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 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 || ===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 [[http://soundclick.com/share?songid=5683734|A Walk Through the Valley of Ashes]] by Iglashion Jones [[http://www.akjmusic.com/audio/boogie_pie.mp3|Boogie Pie]], Aaron Krister Johnson [[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 =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:<h1> --><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 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. <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 "blown" fifths of ~678-cent fifths that (he argued) resulted from "overblowing" 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 "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).<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 "anti-pentatonic") and 4 1 4 1 4 4 1 4 (the "quarter-tone" 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 "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. <br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h2> --><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>
<br />
<!-- ws:start:WikiTextHeadingRule:4:<h3> --><h3 id="toc2"><a name="x23 tone equal temperament-Intervals-INSTRUMENTS"></a><!-- ws:end:WikiTextHeadingRule:4 -->INSTRUMENTS</h3>
<!-- ws:start:WikiTextLocalImageRule:158:<img src="/file/view/Icositriphonic_Bass.JPG/206711470/560x182/Icositriphonic_Bass.JPG" alt="" title="" style="height: 182px; width: 560px;" /> --><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:158 --><br />
<em>An Icositriphonic Bass. 23-EDO Bass by Tútim Deft Wafil.</em><br />
<br />
<!-- ws:start:WikiTextLocalImageRule:159:<img src="/file/view/Icositriphonic_Guitar.PNG/206712964/533x237/Icositriphonic_Guitar.PNG" alt="" title="" style="height: 237px; width: 533px;" /> --><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:159 --><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:<h1> --><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 />
<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.akjmusic.com/audio/boogie_pie.mp3" rel="nofollow">Boogie Pie</a>, Aaron Krister Johnson<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 />
<br />
<!-- ws:start:WikiTextHeadingRule:8:<h1> --><h1 id="toc4"><a name="Books"></a><!-- ws:end:WikiTextHeadingRule:8 -->Books</h1>
<!-- ws:start:WikiTextLocalImageRule:160:<img src="/file/view/Libro_Icositrif%C3%B3nico.PNG/163031733/Libro_Icositrif%C3%B3nico.PNG" alt="" title="" /> --><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:160 --><br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:<h1> --><h1 id="toc5"><a name="Keyboards"></a><!-- ws:end:WikiTextHeadingRule:10 -->Keyboards</h1>
<br />
<!-- ws:start:WikiTextLocalImageRule:161:<img src="/file/view/Teclado_Icositrif%C3%B3nico.PNG/258408436/567x297/Teclado_Icositrif%C3%B3nico.PNG" alt="" title="" style="height: 297px; width: 567px;" /> --><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:161 --><br />
<em>A prototype for Armodue 1/3-tone Keyboard, Armodue-Hornbostel Family Temperaments.</em></body></html>