29edo: Difference between revisions
Wikispaces>guest **Imported revision 242159059 - Original comment: ** |
Wikispaces>Osmiorisbendi **Imported revision 242231023 - Original comment: ** |
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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:Osmiorisbendi|Osmiorisbendi]] and made on <tt>2011-07-21 03:35:12 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>242231023</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></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> | ||
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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 [[consistent]]ly 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]], as well as an alternative to 22edo or 15edo for porcupine. For those who enjoy the bizarre character of Father temperament, 29edo can also be used to support that temperament, if one imagines 11\29 is approximating both 5/4 and 4/3 (ignoring the better approximations at 10\29 and 12\29, respectively). | 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 [[consistent]]ly 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]], as well as an alternative to [[22edo]] or [[15edo]] for porcupine. For those who enjoy the bizarre character of Father temperament, 29edo can also be used to support that temperament, if one imagines 11\29 is approximating both 5/4 and 4/3 (ignoring the better approximations at 10\29 and 12\29, respectively). | ||
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 30: | Line 30: | ||
|| 5 || 206.897 || | || 5 || 206.897 || | ||
|| 6 || 248.276 || | || 6 || 248.276 || | ||
|| | || 7· || 289.655 || | ||
|| 8 || 331.034 || | || 8 || 331.034 || | ||
|| 9 || 372.414 || | || 9 || 372.414 || | ||
|| 10 || 413.793 || | || 10 || 413.793 || | ||
|| 11 || 455.172 || | || 11 || 455.172 || | ||
|| | || 12· || 496.552 || | ||
|| 13 || 537.931 || | || 13 || 537.931 || | ||
|| 14 || 579.310 || | || 14 || 579.310 || | ||
|| 15 || 620.690 || | || 15 || 620.690 || | ||
|| 16 || 662.069 || | || 16 || 662.069 || | ||
|| | || 17· || 703.448 || | ||
|| 18 || 744.828 || | || 18 || 744.828 || | ||
|| 19 || 786.207 || | || 19 || 786.207 || | ||
|| 20 || 827.586 || | || 20 || 827.586 || | ||
|| 21 || 868.966 || | || 21 || 868.966 || | ||
|| | || 22· || 910.345 || | ||
|| 23 || 951.724 || | || 23 || 951.724 || | ||
|| 24 || 993.103 || | || 24 || 993.103 || | ||
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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 <a class="wiki_link" href="/consistent">consistent</a>ly 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>, as well as an alternative to 22edo or 15edo for porcupine. For those who enjoy the bizarre character of Father temperament, 29edo can also be used to support that temperament, if one imagines 11\29 is approximating both 5/4 and 4/3 (ignoring the better approximations at 10\29 and 12\29, respectively). <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 <a class="wiki_link" href="/consistent">consistent</a>ly 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>, as well as an alternative to <a class="wiki_link" href="/22edo">22edo</a> or <a class="wiki_link" href="/15edo">15edo</a> for porcupine. For those who enjoy the bizarre character of Father temperament, 29edo can also be used to support that temperament, if one imagines 11\29 is approximating both 5/4 and 4/3 (ignoring the better approximations at 10\29 and 12\29, respectively).<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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</tr> | </tr> | ||
<tr> | <tr> | ||
<td> | <td>7·<br /> | ||
</td> | </td> | ||
<td>289.655<br /> | <td>289.655<br /> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td> | <td>12·<br /> | ||
</td> | </td> | ||
<td>496.552<br /> | <td>496.552<br /> | ||
| Line 212: | Line 212: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td> | <td>17·<br /> | ||
</td> | </td> | ||
<td>703.448<br /> | <td>703.448<br /> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td> | <td>22·<br /> | ||
</td> | </td> | ||
<td>910.345<br /> | <td>910.345<br /> | ||