22edo: Difference between revisions
Wikispaces>Kosmorsky **Imported revision 336226050 - Original comment: ** |
Wikispaces>Kosmorsky **Imported revision 336226738 - 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:Kosmorsky|Kosmorsky]] and made on <tt>2012-05-16 19: | : This revision was by author [[User:Kosmorsky|Kosmorsky]] and made on <tt>2012-05-16 19:23:45 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>336226738</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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The 22-et system is in fact the third equal division, after 12 and 19, which is capable of approximating the [[5-limit]] to within a TE error of 4 cents/oct. While not an integral or gap edo it at least qualifies as a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak]]. Moreover, there is more to it than just the 5-limit; unlike 12 or 19 it is able to approximate the [[7-limit|7-]] and [[11-limit]]s to within 3 cents/oct of error. While [[31edo|31 equal temperament]] does much better, 22-et still allows the use of these higher-limit harmonies, and in fact 22 is the smallest equal division to represent the 11-limit[[consistent| consistent]]ly. Furthermore, 22-et, unlike 12 and [[19edo|19]], is not a [[Regular Temperaments#meantone|meantone]] system. The net effect is that 22 allows, and to some extent even forces, the exploration of less familiar musical territory, yet is small enough that it can be used in live performances with suitably designed instruments, such as 22-tone guitars and the like. | The 22-et system is in fact the third equal division, after 12 and 19, which is capable of approximating the [[5-limit]] to within a TE error of 4 cents/oct. While not an integral or gap edo it at least qualifies as a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak]]. Moreover, there is more to it than just the 5-limit; unlike 12 or 19 it is able to approximate the [[7-limit|7-]] and [[11-limit]]s to within 3 cents/oct of error. While [[31edo|31 equal temperament]] does much better, 22-et still allows the use of these higher-limit harmonies, and in fact 22 is the smallest equal division to represent the 11-limit[[consistent| consistent]]ly. Furthermore, 22-et, unlike 12 and [[19edo|19]], is not a [[Regular Temperaments#meantone|meantone]] system. The net effect is that 22 allows, and to some extent even forces, the exploration of less familiar musical territory, yet is small enough that it can be used in live performances with suitably designed instruments, such as 22-tone guitars and the like. | ||
22-et can also be treated as adding harmonics 3 and 5 to 11-EDO's 2.7.9.11.15.17 subgroup, making it a (rather accurate) 2.3.5.7.11.17 subgroup temperament. Let us also mind it's approximation of the 31st harmonic is within half a cent, which is fairly accurate | 22-et can also be treated as adding harmonics 3 and 5 to 11-EDO's 2.7.9.11.15.17 subgroup, making it a (rather accurate) 2.3.5.7.11.17 subgroup temperament. Let us also mind it's approximation of the 31st harmonic is within half a cent, which is fairly accurate. | ||
|| Degree || Cents ||= Approximate | || Degree || Cents ||= Approximate | ||
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|| 19 || 1036.36 ||= 20/11, 9/5 || | || 19 || 1036.36 ||= 20/11, 9/5 || | ||
|| 20 || 1090.91 ||= 15/8, 32/17, 17/9 || | || 20 || 1090.91 ||= 15/8, 32/17, 17/9 || | ||
|| 21 || 1145.45 ||= 33/17, 64/33 || | || 21 || 1145.45 ||= 33/17, 64/33, 31/16 || | ||
|| 22 || 1200 ||= 2/1 || | || 22 || 1200 ||= 2/1 || | ||
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The 22-et system is in fact the third equal division, after 12 and 19, which is capable of approximating the <a class="wiki_link" href="/5-limit">5-limit</a> to within a TE error of 4 cents/oct. While not an integral or gap edo it at least qualifies as a <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists">zeta peak</a>. Moreover, there is more to it than just the 5-limit; unlike 12 or 19 it is able to approximate the <a class="wiki_link" href="/7-limit">7-</a> and <a class="wiki_link" href="/11-limit">11-limit</a>s to within 3 cents/oct of error. While <a class="wiki_link" href="/31edo">31 equal temperament</a> does much better, 22-et still allows the use of these higher-limit harmonies, and in fact 22 is the smallest equal division to represent the 11-limit<a class="wiki_link" href="/consistent"> consistent</a>ly. Furthermore, 22-et, unlike 12 and <a class="wiki_link" href="/19edo">19</a>, is not a <a class="wiki_link" href="/Regular%20Temperaments#meantone">meantone</a> system. The net effect is that 22 allows, and to some extent even forces, the exploration of less familiar musical territory, yet is small enough that it can be used in live performances with suitably designed instruments, such as 22-tone guitars and the like.<br /> | The 22-et system is in fact the third equal division, after 12 and 19, which is capable of approximating the <a class="wiki_link" href="/5-limit">5-limit</a> to within a TE error of 4 cents/oct. While not an integral or gap edo it at least qualifies as a <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists">zeta peak</a>. Moreover, there is more to it than just the 5-limit; unlike 12 or 19 it is able to approximate the <a class="wiki_link" href="/7-limit">7-</a> and <a class="wiki_link" href="/11-limit">11-limit</a>s to within 3 cents/oct of error. While <a class="wiki_link" href="/31edo">31 equal temperament</a> does much better, 22-et still allows the use of these higher-limit harmonies, and in fact 22 is the smallest equal division to represent the 11-limit<a class="wiki_link" href="/consistent"> consistent</a>ly. Furthermore, 22-et, unlike 12 and <a class="wiki_link" href="/19edo">19</a>, is not a <a class="wiki_link" href="/Regular%20Temperaments#meantone">meantone</a> system. The net effect is that 22 allows, and to some extent even forces, the exploration of less familiar musical territory, yet is small enough that it can be used in live performances with suitably designed instruments, such as 22-tone guitars and the like.<br /> | ||
<br /> | <br /> | ||
22-et can also be treated as adding harmonics 3 and 5 to 11-EDO's 2.7.9.11.15.17 subgroup, making it a (rather accurate) 2.3.5.7.11.17 subgroup temperament. Let us also mind it's approximation of the 31st harmonic is within half a cent, which is fairly accurate | 22-et can also be treated as adding harmonics 3 and 5 to 11-EDO's 2.7.9.11.15.17 subgroup, making it a (rather accurate) 2.3.5.7.11.17 subgroup temperament. Let us also mind it's approximation of the 31st harmonic is within half a cent, which is fairly accurate.<br /> | ||
<br /> | <br /> | ||
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<td>1145.45<br /> | <td>1145.45<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">33/17, 64/33<br /> | <td style="text-align: center;">33/17, 64/33, 31/16<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||