23edo

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=<span style="color: #007a1b; display: block;">23 tone equal temperament</span>= 

23et, or 23-EDO, is a tuning system which divides the [[octave]] into 23 equal parts of approximately 52.173913 cents. It has good approximations for 5/3, 11/7, 13 and 17, allowing it to represent the 2.5/3.11/7.13.17 [[just intonation subgroup]]. If to this subgroup is added the commas of 17-limit 46et, the larger 17-limit [[k*N subgroups|2*23 subgroup]] 2.9.15.21.33.13.17 is obtained. This is the largest subgroup on which 23 has the same tuning and commas as does 17-limit 46, and may be regarded as a basis for analyzing the harmony of 23-EDO so far as approximations to just intervals goes. 

23-EDO was proposed by ethnomusicologist [[http://en.wikipedia.org/wiki/Erich_von_Hornbostel|Erich von Hornbostel]] as the result of continuing a circle of "blown" fifths of ~678-cent fifths that (he argued) resulted from "overblowing" a bamboo pipe.

23-EDO is also significant in that it is the largest EDO that fails to approximate the 3rd, 5th, and 7th harmonics within 20 cents, which makes it well-suited for musicians seeking to explore unusual harmonic territory. Oddly, despite the fact that it fails to approximate these harmonics, it approximates the intervals between them (5/3, 7/3, and 7/5) very well. The lowest harmonics well-approximated by 23-EDO are 13, 17, 21, and 23. 

Like 9-EDO, 16-EDO, and 25-EDO, one way to treat 23-EDO is as a Pelogic temperament, tempering out the "comma" of 135/128 and equating three sharp 4/3's with 5/1 (related to the Armodue system). This means mapping 3/2 to 13 degrees of 23, and results in a 7-note "anti-diatonic" scale of 3 3 4 3 3 3 4 (in steps of 23-EDO), which extends to 9 notes (3 3 3 1 3 3 3 3 1).

However, one can also map 3/2 to 14 degrees of 23-EDO without significantly increasing the error, taking us to a 7-limit temperament where two 3/2's equals 7/3, meaning 28/27 is tempered out, and six 4/3's octave-reduced equals 5/4, meaning 4096/3645 is tempered out. Both of these are very large commas, so this is not at all an accurate temperament, but it is related to 13-EDO and 18-EDO and produces MOS scales of 5 and 8 notes: 5 5 4 5 4 (the "anti-pentatonic") and 4 1 4 1 4 4 1 4 (the "quarter-tone" version of the Blackwood/[[http://en.wikipedia.org/wiki/Paul_Rapoport_(music_critic)|Rapoport]]/Wilson 13-EDO "subminor" scale). Alternatively we can treat this temperament as a 2.9.21 subgroup, and instead of calling 9 degrees of 23-EDO a "4/3", we can call it 21/16. Here three 21/16's gets us to 9/4, meaning 1029/1024 is tempered out. This allows us to treat a triad of 0-4-9 degrees of 23-EDO as an approximation to 16:18:21, and 0-5-9 as 1/(16:18:21); both of these triads are abundant in the 8-note MOS scale. 

==Intervals== 
|| [[Degree]]s of 23-EDO || [[Cent]]s value ||
|| 0 || 0 ||
|| 1 || 52.1739 ||
|| 2 || 104.3478 ||
|| 3 || 156.5217 ||
|| 4 || 208.6957 ||
|| 5 || 260.8696 ||
|| 6 || 313.0435 ||
|| 7 || 365.2174 ||
|| 8 || 417.3913 ||
|| 9 || 469.5652 ||
|| 10 || 521.7391 ||
|| 11 || 573.913 ||
|| 12 || 626.087 ||
|| 13 || 678.2609 ||
|| 14 || 730.4348 ||
|| 15 || 782.6087 ||
|| 16 || 834.7826 ||
|| 17 || 886.9565 ||
|| 18 || 939.1304 ||
|| 19 || 991.3043 ||
|| 20 || 1043.4783 ||
|| 21 || 1095.6522 ||
|| 22 || 1147.8261 ||
[[image:Ciclo_Icositrifonía.png width="341" height="344" caption="Intervallic Cicle of 23 steps Equal per Octave"]]

===INSTRUMENTS=== 
[[image:Icositriphonic_Bass.JPG width="560" height="182"]]
//An Icositriphonic Bass. 23-EDO Bass by Tútim Deft Wafil.//

[[image:Icositriphonic_Guitar.PNG width="533" height="237"]]
//An Icositriphonic 8-string Guitar. 23-EDO Guitar by Ron Sword.//

**23 tone [[Equal Modes]]:**

10 10 3
9 9 5
8 8 7
7 7 7 2
7 2 7 7
6 6 6 5
6 5 6 6
5 4 5 5 4
5 4 5 4 5
7 1 7 7 1
7 1 7 1 7
5 5 5 5 3
5 3 5 5 5
4 4 4 4 4 3
4 3 4 4 4 4
5 1 5 1 5 1 5
3 3 3 5 3 3 3
4 3 3 3 3 3 4
3 4 3 3 4 3 3
3 3 4 3 3 3 4
3 3 3 4 3 3 4
3 3 3 4 3 4 3
2 5 2 5 2 5 2
4 1 4 4 1 4 4 1
3 3 3 3 3 3 3 2
3 2 3 3 3 3 3 3
**3 3 3 1 3 3 3 3 1**
3 3 1 3 3 3 1 3 3
3 2 3 2 3 2 3 2 3
2 2 3 2 2 3 2 2 2 3
**3 1 3 1 3 1 3 1 3 1 3**
2 2 2 1 2 2 2 1 2 2 2 2 1
2 2 1 2 2 1 2 2 1 2 2 1 2 1
**2 1 2 2 1 2 2 1 2 2 1 2 2 1**
1 1 1 4 1 1 1 1 4 1 1 1 1 4
2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
**2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1**
2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1

=Compositions= 
[[http://home.vicnet.net.au/~epoetry/family.mp3|The Family Supper]] by Warren Burt
__Allegro Moderato__ by Easley Blackwood
[[http://www.youtube.com/watch?v=Hqst8MaRiYM|Icositriphonic Heptatonic MOS]] by Igliashon Jones
[[http://soundclick.com/share?songid=5683734|A Walk Through the Valley of Ashes]] by Iglashion Jones
[[http://www.nonoctave.com/tunes/CosmicChamber.mp3|Cosmic Chamber]] by X. J. Scott
[[http://www.nonoctave.com/tunes/Daisies.mp3|Daisies on the Beach]] by X. J. Scott
<span style="background-attachment: initial; background-clip: initial; background-color: initial; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; cursor: pointer; padding-right: 10px;">[[http://www.akjmusic.com/audio/boogie_pie.mp3|Boogie Pie]]</span>by Aaron Krister Johnson

=Books= 
[[image:Libro_Icositrifónico.PNG]]

=Keyboards= 

[[image:Teclado_Icositrifónico.PNG width="567" height="297"]]
//A prototype for Armodue 1/3-tone Keyboard, Armodue-Hornbostel Family Temperaments.//

Original HTML content:

<html><head><title>23edo</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x23 tone equal temperament"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="color: #007a1b; display: block;">23 tone equal temperament</span></h1>
 <br />
23et, or 23-EDO, is a tuning system which divides the <a class="wiki_link" href="/octave">octave</a> into 23 equal parts of approximately 52.173913 cents. It has good approximations for 5/3, 11/7, 13 and 17, allowing it to represent the 2.5/3.11/7.13.17 <a class="wiki_link" href="/just%20intonation%20subgroup">just intonation subgroup</a>. If to this subgroup is added the commas of 17-limit 46et, the larger 17-limit <a class="wiki_link" href="/k%2AN%20subgroups">2*23 subgroup</a> 2.9.15.21.33.13.17 is obtained. This is the largest subgroup on which 23 has the same tuning and commas as does 17-limit 46, and may be regarded as a basis for analyzing the harmony of 23-EDO so far as approximations to just intervals goes. <br />
<br />
23-EDO was proposed by ethnomusicologist <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Erich_von_Hornbostel" rel="nofollow">Erich von Hornbostel</a> as the result of continuing a circle of &quot;blown&quot; fifths of ~678-cent fifths that (he argued) resulted from &quot;overblowing&quot; a bamboo pipe.<br />
<br />
23-EDO is also significant in that it is the largest EDO that fails to approximate the 3rd, 5th, and 7th harmonics within 20 cents, which makes it well-suited for musicians seeking to explore unusual harmonic territory. Oddly, despite the fact that it fails to approximate these harmonics, it approximates the intervals between them (5/3, 7/3, and 7/5) very well. The lowest harmonics well-approximated by 23-EDO are 13, 17, 21, and 23. <br />
<br />
Like 9-EDO, 16-EDO, and 25-EDO, one way to treat 23-EDO is as a Pelogic temperament, tempering out the &quot;comma&quot; of 135/128 and equating three sharp 4/3's with 5/1 (related to the Armodue system). This means mapping 3/2 to 13 degrees of 23, and results in a 7-note &quot;anti-diatonic&quot; scale of 3 3 4 3 3 3 4 (in steps of 23-EDO), which extends to 9 notes (3 3 3 1 3 3 3 3 1).<br />
<br />
However, one can also map 3/2 to 14 degrees of 23-EDO without significantly increasing the error, taking us to a 7-limit temperament where two 3/2's equals 7/3, meaning 28/27 is tempered out, and six 4/3's octave-reduced equals 5/4, meaning 4096/3645 is tempered out. Both of these are very large commas, so this is not at all an accurate temperament, but it is related to 13-EDO and 18-EDO and produces MOS scales of 5 and 8 notes: 5 5 4 5 4 (the &quot;anti-pentatonic&quot;) and 4 1 4 1 4 4 1 4 (the &quot;quarter-tone&quot; version of the Blackwood/<a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Paul_Rapoport_(music_critic)" rel="nofollow">Rapoport</a>/Wilson 13-EDO &quot;subminor&quot; scale). Alternatively we can treat this temperament as a 2.9.21 subgroup, and instead of calling 9 degrees of 23-EDO a &quot;4/3&quot;, we can call it 21/16. Here three 21/16's gets us to 9/4, meaning 1029/1024 is tempered out. This allows us to treat a triad of 0-4-9 degrees of 23-EDO as an approximation to 16:18:21, and 0-5-9 as 1/(16:18:21); both of these triads are abundant in the 8-note MOS scale. <br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x23 tone equal temperament-Intervals"></a><!-- ws:end:WikiTextHeadingRule:2 -->Intervals</h2>
 

<table class="wiki_table">
    <tr>
        <td><a class="wiki_link" href="/Degree">Degree</a>s of 23-EDO<br />
</td>
        <td><a class="wiki_link" href="/Cent">Cent</a>s value<br />
</td>
    </tr>
    <tr>
        <td>0<br />
</td>
        <td>0<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>52.1739<br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>104.3478<br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>156.5217<br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>208.6957<br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>260.8696<br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>313.0435<br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>365.2174<br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>417.3913<br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>469.5652<br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td>521.7391<br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td>573.913<br />
</td>
    </tr>
    <tr>
        <td>12<br />
</td>
        <td>626.087<br />
</td>
    </tr>
    <tr>
        <td>13<br />
</td>
        <td>678.2609<br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td>730.4348<br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td>782.6087<br />
</td>
    </tr>
    <tr>
        <td>16<br />
</td>
        <td>834.7826<br />
</td>
    </tr>
    <tr>
        <td>17<br />
</td>
        <td>886.9565<br />
</td>
    </tr>
    <tr>
        <td>18<br />
</td>
        <td>939.1304<br />
</td>
    </tr>
    <tr>
        <td>19<br />
</td>
        <td>991.3043<br />
</td>
    </tr>
    <tr>
        <td>20<br />
</td>
        <td>1043.4783<br />
</td>
    </tr>
    <tr>
        <td>21<br />
</td>
        <td>1095.6522<br />
</td>
    </tr>
    <tr>
        <td>22<br />
</td>
        <td>1147.8261<br />
</td>
    </tr>
</table>

<!-- ws:start:WikiTextLocalImageRule:158:&lt;img src=&quot;/file/view/Ciclo_Icositrifon%C3%ADa.png/255957914/341x344/Ciclo_Icositrifon%C3%ADa.png&quot; alt=&quot;Intervallic Cicle of 23 steps Equal per Octave&quot; title=&quot;Intervallic Cicle of 23 steps Equal per Octave&quot; style=&quot;height: 344px; width: 341px;&quot; /&gt; --><table class="captionBox"><tr><td class="captionedImage"><img src="/file/view/Ciclo_Icositrifon%C3%ADa.png/255957914/341x344/Ciclo_Icositrifon%C3%ADa.png" alt="Ciclo_Icositrifonía.png" title="Ciclo_Icositrifonía.png" style="height: 344px; width: 341px;" /></td></tr><tr><td class="imageCaption">Intervallic Cicle of 23 steps Equal per Octave</td></tr></table><!-- ws:end:WikiTextLocalImageRule:158 --><br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="x23 tone equal temperament-Intervals-INSTRUMENTS"></a><!-- ws:end:WikiTextHeadingRule:4 -->INSTRUMENTS</h3>
 <!-- ws:start:WikiTextLocalImageRule:159:&lt;img src=&quot;/file/view/Icositriphonic_Bass.JPG/206711470/560x182/Icositriphonic_Bass.JPG&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 182px; width: 560px;&quot; /&gt; --><img src="/file/view/Icositriphonic_Bass.JPG/206711470/560x182/Icositriphonic_Bass.JPG" alt="Icositriphonic_Bass.JPG" title="Icositriphonic_Bass.JPG" style="height: 182px; width: 560px;" /><!-- ws:end:WikiTextLocalImageRule:159 --><br />
<em>An Icositriphonic Bass. 23-EDO Bass by Tútim Deft Wafil.</em><br />
<br />
<!-- ws:start:WikiTextLocalImageRule:160:&lt;img src=&quot;/file/view/Icositriphonic_Guitar.PNG/206712964/533x237/Icositriphonic_Guitar.PNG&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 237px; width: 533px;&quot; /&gt; --><img src="/file/view/Icositriphonic_Guitar.PNG/206712964/533x237/Icositriphonic_Guitar.PNG" alt="Icositriphonic_Guitar.PNG" title="Icositriphonic_Guitar.PNG" style="height: 237px; width: 533px;" /><!-- ws:end:WikiTextLocalImageRule:160 --><br />
<em>An Icositriphonic 8-string Guitar. 23-EDO Guitar by Ron Sword.</em><br />
<br />
<strong>23 tone <a class="wiki_link" href="/Equal%20Modes">Equal Modes</a>:</strong><br />
<br />
10 10 3<br />
9 9 5<br />
8 8 7<br />
7 7 7 2<br />
7 2 7 7<br />
6 6 6 5<br />
6 5 6 6<br />
5 4 5 5 4<br />
5 4 5 4 5<br />
7 1 7 7 1<br />
7 1 7 1 7<br />
5 5 5 5 3<br />
5 3 5 5 5<br />
4 4 4 4 4 3<br />
4 3 4 4 4 4<br />
5 1 5 1 5 1 5<br />
3 3 3 5 3 3 3<br />
4 3 3 3 3 3 4<br />
3 4 3 3 4 3 3<br />
3 3 4 3 3 3 4<br />
3 3 3 4 3 3 4<br />
3 3 3 4 3 4 3<br />
2 5 2 5 2 5 2<br />
4 1 4 4 1 4 4 1<br />
3 3 3 3 3 3 3 2<br />
3 2 3 3 3 3 3 3<br />
<strong>3 3 3 1 3 3 3 3 1</strong><br />
3 3 1 3 3 3 1 3 3<br />
3 2 3 2 3 2 3 2 3<br />
2 2 3 2 2 3 2 2 2 3<br />
<strong>3 1 3 1 3 1 3 1 3 1 3</strong><br />
2 2 2 1 2 2 2 1 2 2 2 2 1<br />
2 2 1 2 2 1 2 2 1 2 2 1 2 1<br />
<strong>2 1 2 2 1 2 2 1 2 2 1 2 2 1</strong><br />
1 1 1 4 1 1 1 1 4 1 1 1 1 4<br />
2 1 2 1 2 1 2 1 2 1 2 1 2 1 2<br />
<strong>2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1</strong><br />
2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Compositions"></a><!-- ws:end:WikiTextHeadingRule:6 -->Compositions</h1>
 <a class="wiki_link_ext" href="http://home.vicnet.net.au/~epoetry/family.mp3" rel="nofollow">The Family Supper</a> by Warren Burt<br />
<u>Allegro Moderato</u> by Easley Blackwood<br />
<a class="wiki_link_ext" href="http://www.youtube.com/watch?v=Hqst8MaRiYM" rel="nofollow">Icositriphonic Heptatonic MOS</a> by Igliashon Jones<br />
<a class="wiki_link_ext" href="http://soundclick.com/share?songid=5683734" rel="nofollow">A Walk Through the Valley of Ashes</a> by Iglashion Jones<br />
<a class="wiki_link_ext" href="http://www.nonoctave.com/tunes/CosmicChamber.mp3" rel="nofollow">Cosmic Chamber</a> by X. J. Scott<br />
<a class="wiki_link_ext" href="http://www.nonoctave.com/tunes/Daisies.mp3" rel="nofollow">Daisies on the Beach</a> by X. J. Scott<br />
<span style="background-attachment: initial; background-clip: initial; background-color: initial; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; cursor: pointer; padding-right: 10px;"><a class="wiki_link_ext" href="http://www.akjmusic.com/audio/boogie_pie.mp3" rel="nofollow">Boogie Pie</a></span>by Aaron Krister Johnson<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Books"></a><!-- ws:end:WikiTextHeadingRule:8 -->Books</h1>
 <!-- ws:start:WikiTextLocalImageRule:161:&lt;img src=&quot;/file/view/Libro_Icositrif%C3%B3nico.PNG/163031733/Libro_Icositrif%C3%B3nico.PNG&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/Libro_Icositrif%C3%B3nico.PNG/163031733/Libro_Icositrif%C3%B3nico.PNG" alt="Libro_Icositrifónico.PNG" title="Libro_Icositrifónico.PNG" /><!-- ws:end:WikiTextLocalImageRule:161 --><br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Keyboards"></a><!-- ws:end:WikiTextHeadingRule:10 -->Keyboards</h1>
 <br />
<!-- ws:start:WikiTextLocalImageRule:162:&lt;img src=&quot;/file/view/Teclado_Icositrif%C3%B3nico.PNG/258408436/567x297/Teclado_Icositrif%C3%B3nico.PNG&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 297px; width: 567px;&quot; /&gt; --><img src="/file/view/Teclado_Icositrif%C3%B3nico.PNG/258408436/567x297/Teclado_Icositrif%C3%B3nico.PNG" alt="Teclado_Icositrifónico.PNG" title="Teclado_Icositrifónico.PNG" style="height: 297px; width: 567px;" /><!-- ws:end:WikiTextLocalImageRule:162 --><br />
<em>A prototype for Armodue 1/3-tone Keyboard, Armodue-Hornbostel Family Temperaments.</em></body></html>