29edo: Difference between revisions
Wikispaces>genewardsmith **Imported revision 234387598 - Original comment: ** |
Wikispaces>xenwolf **Imported revision 234400422 - Original comment: decimal mantissa of commas: 5 digits** |
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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:xenwolf|xenwolf]] and made on <tt>2011-06-05 16:33:24 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>234400422</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt>decimal mantissa of commas: 5 digits</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> | ||
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=<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 | 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. | 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]]. | 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. | 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. | ||
| Line 50: | Line 50: | ||
|| 27 || 1117.241 || | || 27 || 1117.241 || | ||
|| 28 || 1158.621 || | || 28 || 1158.621 || | ||
=Commas= | =Commas= | ||
29 EDO tempers out the following commas. (Note: This assumes the val < 29 46 67 81 100 107 |.) | 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 || | ||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 || | ||
||= 16875/16384 || | -14 3 4 > ||> 51. | ||= 16875/16384 || | -14 3 4 > ||> 51.120 ||= Negri Comma ||= Double Augmentation Diesis || | ||
||= 250/243 || | 1 -5 3 > ||> 49. | ||= 250/243 || | 1 -5 3 > ||> 49.166 ||= Maximal Diesis ||= Porcupine Comma || | ||
||= 32805/32768 || | -15 8 1 > ||> 1. | ||= 32805/32768 || | -15 8 1 > ||> 1.9537 ||= Schisma ||= || | ||
||= 525/512 || | -9 1 2 1 > ||> 43. | ||= 525/512 || | -9 1 2 1 > ||> 43.408 ||= Avicennma ||= Avicenna's Enharmonic Diesis || | ||
||= 49/48 || | -4 -1 0 2 > ||> 35. | ||= 49/48 || | -4 -1 0 2 > ||> 35.697 ||= Slendro Diesis ||= || | ||
||= 686/675 || | 1 -3 -2 3 > ||> 27. | ||= 686/675 || | 1 -3 -2 3 > ||> 27.985 ||= Senga ||= || | ||
||= 64827/64000 || | -9 3 -3 4 > ||> 22. | ||= 64827/64000 || | -9 3 -3 4 > ||> 22.227 ||= Squalentine ||= || | ||
||= 3125/3087 || | 0 -2 5 -3 > ||> 21. | ||= 3125/3087 || | 0 -2 5 -3 > ||> 21.181 ||= Gariboh ||= || | ||
||= 50421/50000 || | -4 1 -5 5 > ||> 14. | ||= 50421/50000 || | -4 1 -5 5 > ||> 14.516 ||= Trimyna ||= || | ||
||= 4000/3969 || | 5 -4 3 -2 > ||> 13. | ||= 4000/3969 || | 5 -4 3 -2 > ||> 13.469 ||= Octagar ||= || | ||
||= 225/224 || | -5 2 2 -1 > ||> 7. | ||= 225/224 || | -5 2 2 -1 > ||> 7.7115 ||= Septimal Kleisma ||= Marvel Comma || | ||
||= 5120/5103 || | 10 -6 1 -1 > ||> 5. | ||= 5120/5103 || | 10 -6 1 -1 > ||> 5.7578 ||= Hemifamity ||= || | ||
||= 4994735/4983772 || | 25 -14 0 -1 > ||> 3. | ||= 4994735/4983772 || | 25 -14 0 -1 > ||> 3.8041 ||= Garischisma ||= || | ||
||= 100/99 || | 2 -2 2 0 -1 > ||> 17. | ||= 100/99 || | 2 -2 2 0 -1 > ||> 17.399 ||= Ptolemisma ||= || | ||
||= 121/120 || | -3 -1 -1 0 2 > ||> 14. | ||= 121/120 || | -3 -1 -1 0 2 > ||> 14.367 ||= Biyatisma ||= || | ||
||= 896/891 || | 7 -4 0 1 -1 > ||> 9. | ||= 896/891 || | 7 -4 0 1 -1 > ||> 9.6880 ||= Pentacircle ||= || | ||
||= 441/440 || | -3 2 -1 2 -1 > ||> 3. | ||= 441/440 || | -3 2 -1 2 -1 > ||> 3.9302 ||= Werckisma ||= || | ||
||= 4000/3993 || | 5 -1 3 0 -3 > ||> 3. | ||= 4000/3993 || | 5 -1 3 0 -3 > ||> 3.0323 ||= Wizardharry ||= || | ||
||= 9801/9800 || | -3 4 -2 -2 2 > ||> 0. | ||= 9801/9800 || | -3 4 -2 -2 2 > ||> 0.17665 ||= Kalisma ||= Gauss' Comma || | ||
||= 91/90 || | -1 -2 -1 1 0 1 > ||> 19. | ||= 91/90 || | -1 -2 -1 1 0 1 > ||> 19.130 ||= Superleap ||= || | ||
=Music= | =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> | ||
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<!-- 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: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 /> | <br /> | ||
29edo divides the 2:1 octave into 29 equal steps of approximately 41.37931 | 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 /> | <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 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 <a class="wiki_link" href="/19edo">19edo</a> for <a class="wiki_link" href="/Marvel%20temperaments">negri</a>.<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 /> | <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 /> | 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 /> | ||
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</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: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 /> | 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 /> | ||
| Line 297: | Line 300: | ||
<td>| -14 3 4 &gt;<br /> | <td>| -14 3 4 &gt;<br /> | ||
</td> | </td> | ||
<td style="text-align: right;">51. | <td style="text-align: right;">51.120<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Negri Comma<br /> | <td style="text-align: center;">Negri Comma<br /> | ||
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<td>| 1 -5 3 &gt;<br /> | <td>| 1 -5 3 &gt;<br /> | ||
</td> | </td> | ||
<td style="text-align: right;">49. | <td style="text-align: right;">49.166<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Maximal Diesis<br /> | <td style="text-align: center;">Maximal Diesis<br /> | ||
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<td>| -15 8 1 &gt;<br /> | <td>| -15 8 1 &gt;<br /> | ||
</td> | </td> | ||
<td style="text-align: right;">1. | <td style="text-align: right;">1.9537<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Schisma<br /> | <td style="text-align: center;">Schisma<br /> | ||
| Line 333: | Line 336: | ||
<td>| -9 1 2 1 &gt;<br /> | <td>| -9 1 2 1 &gt;<br /> | ||
</td> | </td> | ||
<td style="text-align: right;">43. | <td style="text-align: right;">43.408<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Avicennma<br /> | <td style="text-align: center;">Avicennma<br /> | ||
| Line 345: | Line 348: | ||
<td>| -4 -1 0 2 &gt;<br /> | <td>| -4 -1 0 2 &gt;<br /> | ||
</td> | </td> | ||
<td style="text-align: right;">35. | <td style="text-align: right;">35.697<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Slendro Diesis<br /> | <td style="text-align: center;">Slendro Diesis<br /> | ||
| Line 357: | Line 360: | ||
<td>| 1 -3 -2 3 &gt;<br /> | <td>| 1 -3 -2 3 &gt;<br /> | ||
</td> | </td> | ||
<td style="text-align: right;">27. | <td style="text-align: right;">27.985<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Senga<br /> | <td style="text-align: center;">Senga<br /> | ||
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<td>| -9 3 -3 4 &gt;<br /> | <td>| -9 3 -3 4 &gt;<br /> | ||
</td> | </td> | ||
<td style="text-align: right;">22. | <td style="text-align: right;">22.227<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Squalentine<br /> | <td style="text-align: center;">Squalentine<br /> | ||
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<td>| 0 -2 5 -3 &gt;<br /> | <td>| 0 -2 5 -3 &gt;<br /> | ||
</td> | </td> | ||
<td style="text-align: right;">21. | <td style="text-align: right;">21.181<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Gariboh<br /> | <td style="text-align: center;">Gariboh<br /> | ||
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<td>| -4 1 -5 5 &gt;<br /> | <td>| -4 1 -5 5 &gt;<br /> | ||
</td> | </td> | ||
<td style="text-align: right;">14. | <td style="text-align: right;">14.516<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Trimyna<br /> | <td style="text-align: center;">Trimyna<br /> | ||
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<td>| 5 -4 3 -2 &gt;<br /> | <td>| 5 -4 3 -2 &gt;<br /> | ||
</td> | </td> | ||
<td style="text-align: right;">13. | <td style="text-align: right;">13.469<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Octagar<br /> | <td style="text-align: center;">Octagar<br /> | ||
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<td>| -5 2 2 -1 &gt;<br /> | <td>| -5 2 2 -1 &gt;<br /> | ||
</td> | </td> | ||
<td style="text-align: right;">7. | <td style="text-align: right;">7.7115<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Septimal Kleisma<br /> | <td style="text-align: center;">Septimal Kleisma<br /> | ||
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<td>| 10 -6 1 -1 &gt;<br /> | <td>| 10 -6 1 -1 &gt;<br /> | ||
</td> | </td> | ||
<td style="text-align: right;">5. | <td style="text-align: right;">5.7578<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Hemifamity<br /> | <td style="text-align: center;">Hemifamity<br /> | ||
| Line 441: | Line 444: | ||
<td>| 25 -14 0 -1 &gt;<br /> | <td>| 25 -14 0 -1 &gt;<br /> | ||
</td> | </td> | ||
<td style="text-align: right;">3. | <td style="text-align: right;">3.8041<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Garischisma<br /> | <td style="text-align: center;">Garischisma<br /> | ||
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<td>| 2 -2 2 0 -1 &gt;<br /> | <td>| 2 -2 2 0 -1 &gt;<br /> | ||
</td> | </td> | ||
<td style="text-align: right;">17. | <td style="text-align: right;">17.399<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Ptolemisma<br /> | <td style="text-align: center;">Ptolemisma<br /> | ||
| Line 465: | Line 468: | ||
<td>| -3 -1 -1 0 2 &gt;<br /> | <td>| -3 -1 -1 0 2 &gt;<br /> | ||
</td> | </td> | ||
<td style="text-align: right;">14. | <td style="text-align: right;">14.367<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Biyatisma<br /> | <td style="text-align: center;">Biyatisma<br /> | ||
| Line 477: | Line 480: | ||
<td>| 7 -4 0 1 -1 &gt;<br /> | <td>| 7 -4 0 1 -1 &gt;<br /> | ||
</td> | </td> | ||
<td style="text-align: right;">9. | <td style="text-align: right;">9.6880<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Pentacircle<br /> | <td style="text-align: center;">Pentacircle<br /> | ||
| Line 489: | Line 492: | ||
<td>| -3 2 -1 2 -1 &gt;<br /> | <td>| -3 2 -1 2 -1 &gt;<br /> | ||
</td> | </td> | ||
<td style="text-align: right;">3. | <td style="text-align: right;">3.9302<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Werckisma<br /> | <td style="text-align: center;">Werckisma<br /> | ||
| Line 501: | Line 504: | ||
<td>| 5 -1 3 0 -3 &gt;<br /> | <td>| 5 -1 3 0 -3 &gt;<br /> | ||
</td> | </td> | ||
<td style="text-align: right;">3. | <td style="text-align: right;">3.0323<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Wizardharry<br /> | <td style="text-align: center;">Wizardharry<br /> | ||
| Line 513: | Line 516: | ||
<td>| -3 4 -2 -2 2 &gt;<br /> | <td>| -3 4 -2 -2 2 &gt;<br /> | ||
</td> | </td> | ||
<td style="text-align: right;">0. | <td style="text-align: right;">0.17665<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Kalisma<br /> | <td style="text-align: center;">Kalisma<br /> | ||
| Line 525: | Line 528: | ||
<td>| -1 -2 -1 1 0 1 &gt;<br /> | <td>| -1 -2 -1 1 0 1 &gt;<br /> | ||
</td> | </td> | ||
<td style="text-align: right;">19. | <td style="text-align: right;">19.130<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Superleap<br /> | <td style="text-align: center;">Superleap<br /> | ||
| Line 534: | Line 537: | ||
</table> | </table> | ||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:6 -->Music</h1> | <!-- 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> | <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> | ||