assumes the val &lt; 26 41 60 73 90 96 |.)<br /> | ||
| Line 1,053: | Line 1,060: | ||
</table> | </table> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:13:&lt;h1&gt; --><h1 id="toc6"><a name="Orgone Temperament"></a><!-- ws:end:WikiTextHeadingRule:13 --><a class="wiki_link" href="/Orgonia">Orgone Temperament</a></h1> | ||
<br /> | <br /> | ||
<a class="wiki_link" href="/Andrew%20Heathwaite">Andrew Heathwaite</a> first proposed orgone temperament to take advantage of 26edo's excellent 11 and 7 approximations. 7 degrees of 26edo is a wide minor third of approximately 323.077 cents, and that interval taken as a generator produces 7-tone and 11-tone MOS scales:<br /> | <a class="wiki_link" href="/Andrew%20Heathwaite">Andrew Heathwaite</a> first proposed orgone temperament to take advantage of 26edo's excellent 11 and 7 approximations. 7 degrees of 26edo is a wide minor third of approximately 323.077 cents, and that interval taken as a generator produces 7-tone and 11-tone MOS scales:<br /> | ||
| Line 1,279: | Line 1,286: | ||
Orgone has a minimax tuning which sharpens both 7 and 11 by 1/5 of an orgonisma, or 1.679 cents. This makes the generator g a 77/64 sharp by 2/5 of the orgonisma. From this we may conclude that 24/89 or 31/115 would be reasonable alternatives to the 7/26 generator.<br /> | Orgone has a minimax tuning which sharpens both 7 and 11 by 1/5 of an orgonisma, or 1.679 cents. This makes the generator g a 77/64 sharp by 2/5 of the orgonisma. From this we may conclude that 24/89 or 31/115 would be reasonable alternatives to the 7/26 generator.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:1072:&lt;img src=&quot;/file/view/orgone_heptatonic.jpg/155606933/orgone_heptatonic.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/orgone_heptatonic.jpg/155606933/orgone_heptatonic.jpg" alt="orgone_heptatonic.jpg" title="orgone_heptatonic.jpg" /><!-- ws:end:WikiTextLocalImageRule:1072 --><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:15:&lt;h1&gt; --><h1 id="toc7"><!-- ws:end:WikiTextHeadingRule:15 --> </h1> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:17:&lt;h1&gt; --><h1 id="toc8"><a name="Additional Scalar Bases Available in 26-EDO:"></a><!-- ws:end:WikiTextHeadingRule:17 -->Additional Scalar Bases Available in 26-EDO:</h1> | ||
Since the perfect 5th in 26-EDO spans 15 degrees, it can be divided into three equal parts (each approximately an 8/7) as well as five equal parts (each approximately a 13/12). The former approach produces MOS at 1L+4s, 5L+1s, and 5L+6s (5 5 5 5 6, 5 5 5 5 5 1, and 4 1 4 1 4 1 4 1 4 1 1 respectively), and is excellent for 4:6:7 triads. The latter produces MOS at 1L+7s and 8L+1s (3 3 3 3 3 3 3 5 and 3 3 3 3 3 3 3 3 2 respectively), and is fairly well-supplied with 4:6:7:11:13 pentads. It also works well for more conventional (thought further from Just) 6:7:9 triads, as well as 4:5:6 triads that use the worse mapping for 5 (making 5/4 the 415.38-cent interval).<br /> | Since the perfect 5th in 26-EDO spans 15 degrees, it can be divided into three equal parts (each approximately an 8/7) as well as five equal parts (each approximately a 13/12). The former approach produces MOS at 1L+4s, 5L+1s, and 5L+6s (5 5 5 5 6, 5 5 5 5 5 1, and 4 1 4 1 4 1 4 1 4 1 1 respectively), and is excellent for 4:6:7 triads. The latter produces MOS at 1L+7s and 8L+1s (3 3 3 3 3 3 3 5 and 3 3 3 3 3 3 3 3 2 respectively), and is fairly well-supplied with 4:6:7:11:13 pentads. It also works well for more conventional (thought further from Just) 6:7:9 triads, as well as 4:5:6 triads that use the worse mapping for 5 (making 5/4 the 415.38-cent interval).<br /> | ||
<br /> | <br /> | ||
-Igs<br /> | -Igs<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:19:&lt;h1&gt; --><h1 id="toc9"><a name="Literature"></a><!-- ws:end:WikiTextHeadingRule:19 -->Literature</h1> | ||
<br /> | <br /> | ||
<a class="wiki_link_ext" href="http://www.ronsword.com" rel="nofollow">Sword, Ron. **Icosihexaphonic Scales for Guitar**. IAAA Press. 2010 - A Guitar-scale thesaurus for 26-EDO.</a><br /> | <a class="wiki_link_ext" href="http://www.ronsword.com" rel="nofollow">Sword, Ron. **Icosihexaphonic Scales for Guitar**. IAAA Press. 2010 - A Guitar-scale thesaurus for 26-EDO.</a><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:21:&lt;h1&gt; --><h1 id="toc10"><a name="Compositions"></a><!-- ws:end:WikiTextHeadingRule:21 -->Compositions</h1> | ||
<br /> | <br /> | ||
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20A%20Time-Yellowed%20Photograph%20of%20Cliffs%20Hangs%20in%20the%20Hall.mp3" rel="nofollow">A Time-Yellowed Photo of the Cliffs Hangs on the Wall </a></span> by <a class="wiki_link" href="/IgliashonJones">Igliashon Jones</a><br /> | <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20A%20Time-Yellowed%20Photograph%20of%20Cliffs%20Hangs%20in%20the%20Hall.mp3" rel="nofollow">A Time-Yellowed Photo of the Cliffs Hangs on the Wall </a></span> by <a class="wiki_link" href="/IgliashonJones">Igliashon Jones</a><br /> | ||