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 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 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 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 || =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:6:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:6 --><!-- ws:start:WikiTextTocRule:7: --><a href="#x29 tone equal temperament">29 tone equal temperament</a><!-- ws:end:WikiTextTocRule:7 --><!-- ws:start:WikiTextTocRule:8: --> | <a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --> | <a href="#Music">Music</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: -->
<!-- ws:end:WikiTextTocRule:10 --><br />
<!-- ws:start:WikiTextHeadingRule:0:<h1> --><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 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 <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 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 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:<h1> --><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:<h1> --><h1 id="toc2"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:4 -->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>