29edo: Difference between revisions
Wikispaces>genewardsmith **Imported revision 232648140 - Original comment: ** |
Wikispaces>jdfreivald **Imported revision 233351404 - Original comment: Added comma table.** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:jdfreivald|jdfreivald]] and made on <tt>2011-05-31 22:33:30 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>233351404</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt>Added comma table.</tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]] | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]] | ||
=<span style="color: #ff4700; font-size: 103%;">29 tone equal temperament</span>= | =<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. | 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. | 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. | 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= | =Intervals= | ||
|| Degrees of 29-EDO || Cents value || | || Degrees of 29-EDO || Cents value || | ||
|| 0 || 0 || | || 0 || 0 || | ||
| Line 47: | Line 46: | ||
|| 27 || 1117.241 || | || 27 || 1117.241 || | ||
|| 28 || 1158.621 || | || 28 || 1158.621 || | ||
=Commas= | |||
=Music= | 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]]</pre></div> | [[http://tinyurl.com/45lancy|Paint in the Water 29]] by [[Igliashon Jones]]</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>29edo</title></head><body><!-- ws:start:WikiTextTocRule: | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><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: | <!-- 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> | ||
<!-- 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 /> | <br /> | ||
29edo divides the 2:1 octave into 29 equal steps of approximately 41.37931 cents.<br /> | 29edo divides the 2:1 octave into 29 equal steps of approximately 41.37931 cents.<br /> | ||
<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 /> | 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 /> | <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 /> | 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 /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:2 -->Intervals</h1> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:2 -->Intervals</h1> | ||
<table class="wiki_table"> | <table class="wiki_table"> | ||
| Line 247: | Line 267: | ||
</table> | </table> | ||
<br /> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:4 -->Commas</h1> | ||
<!-- ws:start:WikiTextHeadingRule: | 29 EDO tempers out the following commas. (Note: This assumes the val &lt; 29 46 67 81 100 107 |.)<br /> | ||
<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></pre></div> | |||
<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></pre></div> | |||