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26edo

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=**Structure**= 

The structure of 26-EDO is an interesting beast, and I find three particular manners of understanding the chords and chordal relationships in it;
1. In terms of traditional chord types, that is, as a variant of meantone stretched too far, which yields interesting but perhaps unsatisfying results (due mainly to the dissonance of its thirds, and its major seconds of either approximately [[10_9|10/9]] or [[8_7|8/7]], but NOT [[9_8|9/8]]). This is not meant to say that this approach is without merit.
2. As two parallel 13-EDO scales (And as I've considered, if these two chains be shifted together or apart slightly but appropriately, interesting variants may be constructed) which is suitable for more atonal melodies. In this way its internal dynamics quite resemble those of 14-EDO.
3. 26-EDO nearly perfectly approximates the 7th and 11th harmonics, and an entire system may be constructed analogous to that based on the 3rd and 5th harmonics. The resulting scale is jagged, and it perverts common musical rules and conventions, but this particular organisation has its charm -perhaps because of this- and leads to some intriguing music.

- K.


=Orgone Temperament= 

[[Andrew Heathwaite]] proposes a temperament family which takes advantage of 26edo's excellent 11 & 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:

The 7-tone scale in degrees-in-between: 5 2 5 2 5 2 5. [[MOSScales|MOS]] of type [[4L 3s|4L 3s (mish)]].
The 7-tone scale in cents: 0 231 323 554 646 877 969 1200.

The 11-tone scale in degrees-in-between: 2 3 2 2 3 2 3 2 2 3 2. [[MOSScales|MOS]] of type [[4L 7s]].
The 11-tone scale in cents: 0 92 231 323 415 554 646 785 877 969 1108, 1200.

The primary triad for Orgone Temperament is 8:11:14 and its subharmonic inversion, which these scales have in abundance. 2g approximates [[16_11|16:11]] and 3g approximates [[7_4|7:4]] (and I would call that the definition of Orgone Temperament). That also implies that g approximates the difference between 7:4 and 16:11, which is 77:64, about 320.1 cents.

[[37edo]] is another excellent Orgone tuning. [[11edo]] is passable, but barely. If we take 11 and 26 to be the edges of the Orgone Spectrum, we may fill in the rest of the spectrum thus:

|| 3\11 ||   ||   ||   ||   ||
||   ||   ||   ||   || 19\70 ||
||   ||   ||   || 16\59 ||   ||
||   ||   ||   ||   || 29\107 ||
||   ||   || 13\48 ||   ||   ||
||   ||   ||   ||   || 36\133 ||
||   ||   ||   || 23\85 ||   ||
||   ||   ||   ||   || 33\122 ||
||   || 10\37 ||   ||   ||   ||
||   ||   ||   ||   || 37\137 ||
||   ||   ||   || 27\100 ||   ||
||   ||   ||   ||   || 44\163 ||
||   ||   || 17\63 ||   ||   ||
||   ||   ||   ||   || 41\152 ||
||   ||   ||   || 24\89 ||   ||
||   ||   ||   ||   || 31\115 ||
|| 7\26 ||   ||   ||   ||   ||

Orgone tempers out 65539/65219 = |16 0 0 -2 -3>, and has a minimax tuning which sharpens both 7 and 11 by 1/5 of this comma, or 1.679 cents. This makes the generator g a 77/64 sharp by 2/5 of the orgone comma. From this we may conclude that 24/89 or 31/115 would be reasonable alternatives to the 7/26 generator.

If a name already exists for this temperament, I'd be interested to know about it. As far as I know, temperaments which ignore lower primes (in this case 3 and 5) in favor of higher ones (in this case 7 and 11) are still largely uncharted, but I am interested in finding out if someone has walked this path before.

[[image:orgone_heptatonic.jpg]]

=Intervals= 

|| degree || cents ||
|| 0 || 0 ||
|| 1 || 46.15 ||
|| 2 || 92.31 ||
|| 3 || 138.46 ||
|| 4 || 184.62 ||
|| 5 || 230.77 ||
|| 6 || 276.92 ||
|| 7 || 323.08 ||
|| 8 || 369.23 ||
|| 9 || 415.38 ||
|| 10 || 461.54 ||
|| 11 || 507.69 ||
|| 12 || 553.85 ||
|| 13 || 600.00 ||
|| 14 || 646.15 ||
|| 15 || 692.31 ||
|| 16 || 738.46 ||
|| 17 || 784.62 ||
|| 18 || 830.77 ||
|| 19 || 876.92 ||
|| 20 || 923.08 ||
|| 21 || 969.23 ||
|| 22 || 1015.38 ||
|| 23 || 1061.54 ||
|| 24 || 1107.69 ||
|| 25 || 1153.85 ||

=Additional Scalar Bases Available in 26-EDO:= 
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).

-Igs

=Literature= 

[[http://www.ronsword.com|Sword, Ron. **Icosihexaphonic Scales for Guitar**. IAAA Press. 2010 - A Guitar-scale thesaurus for 26-EDO.]]

=Compositions= 

[[http://soundclick.com/share?songid=5683791|A Time-Yellowed Photo of the Cliffs Hangs on the Wall ]] by [[Igliashon Jones]]
[[http://www.io.com/%7Ehmiller/midi/26tet.mid|Etude in 26-tone equal tuning]] by [[Herman Miller]]
[[http://danielthompson.blogspot.com/2007/04/new-version-of-organ-study-1.html|A New Recording of Organ Study #1]] by [[Daniel Thompson]]

Original HTML content:

<html><head><title>26edo</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Structure"></a><!-- ws:end:WikiTextHeadingRule:0 --><strong>Structure</strong></h1>
 <br />
The structure of 26-EDO is an interesting beast, and I find three particular manners of understanding the chords and chordal relationships in it;<br />
1. In terms of traditional chord types, that is, as a variant of meantone stretched too far, which yields interesting but perhaps unsatisfying results (due mainly to the dissonance of its thirds, and its major seconds of either approximately <a class="wiki_link" href="/10_9">10/9</a> or <a class="wiki_link" href="/8_7">8/7</a>, but NOT <a class="wiki_link" href="/9_8">9/8</a>). This is not meant to say that this approach is without merit.<br />
2. As two parallel 13-EDO scales (And as I've considered, if these two chains be shifted together or apart slightly but appropriately, interesting variants may be constructed) which is suitable for more atonal melodies. In this way its internal dynamics quite resemble those of 14-EDO.<br />
3. 26-EDO nearly perfectly approximates the 7th and 11th harmonics, and an entire system may be constructed analogous to that based on the 3rd and 5th harmonics. The resulting scale is jagged, and it perverts common musical rules and conventions, but this particular organisation has its charm -perhaps because of this- and leads to some intriguing music.<br />
<br />
- K.<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Orgone Temperament"></a><!-- ws:end:WikiTextHeadingRule:2 -->Orgone Temperament</h1>
 <br />
<a class="wiki_link" href="/Andrew%20Heathwaite">Andrew Heathwaite</a> proposes a temperament family which takes advantage of 26edo's excellent 11 &amp; 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 />
<br />
The 7-tone scale in degrees-in-between: 5 2 5 2 5 2 5. <a class="wiki_link" href="/MOSScales">MOS</a> of type <a class="wiki_link" href="/4L%203s">4L 3s (mish)</a>.<br />
The 7-tone scale in cents: 0 231 323 554 646 877 969 1200.<br />
<br />
The 11-tone scale in degrees-in-between: 2 3 2 2 3 2 3 2 2 3 2. <a class="wiki_link" href="/MOSScales">MOS</a> of type <a class="wiki_link" href="/4L%207s">4L 7s</a>.<br />
The 11-tone scale in cents: 0 92 231 323 415 554 646 785 877 969 1108, 1200.<br />
<br />
The primary triad for Orgone Temperament is 8:11:14 and its subharmonic inversion, which these scales have in abundance. 2g approximates <a class="wiki_link" href="/16_11">16:11</a> and 3g approximates <a class="wiki_link" href="/7_4">7:4</a> (and I would call that the definition of Orgone Temperament). That also implies that g approximates the difference between 7:4 and 16:11, which is 77:64, about 320.1 cents.<br />
<br />
<a class="wiki_link" href="/37edo">37edo</a> is another excellent Orgone tuning. <a class="wiki_link" href="/11edo">11edo</a> is passable, but barely. If we take 11 and 26 to be the edges of the Orgone Spectrum, we may fill in the rest of the spectrum thus:<br />
<br />


<table class="wiki_table">
    <tr>
        <td>3\11<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>19\70<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>16\59<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>29\107<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td>13\48<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>36\133<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>23\85<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>33\122<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>10\37<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>37\137<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>27\100<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>44\163<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td>17\63<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>41\152<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>24\89<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>31\115<br />
</td>
    </tr>
    <tr>
        <td>7\26<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
</table>

<br />
Orgone tempers out 65539/65219 = |16 0 0 -2 -3&gt;, and has a minimax tuning which sharpens both 7 and 11 by 1/5 of this comma, or 1.679 cents. This makes the generator g a 77/64 sharp by 2/5 of the orgone comma. From this we may conclude that 24/89 or 31/115 would be reasonable alternatives to the 7/26 generator.<br />
<br />
If a name already exists for this temperament, I'd be interested to know about it. As far as I know, temperaments which ignore lower primes (in this case 3 and 5) in favor of higher ones (in this case 7 and 11) are still largely uncharted, but I am interested in finding out if someone has walked this path before.<br />
<br />
<!-- ws:start:WikiTextLocalImageRule:382:&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:382 --><br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:4 -->Intervals</h1>
 <br />


<table class="wiki_table">
    <tr>
        <td>degree<br />
</td>
        <td>cents<br />
</td>
    </tr>
    <tr>
        <td>0<br />
</td>
        <td>0<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>46.15<br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>92.31<br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>138.46<br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>184.62<br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>230.77<br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>276.92<br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>323.08<br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>369.23<br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>415.38<br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td>461.54<br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td>507.69<br />
</td>
    </tr>
    <tr>
        <td>12<br />
</td>
        <td>553.85<br />
</td>
    </tr>
    <tr>
        <td>13<br />
</td>
        <td>600.00<br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td>646.15<br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td>692.31<br />
</td>
    </tr>
    <tr>
        <td>16<br />
</td>
        <td>738.46<br />
</td>
    </tr>
    <tr>
        <td>17<br />
</td>
        <td>784.62<br />
</td>
    </tr>
    <tr>
        <td>18<br />
</td>
        <td>830.77<br />
</td>
    </tr>
    <tr>
        <td>19<br />
</td>
        <td>876.92<br />
</td>
    </tr>
    <tr>
        <td>20<br />
</td>
        <td>923.08<br />
</td>
    </tr>
    <tr>
        <td>21<br />
</td>
        <td>969.23<br />
</td>
    </tr>
    <tr>
        <td>22<br />
</td>
        <td>1015.38<br />
</td>
    </tr>
    <tr>
        <td>23<br />
</td>
        <td>1061.54<br />
</td>
    </tr>
    <tr>
        <td>24<br />
</td>
        <td>1107.69<br />
</td>
    </tr>
    <tr>
        <td>25<br />
</td>
        <td>1153.85<br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Additional Scalar Bases Available in 26-EDO:"></a><!-- ws:end:WikiTextHeadingRule:6 -->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 />
<br />
-Igs<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Literature"></a><!-- ws:end:WikiTextHeadingRule:8 -->Literature</h1>
 <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 />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Compositions"></a><!-- ws:end:WikiTextHeadingRule:10 -->Compositions</h1>
 <br />
<a class="wiki_link_ext" href="http://soundclick.com/share?songid=5683791" rel="nofollow">A Time-Yellowed Photo of the Cliffs Hangs on the Wall </a> by <a class="wiki_link" href="/Igliashon%20Jones">Igliashon Jones</a><br />
<a class="wiki_link_ext" href="http://www.io.com/%7Ehmiller/midi/26tet.mid" rel="nofollow">Etude in 26-tone equal tuning</a> by <a class="wiki_link" href="/Herman%20Miller">Herman Miller</a><br />
<a class="wiki_link_ext" href="http://danielthompson.blogspot.com/2007/04/new-version-of-organ-study-1.html" rel="nofollow">A New Recording of Organ Study #1</a> by <a class="wiki_link" href="/Daniel%20Thompson">Daniel Thompson</a></body></html>
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