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, and the medium magic diesis. 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 or grave minor seventh 1280/729 approximation (whichever you please). The harmonic occurs at 968.826 cents while the 21-tone 7/4 approximation is 971.4286 cents, or 2.6 cents off.

Twenty-one has a period of 1/3 of the octave as opposed to 1/4 found in diminished families like 12-tet and 16-tet. Some sources claim that cultures in North and South Africa ( Zambezi / Angola / Chopi, etc), as well as ancient traditional Chinese music used 7-edo (or slight alterations within the pitch sets).

0. 1/1 C
1. 57.143 cents C^ Dvv
2. 114.286 cents C^^ Dv
3. 171.429 cents D
4. 228.571 cents D^ Evv
5. 285.714 cents D^^ Ev
6. 342.857 cents E
7. 400.000 cents E^ Fvv
8. 457.143 cents E^^ Fv
9. 514.286 cents F
10. 571.429 cents F^ Gvv
11. 628.571 cents F^^ Gv
12. 685.714 cents G
13. 742.857 cents G^ Avv
14. 800.000 cents G^^ Av
15. 857.143 cents A
16. 914.286 cents A^ Bvv
17. 971.429 cents A^^ Bv
18. 1028.571 cents B
19. 1085.714 cents B^ Cvv
20. 1142.857 cents B^^ Cv
21. 2/1 C

Some 21-tone scales:

21-edo Narrow (minor) whole tone
21-edo Undecimal minor
21-edo Septimal neutral major
21-edo Quasi-equal major
21-edo Undecimal minor
21-edo Septimal neutral major
21-edo Quasi-equal major
[[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)//**

==Commas== 
21 EDO tempers out the following 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 ||=   ||
==**Progressions / Chords / Pitch Space Divisions:**== 
2 octaves into 7 parts = 6 6 6 6 6 6
3 octaves into 7 parts = 9 9 9 9 9 9
4 octaves into 7 parts = 12 12 12 12 12 12 12
5 octaves into 7 parts = 15 15 15 15 15 15 15
6 octaves into 7 parts = 18 18 18 18 18 18 18
7 octaves into 7 parts = 21 21 21 21 21 21 21 = octaves
8 octaves into 7 parts = 24 24 24 24 24 24 24
9 octaves into 7 parts = 27 27 27 27 27 27 27
10 octaves into 7 parts = 30 30 30 30 30 30 30

2 octaves into 3 parts = 14 14 14
3 octaves into 3 parts = 21 21 21 = octave
4 octaves into 3 parts = 28 28 28
5 octaves into 3 parts = 35 35 35
6 octaves into 3 parts = 42 42 42 = 2 octaves
7 octaves into 3 parts = 49 49 49
8 octaves into 3 parts = 56 56 56
9 octaves into 3 parts = 63 63 63 = 3 octaves
10 octaves into 3 parts = 70 70 70


=**Books / Literature:**= 
Sword, Ron. "Icosihenaphonic Scales for Guitar". IAAA Press. 1st ed: July 2009.

=**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

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, and the medium magic diesis. 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 or grave minor seventh 1280/729 approximation (whichever you please). The harmonic occurs at 968.826 cents while the 21-tone 7/4 approximation is 971.4286 cents, or 2.6 cents off.<br />
<br />
Twenty-one has a period of 1/3 of the octave as opposed to 1/4 found in diminished families like 12-tet and 16-tet. Some sources claim that cultures in North and South Africa ( Zambezi / Angola / Chopi, etc), as well as ancient traditional Chinese music used 7-edo (or slight alterations within the pitch sets).<br />
<br />
0. 1/1 C<br />
1. 57.143 cents C^ Dvv<br />
2. 114.286 cents C^^ Dv<br />
3. 171.429 cents D<br />
4. 228.571 cents D^ Evv<br />
5. 285.714 cents D^^ Ev<br />
6. 342.857 cents E<br />
7. 400.000 cents E^ Fvv<br />
8. 457.143 cents E^^ Fv<br />
9. 514.286 cents F<br />
10. 571.429 cents F^ Gvv<br />
11. 628.571 cents F^^ Gv<br />
12. 685.714 cents G<br />
13. 742.857 cents G^ Avv<br />
14. 800.000 cents G^^ Av<br />
15. 857.143 cents A<br />
16. 914.286 cents A^ Bvv<br />
17. 971.429 cents A^^ Bv<br />
18. 1028.571 cents B<br />
19. 1085.714 cents B^ Cvv<br />
20. 1142.857 cents B^^ Cv<br />
21. 2/1 C<br />
<br />
Some 21-tone scales:<br />
<br />
21-edo Narrow (minor) whole tone<br />
21-edo Undecimal minor<br />
21-edo Septimal neutral major<br />
21-edo Quasi-equal major<br />
21-edo Undecimal minor<br />
21-edo Septimal neutral major<br />
21-edo Quasi-equal major<br />
<!-- ws:start:WikiTextRemoteImageRule:204:&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:204 --><!-- ws:start:WikiTextRemoteImageRule:205:&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:205 --><br />
<strong><em>21-edo Icosihenaphonic Acoustic Guitar (Ron Sword)</em></strong><br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x21 equal divisions of the octave-Commas"></a><!-- ws:end:WikiTextHeadingRule:2 -->Commas</h2>
 21 EDO tempers out the following 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>

<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x21 equal divisions of the octave-Progressions / Chords / Pitch Space Divisions:"></a><!-- ws:end:WikiTextHeadingRule:4 --><strong>Progressions / Chords / Pitch Space Divisions:</strong></h2>
 2 octaves into 7 parts = 6 6 6 6 6 6<br />
3 octaves into 7 parts = 9 9 9 9 9 9<br />
4 octaves into 7 parts = 12 12 12 12 12 12 12<br />
5 octaves into 7 parts = 15 15 15 15 15 15 15<br />
6 octaves into 7 parts = 18 18 18 18 18 18 18<br />
7 octaves into 7 parts = 21 21 21 21 21 21 21 = octaves<br />
8 octaves into 7 parts = 24 24 24 24 24 24 24<br />
9 octaves into 7 parts = 27 27 27 27 27 27 27<br />
10 octaves into 7 parts = 30 30 30 30 30 30 30<br />
<br />
2 octaves into 3 parts = 14 14 14<br />
3 octaves into 3 parts = 21 21 21 = octave<br />
4 octaves into 3 parts = 28 28 28<br />
5 octaves into 3 parts = 35 35 35<br />
6 octaves into 3 parts = 42 42 42 = 2 octaves<br />
7 octaves into 3 parts = 49 49 49<br />
8 octaves into 3 parts = 56 56 56<br />
9 octaves into 3 parts = 63 63 63 = 3 octaves<br />
10 octaves into 3 parts = 70 70 70<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 />
<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 Ron Sword<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</body></html>