21edo

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=21 equal divisions of the octave= 

Twenty one equal divisions of the octave provides the sonic fingerprint of the augmented and 7-edo family, while also giving a some higher harmony possibilities and fun intervals like the apotome. The system can be treated as three intertwining 7-edo or "equi-heptatonic" scales, or as seven 3-edo ''augmented'' triads. Some other cool things about 21-edo: it has an 11-limit minor third/wide sixth, 7-limit neutral third and sixth, a 7/4 harmonic seventh and grave seventh. The 7/4 at 968.826 cents is only off in 21-tone by 2.6 cents.

==Harmony in 21-EDO:== 

21-EDO can be treated as a 13-limit temperament, but of harmonics 3, 5, 7, 11, and 13, the only harmonic 21-EDO approximates with anything approaching a near-Just flavor is the 7th harmonic. On the other hand, 21-EDO provides exceptionally accurate tunings of the 15th, 23rd, and 29th harmonics (within 3 cents or less), as well as a very reasonable approximation of the 27th harmonic (around 8 cents sharp). As such, treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament allows for a more accurate rationalization of the tuning, since almost every interval in 21-EDO can be described as a ratio within the 29-odd-limit.
|| Degree || Cents
Value || Name ||= Approximate
Ratios* ||
|| 0 || 0 || C ||= 1/1 ||
|| 1 || 57.143 || C^/Dvv ||= 28/27, 30/29 ||
|| 2 || 114.286 || C^^/Dv ||= 16/15, 15/14, 29/27 ||
|| 3 || 171.429 || D ||= 10/9, 32/29 ||
|| 4 || 228.571 || D^/Evv ||= 8/7 ||
|| 5 || 285.714 || D^^/Ev ||= 27/23, 32/27 ||
|| 6 || 342.857 || E ||= 28/23 ||
|| 7 || 400 || E^/Fvv ||= 29/23 ||
|| 8 || 457.143 || E^^/Fv ||= 30/23 ||
|| 9 || 514.286 || F ||= 161/120, 256/189 ||
|| 10 || 571.429 || F^/Gvv ||= 32/23 ||
|| 11 || 628.571 || F^^/Gv ||= 23/16 ||
|| 12 || 685.714 || G ||= 189/128, 240/161 ||
|| 13 || 742.857 || G^/Avv ||= 23/15 ||
|| 14 || 800 || G^^/Av ||= 46/29 ||
|| 15 || 857.143 || A ||= 23/14 ||
|| 16 || 914.286 || A^/Bvv ||= 27/16, 46/27 ||
|| 17 || 971.429 || A^^/Bv ||= 7/4 ||
|| 18 || 1028.571 || B ||= 29/16, 9/5 ||
|| 19 || 1085.714 || B^/Cvv ||= 15/8 ||
|| 20 || 1142.857 || B^^/Cv ||= 27/14, 29/15 ||
|| 21 || 1200 || C ||= 2/1 ||

*based on treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament; other approaches are possible.

**21-tone scales:**
[[augment6]]
[[augment9]]
[[augment12]]



==13-limit Commas== 
21 EDO tempers out the following 13-limit commas. (Note: This assumes the val < 21 33 49 59 73 78 |.)
||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||
||= 2187/2048 ||< | -11 7 > ||> 113.69 ||= Apotome ||=   ||
||= 128/125 ||< | 7 0 -3 > ||> 41.06 ||= Diesis ||= Augmented Comma ||
||= 9931568/9752117 ||< | -25 7 6 > ||> 31.57 ||= Ampersand's Comma ||   ||
||= 9193891/9143623 ||< | 32 -7 -9 > ||> 9.49 ||= Escapade Comma ||   ||
||= 1029/1000 ||< | -3 1 -3 3 > ||> 49.49 ||= Keega ||=   ||
||= 36/35 ||< | 2 2 -1 -1 > ||> 48.77 ||= Septimal Quarter Tone ||   ||
||= 9859966/9733137 ||< | -10 7 8 -7 > ||> 22.41 ||= Blackjackisma ||=   ||
||= 1029/1024 ||< | -10 1 0 3 > ||> 8.43 ||= Gamelisma ||=   ||
||= 225/224 ||< | -5 2 2 -1 > ||> 7.71 ||= Septimal Kleisma ||= Marvel Comma ||
||= 16875/16807 ||< | 0 3 4 -5 > ||> 6.99 ||= Mirkwai ||=   ||
||= 2401/2400 ||< | -5 -1 -2 4 > ||> 0.72 ||= Breedsma ||=   ||
||= 394839/394762 ||< | 47 -7 -7 -7 > ||> 0.34 ||= Akjaysma ||= 5\7 Octave Comma ||
||= 99/98 ||< | -1 2 0 -2 1 > ||> 17.58 ||= Mothwellsma ||=   ||
||= 176/175 ||< | 4 0 -2 -1 1 > ||> 9.86 ||= Valinorsma ||=   ||
||= 4000/3993 ||< | 5 -1 3 0 -3 > ||> 3.03 ||= Wizardharry ||=   ||



=**Books / Literature:**= 
Sword, Ron. "Icosihenaphonic Scales for Guitar". IAAA Press. 1st ed: July 2009.
[[image:http://www.ronsword.com/images/ron1.jpg width="254" height="188"]][[image:http://www.swordguitars.com/21tetsm.JPG width="363" height="191"]]
**//21-edo Icosihenaphonic Acoustic Guitar (Ron Sword)//**

=**Compositions/Listening:**= 
[[@http://www.ronsword.com/sounds/21_improv.mp3|Short Clip of 21-edo Acoustic]] by [[Ron Sword]]
[[@http://www.ronsword.com/sounds/Ron_Sword_21_Tone_improv.mp3|Open tuning Drone Improvisation in 21-edo]] by Ron Sword
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&songID=933715|Anomalous Readings]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+anomalousreadingsin21tet.mp3|play]] by [[Andrew Heathwaite]]

Original HTML content:

<html><head><title>21edo</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x21 equal divisions of the octave"></a><!-- ws:end:WikiTextHeadingRule:0 -->21 equal divisions of the octave</h1>
 <br />
Twenty one equal divisions of the octave provides the sonic fingerprint of the augmented and 7-edo family, while also giving a some higher harmony possibilities and fun intervals like the apotome. The system can be treated as three intertwining 7-edo or &quot;equi-heptatonic&quot; scales, or as seven 3-edo ''augmented'' triads. Some other cool things about 21-edo: it has an 11-limit minor third/wide sixth, 7-limit neutral third and sixth, a 7/4 harmonic seventh and grave seventh. The 7/4 at 968.826 cents is only off in 21-tone by 2.6 cents.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x21 equal divisions of the octave-Harmony in 21-EDO:"></a><!-- ws:end:WikiTextHeadingRule:2 -->Harmony in 21-EDO:</h2>
 <br />
21-EDO can be treated as a 13-limit temperament, but of harmonics 3, 5, 7, 11, and 13, the only harmonic 21-EDO approximates with anything approaching a near-Just flavor is the 7th harmonic. On the other hand, 21-EDO provides exceptionally accurate tunings of the 15th, 23rd, and 29th harmonics (within 3 cents or less), as well as a very reasonable approximation of the 27th harmonic (around 8 cents sharp). As such, treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament allows for a more accurate rationalization of the tuning, since almost every interval in 21-EDO can be described as a ratio within the 29-odd-limit.<br />


<table class="wiki_table">
    <tr>
        <td>Degree<br />
</td>
        <td>Cents<br />
Value<br />
</td>
        <td>Name<br />
</td>
        <td style="text-align: center;">Approximate<br />
Ratios*<br />
</td>
    </tr>
    <tr>
        <td>0<br />
</td>
        <td>0<br />
</td>
        <td>C<br />
</td>
        <td style="text-align: center;">1/1<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>57.143<br />
</td>
        <td>C^/Dvv<br />
</td>
        <td style="text-align: center;">28/27, 30/29<br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>114.286<br />
</td>
        <td>C^^/Dv<br />
</td>
        <td style="text-align: center;">16/15, 15/14, 29/27<br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>171.429<br />
</td>
        <td>D<br />
</td>
        <td style="text-align: center;">10/9, 32/29<br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>228.571<br />
</td>
        <td>D^/Evv<br />
</td>
        <td style="text-align: center;">8/7<br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>285.714<br />
</td>
        <td>D^^/Ev<br />
</td>
        <td style="text-align: center;">27/23, 32/27<br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>342.857<br />
</td>
        <td>E<br />
</td>
        <td style="text-align: center;">28/23<br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>400<br />
</td>
        <td>E^/Fvv<br />
</td>
        <td style="text-align: center;">29/23<br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>457.143<br />
</td>
        <td>E^^/Fv<br />
</td>
        <td style="text-align: center;">30/23<br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>514.286<br />
</td>
        <td>F<br />
</td>
        <td style="text-align: center;">161/120, 256/189<br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td>571.429<br />
</td>
        <td>F^/Gvv<br />
</td>
        <td style="text-align: center;">32/23<br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td>628.571<br />
</td>
        <td>F^^/Gv<br />
</td>
        <td style="text-align: center;">23/16<br />
</td>
    </tr>
    <tr>
        <td>12<br />
</td>
        <td>685.714<br />
</td>
        <td>G<br />
</td>
        <td style="text-align: center;">189/128, 240/161<br />
</td>
    </tr>
    <tr>
        <td>13<br />
</td>
        <td>742.857<br />
</td>
        <td>G^/Avv<br />
</td>
        <td style="text-align: center;">23/15<br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td>800<br />
</td>
        <td>G^^/Av<br />
</td>
        <td style="text-align: center;">46/29<br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td>857.143<br />
</td>
        <td>A<br />
</td>
        <td style="text-align: center;">23/14<br />
</td>
    </tr>
    <tr>
        <td>16<br />
</td>
        <td>914.286<br />
</td>
        <td>A^/Bvv<br />
</td>
        <td style="text-align: center;">27/16, 46/27<br />
</td>
    </tr>
    <tr>
        <td>17<br />
</td>
        <td>971.429<br />
</td>
        <td>A^^/Bv<br />
</td>
        <td style="text-align: center;">7/4<br />
</td>
    </tr>
    <tr>
        <td>18<br />
</td>
        <td>1028.571<br />
</td>
        <td>B<br />
</td>
        <td style="text-align: center;">29/16, 9/5<br />
</td>
    </tr>
    <tr>
        <td>19<br />
</td>
        <td>1085.714<br />
</td>
        <td>B^/Cvv<br />
</td>
        <td style="text-align: center;">15/8<br />
</td>
    </tr>
    <tr>
        <td>20<br />
</td>
        <td>1142.857<br />
</td>
        <td>B^^/Cv<br />
</td>
        <td style="text-align: center;">27/14, 29/15<br />
</td>
    </tr>
    <tr>
        <td>21<br />
</td>
        <td>1200<br />
</td>
        <td>C<br />
</td>
        <td style="text-align: center;">2/1<br />
</td>
    </tr>
</table>

<br />
*based on treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament; other approaches are possible.<br />
<br />
<strong>21-tone scales:</strong><br />
<a class="wiki_link" href="/augment6">augment6</a><br />
<a class="wiki_link" href="/augment9">augment9</a><br />
<a class="wiki_link" href="/augment12">augment12</a><br />
<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x21 equal divisions of the octave-13-limit Commas"></a><!-- ws:end:WikiTextHeadingRule:4 -->13-limit Commas</h2>
 21 EDO tempers out the following 13-limit commas. (Note: This assumes the val &lt; 21 33 49 59 73 78 |.)<br />


<table class="wiki_table">
    <tr>
        <th>Comma<br />
</th>
        <th>Monzo<br />
</th>
        <th>Value (Cents)<br />
</th>
        <th>Name 1<br />
</th>
        <th>Name 2<br />
</th>
    </tr>
    <tr>
        <td style="text-align: center;">2187/2048<br />
</td>
        <td style="text-align: left;">| -11 7 &gt;<br />
</td>
        <td style="text-align: right;">113.69<br />
</td>
        <td style="text-align: center;">Apotome<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">128/125<br />
</td>
        <td style="text-align: left;">| 7 0 -3 &gt;<br />
</td>
        <td style="text-align: right;">41.06<br />
</td>
        <td style="text-align: center;">Diesis<br />
</td>
        <td style="text-align: center;">Augmented Comma<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">9931568/9752117<br />
</td>
        <td style="text-align: left;">| -25 7 6 &gt;<br />
</td>
        <td style="text-align: right;">31.57<br />
</td>
        <td style="text-align: center;">Ampersand's Comma<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">9193891/9143623<br />
</td>
        <td style="text-align: left;">| 32 -7 -9 &gt;<br />
</td>
        <td style="text-align: right;">9.49<br />
</td>
        <td style="text-align: center;">Escapade Comma<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1029/1000<br />
</td>
        <td style="text-align: left;">| -3 1 -3 3 &gt;<br />
</td>
        <td style="text-align: right;">49.49<br />
</td>
        <td style="text-align: center;">Keega<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">36/35<br />
</td>
        <td style="text-align: left;">| 2 2 -1 -1 &gt;<br />
</td>
        <td style="text-align: right;">48.77<br />
</td>
        <td style="text-align: center;">Septimal Quarter Tone<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">9859966/9733137<br />
</td>
        <td style="text-align: left;">| -10 7 8 -7 &gt;<br />
</td>
        <td style="text-align: right;">22.41<br />
</td>
        <td style="text-align: center;">Blackjackisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1029/1024<br />
</td>
        <td style="text-align: left;">| -10 1 0 3 &gt;<br />
</td>
        <td style="text-align: right;">8.43<br />
</td>
        <td style="text-align: center;">Gamelisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">225/224<br />
</td>
        <td style="text-align: left;">| -5 2 2 -1 &gt;<br />
</td>
        <td style="text-align: right;">7.71<br />
</td>
        <td style="text-align: center;">Septimal Kleisma<br />
</td>
        <td style="text-align: center;">Marvel Comma<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">16875/16807<br />
</td>
        <td style="text-align: left;">| 0 3 4 -5 &gt;<br />
</td>
        <td style="text-align: right;">6.99<br />
</td>
        <td style="text-align: center;">Mirkwai<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">2401/2400<br />
</td>
        <td style="text-align: left;">| -5 -1 -2 4 &gt;<br />
</td>
        <td style="text-align: right;">0.72<br />
</td>
        <td style="text-align: center;">Breedsma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">394839/394762<br />
</td>
        <td style="text-align: left;">| 47 -7 -7 -7 &gt;<br />
</td>
        <td style="text-align: right;">0.34<br />
</td>
        <td style="text-align: center;">Akjaysma<br />
</td>
        <td style="text-align: center;">5\7 Octave Comma<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">99/98<br />
</td>
        <td style="text-align: left;">| -1 2 0 -2 1 &gt;<br />
</td>
        <td style="text-align: right;">17.58<br />
</td>
        <td style="text-align: center;">Mothwellsma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">176/175<br />
</td>
        <td style="text-align: left;">| 4 0 -2 -1 1 &gt;<br />
</td>
        <td style="text-align: right;">9.86<br />
</td>
        <td style="text-align: center;">Valinorsma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">4000/3993<br />
</td>
        <td style="text-align: left;">| 5 -1 3 0 -3 &gt;<br />
</td>
        <td style="text-align: right;">3.03<br />
</td>
        <td style="text-align: center;">Wizardharry<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
</table>

<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Books / Literature:"></a><!-- ws:end:WikiTextHeadingRule:6 --><strong>Books / Literature:</strong></h1>
 Sword, Ron. &quot;Icosihenaphonic Scales for Guitar&quot;. IAAA Press. 1st ed: July 2009.<br />
<!-- ws:start:WikiTextRemoteImageRule:436:&lt;img src=&quot;http://www.ronsword.com/images/ron1.jpg&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 188px; width: 254px;&quot; /&gt; --><img src="http://www.ronsword.com/images/ron1.jpg" alt="external image ron1.jpg" title="external image ron1.jpg" style="height: 188px; width: 254px;" /><!-- ws:end:WikiTextRemoteImageRule:436 --><!-- ws:start:WikiTextRemoteImageRule:437:&lt;img src=&quot;http://www.swordguitars.com/21tetsm.JPG&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 191px; width: 363px;&quot; /&gt; --><img src="http://www.swordguitars.com/21tetsm.JPG" alt="external image 21tetsm.JPG" title="external image 21tetsm.JPG" style="height: 191px; width: 363px;" /><!-- ws:end:WikiTextRemoteImageRule:437 --><br />
<strong><em>21-edo Icosihenaphonic Acoustic Guitar (Ron Sword)</em></strong><br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Compositions/Listening:"></a><!-- ws:end:WikiTextHeadingRule:8 --><strong>Compositions/Listening:</strong></h1>
 <a class="wiki_link_ext" href="http://www.ronsword.com/sounds/21_improv.mp3" rel="nofollow" target="_blank">Short Clip of 21-edo Acoustic</a> by <a class="wiki_link" href="/Ron%20Sword">Ron Sword</a><br />
<a class="wiki_link_ext" href="http://www.ronsword.com/sounds/Ron_Sword_21_Tone_improv.mp3" rel="nofollow" target="_blank">Open tuning Drone Improvisation in 21-edo</a> by Ron Sword<br />
<a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933715" rel="nofollow">Anomalous Readings</a> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+anomalousreadingsin21tet.mp3" rel="nofollow">play</a> by <a class="wiki_link" href="/Andrew%20Heathwaite">Andrew Heathwaite</a></body></html>