29edo

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This revision was by author xenwolf and made on 2011-06-05 16:33:24 UTC.

The original revision id was 234400422.

The revision comment was: decimal mantissa of commas: 5 digits

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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 [[cent]]s.

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 29edo approximates very closely, and it does so quite well. Nonetheless, and rather surprisingly, 29 is the smallest equal division which consistently represents the 15 odd limit. It is able to do this since it has an accurate 3, and the 5, 7, 11 and 13, while not very accurate, are all tuned flatly. Hence it tempers out a succession of fairly large commas: 250/243 in the [[5-limit]], 49/48 in the [[7-limit]], 55/54 in the [[11-limit]], and 65/64 in the [[13-limit]]. If using these approximations is desired, 29edo actually shines, and it can be used for such things as an alternative to [[19edo]] for [[Marvel temperaments|negri]].

Another possible use for 29edo is as an equally tempered para-pythagorean scale. Using its fifth as a generator leads to a variant of [[Schismatic family|garibaldi temperament]] which is not very accurate but which has relatively low 13-limit complexity.

Moreover, 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 |, cent values rounded to 5 digits.)
||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||
||= 16875/16384 || | -14 3 4 > ||> 51.120 ||= Negri Comma ||= Double Augmentation Diesis ||
||= 250/243 || | 1 -5 3 > ||> 49.166 ||= Maximal Diesis ||= Porcupine Comma ||
||= 32805/32768 || | -15 8 1 > ||> 1.9537 ||= Schisma ||=   ||
||= 525/512 || | -9 1 2 1 > ||> 43.408 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||
||= 49/48 || | -4 -1 0 2 > ||> 35.697 ||= Slendro Diesis ||=   ||
||= 686/675 || | 1 -3 -2 3 > ||> 27.985 ||= Senga ||=   ||
||= 64827/64000 || | -9 3 -3 4 > ||> 22.227 ||= Squalentine ||=   ||
||= 3125/3087 || | 0 -2 5 -3 > ||> 21.181 ||= Gariboh ||=   ||
||= 50421/50000 || | -4 1 -5 5 > ||> 14.516 ||= Trimyna ||=   ||
||= 4000/3969 || | 5 -4 3 -2 > ||> 13.469 ||= Octagar ||=   ||
||= 225/224 || | -5 2 2 -1 > ||> 7.7115 ||= Septimal Kleisma ||= Marvel Comma ||
||= 5120/5103 || | 10 -6 1 -1 > ||> 5.7578 ||= Hemifamity ||=   ||
||= 4994735/4983772 || | 25 -14 0 -1 > ||> 3.8041 ||= Garischisma ||=   ||
||= 100/99 || | 2 -2 2 0 -1 > ||> 17.399 ||= Ptolemisma ||=   ||
||= 121/120 || | -3 -1 -1 0 2 > ||> 14.367 ||= Biyatisma ||=   ||
||= 896/891 || | 7 -4 0 1 -1 > ||> 9.6880 ||= Pentacircle ||=   ||
||= 441/440 || | -3 2 -1 2 -1 > ||> 3.9302 ||= Werckisma ||=   ||
||= 4000/3993 || | 5 -1 3 0 -3 > ||> 3.0323 ||= Wizardharry ||=   ||
||= 9801/9800 || | -3 4 -2 -2 2 > ||> 0.17665 ||= Kalisma ||= Gauss' Comma ||
||= 91/90 || | -1 -2 -1 1 0 1 > ||> 19.130 ||= 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 <a class="wiki_link" href="/octave">octave</a> into 29 equal steps of approximately 41.37931 <a class="wiki_link" href="/cent">cent</a>s.<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 29edo approximates very closely, and it does so quite well. Nonetheless, and rather surprisingly, 29 is the smallest equal division which consistently represents the 15 odd limit. It is able to do this since it has an accurate 3, and the 5, 7, 11 and 13, while not very accurate, are all tuned flatly. Hence it tempers out a succession of fairly large commas: 250/243 in the <a class="wiki_link" href="/5-limit">5-limit</a>, 49/48 in the <a class="wiki_link" href="/7-limit">7-limit</a>, 55/54 in the <a class="wiki_link" href="/11-limit">11-limit</a>, and 65/64 in the <a class="wiki_link" href="/13-limit">13-limit</a>. If using these approximations is desired, 29edo actually shines, and it can be used for such things as an alternative to <a class="wiki_link" href="/19edo">19edo</a> for <a class="wiki_link" href="/Marvel%20temperaments">negri</a>.<br />
<br />
Another possible use for 29edo is as an equally tempered para-pythagorean scale. Using its fifth as a generator leads to a variant of <a class="wiki_link" href="/Schismatic%20family">garibaldi temperament</a> which is not very accurate but which has relatively low 13-limit complexity.<br />
<br />
Moreover, 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>

<br />
<!-- 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 |, cent values rounded to 5 digits.)<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;">16875/16384<br />
</td>
        <td>| -14 3 4 &gt;<br />
</td>
        <td style="text-align: right;">51.120<br />
</td>
        <td style="text-align: center;">Negri Comma<br />
</td>
        <td style="text-align: center;">Double Augmentation Diesis<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">250/243<br />
</td>
        <td>| 1 -5 3 &gt;<br />
</td>
        <td style="text-align: right;">49.166<br />
</td>
        <td style="text-align: center;">Maximal Diesis<br />
</td>
        <td style="text-align: center;">Porcupine Comma<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">32805/32768<br />
</td>
        <td>| -15 8 1 &gt;<br />
</td>
        <td style="text-align: right;">1.9537<br />
</td>
        <td style="text-align: center;">Schisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">525/512<br />
</td>
        <td>| -9 1 2 1 &gt;<br />
</td>
        <td style="text-align: right;">43.408<br />
</td>
        <td style="text-align: center;">Avicennma<br />
</td>
        <td style="text-align: center;">Avicenna's Enharmonic Diesis<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">49/48<br />
</td>
        <td>| -4 -1 0 2 &gt;<br />
</td>
        <td style="text-align: right;">35.697<br />
</td>
        <td style="text-align: center;">Slendro Diesis<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">686/675<br />
</td>
        <td>| 1 -3 -2 3 &gt;<br />
</td>
        <td style="text-align: right;">27.985<br />
</td>
        <td style="text-align: center;">Senga<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">64827/64000<br />
</td>
        <td>| -9 3 -3 4 &gt;<br />
</td>
        <td style="text-align: right;">22.227<br />
</td>
        <td style="text-align: center;">Squalentine<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">3125/3087<br />
</td>
        <td>| 0 -2 5 -3 &gt;<br />
</td>
        <td style="text-align: right;">21.181<br />
</td>
        <td style="text-align: center;">Gariboh<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">50421/50000<br />
</td>
        <td>| -4 1 -5 5 &gt;<br />
</td>
        <td style="text-align: right;">14.516<br />
</td>
        <td style="text-align: center;">Trimyna<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">4000/3969<br />
</td>
        <td>| 5 -4 3 -2 &gt;<br />
</td>
        <td style="text-align: right;">13.469<br />
</td>
        <td style="text-align: center;">Octagar<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">225/224<br />
</td>
        <td>| -5 2 2 -1 &gt;<br />
</td>
        <td style="text-align: right;">7.7115<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;">5120/5103<br />
</td>
        <td>| 10 -6 1 -1 &gt;<br />
</td>
        <td style="text-align: right;">5.7578<br />
</td>
        <td style="text-align: center;">Hemifamity<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">4994735/4983772<br />
</td>
        <td>| 25 -14 0 -1 &gt;<br />
</td>
        <td style="text-align: right;">3.8041<br />
</td>
        <td style="text-align: center;">Garischisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">100/99<br />
</td>
        <td>| 2 -2 2 0 -1 &gt;<br />
</td>
        <td style="text-align: right;">17.399<br />
</td>
        <td style="text-align: center;">Ptolemisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">121/120<br />
</td>
        <td>| -3 -1 -1 0 2 &gt;<br />
</td>
        <td style="text-align: right;">14.367<br />
</td>
        <td style="text-align: center;">Biyatisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">896/891<br />
</td>
        <td>| 7 -4 0 1 -1 &gt;<br />
</td>
        <td style="text-align: right;">9.6880<br />
</td>
        <td style="text-align: center;">Pentacircle<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">441/440<br />
</td>
        <td>| -3 2 -1 2 -1 &gt;<br />
</td>
        <td style="text-align: right;">3.9302<br />
</td>
        <td style="text-align: center;">Werckisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">4000/3993<br />
</td>
        <td>| 5 -1 3 0 -3 &gt;<br />
</td>
        <td style="text-align: right;">3.0323<br />
</td>
        <td style="text-align: center;">Wizardharry<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">9801/9800<br />
</td>
        <td>| -3 4 -2 -2 2 &gt;<br />
</td>
        <td style="text-align: right;">0.17665<br />
</td>
        <td style="text-align: center;">Kalisma<br />
</td>
        <td style="text-align: center;">Gauss' Comma<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">91/90<br />
</td>
        <td>| -1 -2 -1 1 0 1 &gt;<br />
</td>
        <td style="text-align: right;">19.130<br />
</td>
        <td style="text-align: center;">Superleap<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
</table>

<br />
<!-- 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>