29edo

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[[toc|flat]]
=<span style="color: #ff4700; font-size: 103%;">29 tone equal temperament</span>= 

29edo divides the 2:1 octave into 29 equal steps of approximately 41.37931 cents.

29 is the lowest edo which approximates the [[3_2|3:2]] just fifth more accurately than [[12edo]]: 3/2 = 701.955... cents; 17 degrees of 29edo = 703.448... cents. Since the fifth is slightly sharp, 29edo is a [[positive temperament]] -- a Superpythagorean instead of a Meantone system.

The 3 is the only harmonic, of the intelligibly low ones anyway, that 29-edo approximates, and it does so quite well. Hence one possible use for 29edo is as an equally tempered pythagorean scale. However, it represents the 2.3.11/5.13/5 subgroup to very high accuracy, and 2.3.7/5.11/5.13/5 to a lesser but still good accuracy, and so can be used with this subgroup, which is liberally supplied with chords such as the 1-11/7-13/7 chord, the [[The Archipelago|barbados triad]] 1-13/10-3/2, the minor barbados triad 1-15/13-3/2, the 1-14/11-3/2 triad and the 1-13/11-3/2 triad. 29 tempers out 352/351, 676/675 and 4000/3993 from the 2.3.11/5.13/5 subgroup, and in addition 196/195 and 364/363 from the 2.3.7/5.11/5.13/5 subgroup, so we have various relationships from the tempering, such as the fact that the 1-13/11-3/2 chord and the 1-14/11-3/2 chord are inverses of each other, a major-minor pairing. A larger subgroup containing both of these subgroups is the [[k*N subgroups|3*29 subgroup]] 2.3.125.175.275.325; on this subgroup 29 tunes the same as 87, and the commas of 29 on this subgroup are the same as the 13-limit commas of 87. Still another subgroup of interest is the [[k*N subgroups|2*29 subgroup]] 2.3.25.35.55.65.85; on this subgroup 29 tunes the same as 58 and has the same 17-limit commas.

=Intervals= 
|| Degrees of 29-EDO || Cents value ||
|| 0 || 0 ||
|| 1 || 41.379 ||
|| 2 || 82.759 ||
|| 3 || 124.138 ||
|| 4 || 165.517 ||
|| 5 || 206.897 ||
|| 6 || 248.276 ||
|| 7 || 289.655 ||
|| 8 || 331.034 ||
|| 9 || 372.414 ||
|| 10 || 413.793 ||
|| 11 || 455.172 ||
|| 12 || 496.552 ||
|| 13 || 537.931 ||
|| 14 || 579.310 ||
|| 15 || 620.690 ||
|| 16 || 662.069 ||
|| 17 || 703.448 ||
|| 18 || 744.828 ||
|| 19 || 786.207 ||
|| 20 || 827.586 ||
|| 21 || 868.966 ||
|| 22 || 910.345 ||
|| 23 || 951.724 ||
|| 24 || 993.103 ||
|| 25 || 1034.483 ||
|| 26 || 1075.862 ||
|| 27 || 1117.241 ||
|| 28 || 1158.621 ||
=Commas= 
29 EDO tempers out the following commas. (Note: This assumes the val < 29 46 67 81 100 107 |.)
||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||
|| 16875/16384 || | -14 3 4 > || 51.12 || Negri Comma || Double Augmentation Diesis ||
|| 250/243 || | 1 -5 3 > || 49.17 || Maximal Diesis || Porcupine Comma ||
|| 32805/32768 || | -15 8 1 > || 1.95 || Schisma ||   ||
|| 525/512 || | -9 1 2 1 > || 43.41 || Avicennma || Avicenna's Enharmonic Diesis ||
|| 49/48 || | -4 -1 0 2 > || 35.70 || Slendro Diesis ||   ||
|| 686/675 || | 1 -3 -2 3 > || 27.99 || Senga ||   ||
|| 64827/64000 || | -9 3 -3 4 > || 22.23 || Squalentine ||   ||
|| 3125/3087 || | 0 -2 5 -3 > || 21.18 || Gariboh ||   ||
|| 50421/50000 || | -4 1 -5 5 > || 14.52 || Trimyna ||   ||
|| 4000/3969 || | 5 -4 3 -2 > || 13.47 || Octagar ||   ||
|| 225/224 || | -5 2 2 -1 > || 7.71 || Septimal Kleisma || Marvel Comma ||
|| 5120/5103 || | 10 -6 1 -1 > || 5.76 || Hemifamity ||   ||
|| 4994735/4983772 || | 25 -14 0 -1 > || 3.80 || Garischisma ||   ||
|| 100/99 || | 2 -2 2 0 -1 > || 17.40 || Ptolemisma ||   ||
|| 121/120 || | -3 -1 -1 0 2 > || 14.37 || Biyatisma ||   ||
|| 896/891 || | 7 -4 0 1 -1 > || 9.69 || Pentacircle ||   ||
|| 441/440 || | -3 2 -1 2 -1 > || 3.93 || Werckisma ||   ||
|| 4000/3993 || | 5 -1 3 0 -3 > || 3.03 || Wizardharry ||   ||
|| 9801/9800 || | -3 4 -2 -2 2 > || 0.18 || Kalisma || Gauss' Comma ||
|| 91/90 || | -1 -2 -1 1 0 1 > || 19.13 || Superleap ||   ||
=Music= 
[[http://tinyurl.com/45lancy|Paint in the Water 29]] by [[Igliashon Jones]]

Original HTML content:

<html><head><title>29edo</title></head><body><!-- ws:start:WikiTextTocRule:8:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --><a href="#x29 tone equal temperament">29 tone equal temperament</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --> | <a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --> | <a href="#Commas">Commas</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#Music">Music</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: -->
<!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x29 tone equal temperament"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="color: #ff4700; font-size: 103%;">29 tone equal temperament</span></h1>
 <br />
29edo divides the 2:1 octave into 29 equal steps of approximately 41.37931 cents.<br />
<br />
29 is the lowest edo which approximates the <a class="wiki_link" href="/3_2">3:2</a> just fifth more accurately than <a class="wiki_link" href="/12edo">12edo</a>: 3/2 = 701.955... cents; 17 degrees of 29edo = 703.448... cents. Since the fifth is slightly sharp, 29edo is a <a class="wiki_link" href="/positive%20temperament">positive temperament</a> -- a Superpythagorean instead of a Meantone system.<br />
<br />
The 3 is the only harmonic, of the intelligibly low ones anyway, that 29-edo approximates, and it does so quite well. Hence one possible use for 29edo is as an equally tempered pythagorean scale. However, it represents the 2.3.11/5.13/5 subgroup to very high accuracy, and 2.3.7/5.11/5.13/5 to a lesser but still good accuracy, and so can be used with this subgroup, which is liberally supplied with chords such as the 1-11/7-13/7 chord, the <a class="wiki_link" href="/The%20Archipelago">barbados triad</a> 1-13/10-3/2, the minor barbados triad 1-15/13-3/2, the 1-14/11-3/2 triad and the 1-13/11-3/2 triad. 29 tempers out 352/351, 676/675 and 4000/3993 from the 2.3.11/5.13/5 subgroup, and in addition 196/195 and 364/363 from the 2.3.7/5.11/5.13/5 subgroup, so we have various relationships from the tempering, such as the fact that the 1-13/11-3/2 chord and the 1-14/11-3/2 chord are inverses of each other, a major-minor pairing. A larger subgroup containing both of these subgroups is the <a class="wiki_link" href="/k%2AN%20subgroups">3*29 subgroup</a> 2.3.125.175.275.325; on this subgroup 29 tunes the same as 87, and the commas of 29 on this subgroup are the same as the 13-limit commas of 87. Still another subgroup of interest is the <a class="wiki_link" href="/k%2AN%20subgroups">2*29 subgroup</a> 2.3.25.35.55.65.85; on this subgroup 29 tunes the same as 58 and has the same 17-limit commas.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:2 -->Intervals</h1>
 

<table class="wiki_table">
    <tr>
        <td>Degrees of 29-EDO<br />
</td>
        <td>Cents value<br />
</td>
    </tr>
    <tr>
        <td>0<br />
</td>
        <td>0<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>41.379<br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>82.759<br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>124.138<br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>165.517<br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>206.897<br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>248.276<br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>289.655<br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>331.034<br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>372.414<br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td>413.793<br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td>455.172<br />
</td>
    </tr>
    <tr>
        <td>12<br />
</td>
        <td>496.552<br />
</td>
    </tr>
    <tr>
        <td>13<br />
</td>
        <td>537.931<br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td>579.310<br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td>620.690<br />
</td>
    </tr>
    <tr>
        <td>16<br />
</td>
        <td>662.069<br />
</td>
    </tr>
    <tr>
        <td>17<br />
</td>
        <td>703.448<br />
</td>
    </tr>
    <tr>
        <td>18<br />
</td>
        <td>744.828<br />
</td>
    </tr>
    <tr>
        <td>19<br />
</td>
        <td>786.207<br />
</td>
    </tr>
    <tr>
        <td>20<br />
</td>
        <td>827.586<br />
</td>
    </tr>
    <tr>
        <td>21<br />
</td>
        <td>868.966<br />
</td>
    </tr>
    <tr>
        <td>22<br />
</td>
        <td>910.345<br />
</td>
    </tr>
    <tr>
        <td>23<br />
</td>
        <td>951.724<br />
</td>
    </tr>
    <tr>
        <td>24<br />
</td>
        <td>993.103<br />
</td>
    </tr>
    <tr>
        <td>25<br />
</td>
        <td>1034.483<br />
</td>
    </tr>
    <tr>
        <td>26<br />
</td>
        <td>1075.862<br />
</td>
    </tr>
    <tr>
        <td>27<br />
</td>
        <td>1117.241<br />
</td>
    </tr>
    <tr>
        <td>28<br />
</td>
        <td>1158.621<br />
</td>
    </tr>
</table>

<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:4 -->Commas</h1>
 29 EDO tempers out the following commas. (Note: This assumes the val &lt; 29 46 67 81 100 107 |.)<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>16875/16384<br />
</td>
        <td>| -14 3 4 &gt;<br />
</td>
        <td>51.12<br />
</td>
        <td>Negri Comma<br />
</td>
        <td>Double Augmentation Diesis<br />
</td>
    </tr>
    <tr>
        <td>250/243<br />
</td>
        <td>| 1 -5 3 &gt;<br />
</td>
        <td>49.17<br />
</td>
        <td>Maximal Diesis<br />
</td>
        <td>Porcupine Comma<br />
</td>
    </tr>
    <tr>
        <td>32805/32768<br />
</td>
        <td>| -15 8 1 &gt;<br />
</td>
        <td>1.95<br />
</td>
        <td>Schisma<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>525/512<br />
</td>
        <td>| -9 1 2 1 &gt;<br />
</td>
        <td>43.41<br />
</td>
        <td>Avicennma<br />
</td>
        <td>Avicenna's Enharmonic Diesis<br />
</td>
    </tr>
    <tr>
        <td>49/48<br />
</td>
        <td>| -4 -1 0 2 &gt;<br />
</td>
        <td>35.70<br />
</td>
        <td>Slendro Diesis<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>686/675<br />
</td>
        <td>| 1 -3 -2 3 &gt;<br />
</td>
        <td>27.99<br />
</td>
        <td>Senga<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>64827/64000<br />
</td>
        <td>| -9 3 -3 4 &gt;<br />
</td>
        <td>22.23<br />
</td>
        <td>Squalentine<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>3125/3087<br />
</td>
        <td>| 0 -2 5 -3 &gt;<br />
</td>
        <td>21.18<br />
</td>
        <td>Gariboh<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>50421/50000<br />
</td>
        <td>| -4 1 -5 5 &gt;<br />
</td>
        <td>14.52<br />
</td>
        <td>Trimyna<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>4000/3969<br />
</td>
        <td>| 5 -4 3 -2 &gt;<br />
</td>
        <td>13.47<br />
</td>
        <td>Octagar<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>225/224<br />
</td>
        <td>| -5 2 2 -1 &gt;<br />
</td>
        <td>7.71<br />
</td>
        <td>Septimal Kleisma<br />
</td>
        <td>Marvel Comma<br />
</td>
    </tr>
    <tr>
        <td>5120/5103<br />
</td>
        <td>| 10 -6 1 -1 &gt;<br />
</td>
        <td>5.76<br />
</td>
        <td>Hemifamity<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>4994735/4983772<br />
</td>
        <td>| 25 -14 0 -1 &gt;<br />
</td>
        <td>3.80<br />
</td>
        <td>Garischisma<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>100/99<br />
</td>
        <td>| 2 -2 2 0 -1 &gt;<br />
</td>
        <td>17.40<br />
</td>
        <td>Ptolemisma<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>121/120<br />
</td>
        <td>| -3 -1 -1 0 2 &gt;<br />
</td>
        <td>14.37<br />
</td>
        <td>Biyatisma<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>896/891<br />
</td>
        <td>| 7 -4 0 1 -1 &gt;<br />
</td>
        <td>9.69<br />
</td>
        <td>Pentacircle<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>441/440<br />
</td>
        <td>| -3 2 -1 2 -1 &gt;<br />
</td>
        <td>3.93<br />
</td>
        <td>Werckisma<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>4000/3993<br />
</td>
        <td>| 5 -1 3 0 -3 &gt;<br />
</td>
        <td>3.03<br />
</td>
        <td>Wizardharry<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>9801/9800<br />
</td>
        <td>| -3 4 -2 -2 2 &gt;<br />
</td>
        <td>0.18<br />
</td>
        <td>Kalisma<br />
</td>
        <td>Gauss' Comma<br />
</td>
    </tr>
    <tr>
        <td>91/90<br />
</td>
        <td>| -1 -2 -1 1 0 1 &gt;<br />
</td>
        <td>19.13<br />
</td>
        <td>Superleap<br />
</td>
        <td><br />
</td>
    </tr>
</table>

<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:6 -->Music</h1>
 <a class="wiki_link_ext" href="http://tinyurl.com/45lancy" rel="nofollow">Paint in the Water 29</a> by <a class="wiki_link" href="/Igliashon%20Jones">Igliashon Jones</a></body></html>