26edo: Difference between revisions
Wikispaces>phylingual **Imported revision 352177426 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 491236292 - 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: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-02-22 15:01:04 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>491236292</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> | ||
<h4>Original Wikitext content:</h4> | <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">[[toc | <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">[[toc]] | ||
//26edo// divides the [[octave]] into 26 equal parts of 46.154 [[cent]]s each. It tempers out 81/80 in the [[5-limit]], making it a meantone tuning with a very flat fifth. In the [[7-limit]], it tempers out 50/49, 525/512 and 875/864, and supports [[Meantone family|injera]], [[Meantone family|flattone]], [[Jubilismic clan#Lemba|lemba]] and [[Jubilismic clan#Doublewide|doublewide]] temperaments. It really comes into its own as a higher-limit temperament, being the smallest equal division which represents the [[13-limit|13 odd limit]] [[consistent|consistently]]. 26edo has a very good approximation of the harmonic seventh ([[7_4|7/4]]). | //26edo// divides the [[octave]] into 26 equal parts of 46.154 [[cent]]s each. It tempers out 81/80 in the [[5-limit]], making it a meantone tuning with a very flat fifth. In the [[7-limit]], it tempers out 50/49, 525/512 and 875/864, and supports [[Meantone family|injera]], [[Meantone family|flattone]], [[Jubilismic clan#Lemba|lemba]] and [[Jubilismic clan#Doublewide|doublewide]] temperaments. It really comes into its own as a higher-limit temperament, being the smallest equal division which represents the [[13-limit|13 odd limit]] [[consistent|consistently]]. 26edo has a very good approximation of the harmonic seventh ([[7_4|7/4]]). | ||
| Line 18: | Line 18: | ||
=<span style="font-size: 1.4em;">Intervals</span>= | =<span style="font-size: 1.4em;">Intervals</span>= | ||
|| degree || [[cent]]s ||= Approximate | || degree || [[cent]]s ||= Approximate | ||
Ratios* || Solfege || | Ratios* || Solfege || | ||
| Line 49: | Line 48: | ||
|| 26 || 1200 ||= 2/1 || do || | || 26 || 1200 ||= 2/1 || do || | ||
*based on treating 26-EDO as a 13-limit temperament; other approaches are possible. | *based on treating 26-EDO as a 13-limit temperament; other approaches are possible. | ||
==Intervals by patent val error== | |||
|| Interval || Error || | |||
|| 13/12 || -0.111 || | |||
|| 8/7 || -0.405 || | |||
|| 14/11 || -2.123 || | |||
|| 10/9 || 2.212 || | |||
|| 11/8 || 2.528 || | |||
|| 13/10 || 7.325 || | |||
|| 6/5 || 7.436 || | |||
|| 18/13 || -9.536 || | |||
|| 4/3 || 9.647 || | |||
|| 16/13 || 9.758 || | |||
|| 7/6 || 10.052 || | |||
|| 14/13 || 10.163 || | |||
|| 12/11 || -12.176 || | |||
|| 13/11 || -12.287 || | |||
|| 15/13 || -16.972 || | |||
|| 5/4 || -17.083 || | |||
|| 7/5 || 17.488 || | |||
|| 9/8 || -19.295 || | |||
|| 11/10 || 19.611 || | |||
|| 9/7 || -19.699 || | |||
|| 11/9 || 21.823 || | |||
|| 16/15 || 26.73 || | |||
|| 15/14 || -27.135 || | |||
|| 15/11 || -29.258 || | |||
=Rank two temperaments= | =Rank two temperaments= | ||
| Line 151: | Line 177: | ||
////[[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:21:&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:21 --><!-- ws:start:WikiTextTocRule:22: --><div style="margin-left: 1em;"><a href="#Structure">Structure</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --><div style="margin-left: 1em;"><a href="#Intervals">Intervals</a></div> | ||
<!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --><div style="margin-left: 2em;"><a href="#Intervals-Intervals by patent val error">Intervals by patent val error</a></div> | |||
<!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --><div style="margin-left: 1em;"><a href="#Rank two temperaments">Rank two temperaments</a></div> | |||
<!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --><div style="margin-left: 1em;"><a href="#Commas">Commas</a></div> | |||
<!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><div style="margin-left: 1em;"><a href="#Orgone Temperament">Orgone Temperament</a></div> | |||
<!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --><div style="margin-left: 1em;"><a href="#toc6"> </a></div> | |||
<!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><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:29 --><!-- ws:start:WikiTextTocRule:30: --><div style="margin-left: 1em;"><a href="#Literature">Literature</a></div> | |||
<!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><div style="margin-left: 1em;"><a href="#Compositions">Compositions</a></div> | |||
<!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --></div> | |||
<!-- ws:end:WikiTextTocRule:32 --><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 163: | Line 199: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:3:&lt;h1&gt; --><h1 id="toc1"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:3 --><span style="font-size: 1.4em;">Intervals</span></h1> | <!-- ws:start:WikiTextHeadingRule:3:&lt;h1&gt; --><h1 id="toc1"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:3 --><span style="font-size: 1.4em;">Intervals</span></h1> | ||
<table class="wiki_table"> | <table class="wiki_table"> | ||
| Line 452: | Line 487: | ||
*based on treating 26-EDO as a 13-limit temperament; other approaches are possible.<br /> | *based on treating 26-EDO as a 13-limit temperament; other approaches are possible.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:5:&lt;h1&gt; --><h1 id=" | <!-- ws:start:WikiTextHeadingRule:5:&lt;h2&gt; --><h2 id="toc2"><a name="Intervals-Intervals by patent val error"></a><!-- ws:end:WikiTextHeadingRule:5 -->Intervals by patent val error</h2> | ||
<table class="wiki_table"> | |||
<tr> | |||
<td>Interval<br /> | |||
</td> | |||
<td>Error<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>13/12<br /> | |||
</td> | |||
<td>-0.111<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>8/7<br /> | |||
</td> | |||
<td>-0.405<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>14/11<br /> | |||
</td> | |||
<td>-2.123<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>10/9<br /> | |||
</td> | |||
<td>2.212<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>11/8<br /> | |||
</td> | |||
<td>2.528<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>13/10<br /> | |||
</td> | |||
<td>7.325<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>6/5<br /> | |||
</td> | |||
<td>7.436<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>18/13<br /> | |||
</td> | |||
<td>-9.536<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>4/3<br /> | |||
</td> | |||
<td>9.647<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>16/13<br /> | |||
</td> | |||
<td>9.758<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>7/6<br /> | |||
</td> | |||
<td>10.052<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>14/13<br /> | |||
</td> | |||
<td>10.163<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>12/11<br /> | |||
</td> | |||
<td>-12.176<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>13/11<br /> | |||
</td> | |||
<td>-12.287<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>15/13<br /> | |||
</td> | |||
<td>-16.972<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>5/4<br /> | |||
</td> | |||
<td>-17.083<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>7/5<br /> | |||
</td> | |||
<td>17.488<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>9/8<br /> | |||
</td> | |||
<td>-19.295<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>11/10<br /> | |||
</td> | |||
<td>19.611<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>9/7<br /> | |||
</td> | |||
<td>-19.699<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>11/9<br /> | |||
</td> | |||
<td>21.823<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>16/15<br /> | |||
</td> | |||
<td>26.73<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>15/14<br /> | |||
</td> | |||
<td>-27.135<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>15/11<br /> | |||
</td> | |||
<td>-29.258<br /> | |||
</td> | |||
</tr> | |||
</table> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:7:&lt;h1&gt; --><h1 id="toc3"><a name="Rank two temperaments"></a><!-- ws:end:WikiTextHeadingRule:7 -->Rank two temperaments</h1> | |||
<a class="wiki_link" href="/List%20of%2026et%20rank%20two%20temperaments%20by%20badness">List of 26et rank two temperaments by badness</a><br /> | <a class="wiki_link" href="/List%20of%2026et%20rank%20two%20temperaments%20by%20badness">List of 26et rank two temperaments by badness</a><br /> | ||
<a class="wiki_link" href="/List%20of%20edo-distinct%2026et%20rank%20two%20temperaments">List of edo-distinct 26et rank two temperaments</a><br /> | <a class="wiki_link" href="/List%20of%20edo-distinct%2026et%20rank%20two%20temperaments">List of edo-distinct 26et rank two temperaments</a><br /> | ||
| Line 574: | Line 766: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:9:&lt;h1&gt; --><h1 id="toc4"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:9 -->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 861: | Line 1,053: | ||
</table> | </table> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:11:&lt;h1&gt; --><h1 id="toc5"><a name="Orgone Temperament"></a><!-- ws:end:WikiTextHeadingRule:11 --><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 1,087: | Line 1,279: | ||
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:1069:&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:1069 --><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:13:&lt;h1&gt; --><h1 id="toc6"><!-- ws:end:WikiTextHeadingRule:13 --> </h1> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:15:&lt;h1&gt; --><h1 id="toc7"><a name="Additional Scalar Bases Available in 26-EDO:"></a><!-- ws:end:WikiTextHeadingRule:15 -->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:17:&lt;h1&gt; --><h1 id="toc8"><a name="Literature"></a><!-- ws:end:WikiTextHeadingRule:17 -->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:19:&lt;h1&gt; --><h1 id="toc9"><a name="Compositions"></a><!-- ws:end:WikiTextHeadingRule:19 -->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 /> | ||