36edo: Difference between revisions
Wikispaces>hearneg **Imported revision 484176582 - Original comment: ** |
Wikispaces>hearneg **Imported revision 484177536 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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:hearneg|hearneg]] and made on <tt>2014-01-21 01: | : This revision was by author [[User:hearneg|hearneg]] and made on <tt>2014-01-21 01:50:28 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>484177536</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> | ||
| Line 17: | Line 17: | ||
For those interested in approximations to just intonation, 36edo offers no improvement over 12edo in the 5-limit, since its nearest approximation to 5:4 is the overly-familiar 400-cent sharp third. However, it excels at approximations involving 3 and 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[just intonation subgroup]], 36edo's single degree of around 33 cents serves a double function as 49:48, the so-called [[http://en.wikipedia.org/wiki/Septimal_diesis|Slendro diesis]] of around 36 cents, and as 64:63, the so-called [[http://en.wikipedia.org/wiki/Septimal_comma|septimal comma]] of around 27 cents. Meanwhile, its second degree functions as 28:27, the so-called [[http://en.wikipedia.org/wiki/Septimal_third-tone|Septimal third-tone]] (which = 49:48 x 64:63). The 2.3.7 subgroup can be extended to the [[k*N subgroups|2*36 subgroup]] 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as [[72edo]] does in the full [[17-limit]]. | For those interested in approximations to just intonation, 36edo offers no improvement over 12edo in the 5-limit, since its nearest approximation to 5:4 is the overly-familiar 400-cent sharp third. However, it excels at approximations involving 3 and 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[just intonation subgroup]], 36edo's single degree of around 33 cents serves a double function as 49:48, the so-called [[http://en.wikipedia.org/wiki/Septimal_diesis|Slendro diesis]] of around 36 cents, and as 64:63, the so-called [[http://en.wikipedia.org/wiki/Septimal_comma|septimal comma]] of around 27 cents. Meanwhile, its second degree functions as 28:27, the so-called [[http://en.wikipedia.org/wiki/Septimal_third-tone|Septimal third-tone]] (which = 49:48 x 64:63). The 2.3.7 subgroup can be extended to the [[k*N subgroups|2*36 subgroup]] 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as [[72edo]] does in the full [[17-limit]]. | ||
36 tempers out, like 12, 81/80, 128/125 and 648/625 in the 5-limit. It departs from 12 in the 7-limit, tempering out 686/675, and as a no-fives temperament, 1029/1024 and 118098/117649. In the 11-limit, it tempers out 56/55, 245/242 and 540/539, and is the [[optimal patent val]] for the rank four temperament tempering out 56/55, as well as the rank three temperament [[Didymus rank three family|melpomene]] tempering out 81/80 and 56/55. In the 13-limit, it tempers out 78/77, in the 17-limit 51/50, and in the 19-limit 76/75, 91/90 and 96/95. | 36 tempers out, like 12, 81/80, 128/125 and 648/625 in the 5-limit. It departs from 12 in the 7-limit, tempering out 686/675, and as a no-fives temperament, 1029/1024 and 118098/117649. The no-fives temperament tempering out 1029/1024, [[xenharmonic/slendric|slendric]], is well supported by 36edo, it's generator of ~8/7 represented by 7 steps of 36edo. | ||
In the 11-limit, it tempers out 56/55, 245/242 and 540/539, and is the [[optimal patent val]] for the rank four temperament tempering out 56/55, as well as the rank three temperament [[Didymus rank three family|melpomene]] tempering out 81/80 and 56/55. In the 13-limit, it tempers out 78/77, in the 17-limit 51/50, and in the 19-limit 76/75, 91/90 and 96/95. | |||
As a 5-limit temperament, the patent val for 36edo is [[Wedgies and Multivals|contorted]], meaning there are notes of it which cannot be reached from the unison using only 5-limit intervals. A curious alternative val for the 5-limit is <36 65 116| which is not contorted. It is also a meantone val, in the sense that 81/80 is tempered out. However, the "comma" |29 0 -9> is also tempered out, and the "fifth", 29\36, is actually approximately 7/4, whereas the "major third", 44\36, is actually approximately 7/3. Any 5-limit musical piece or scale which is a [[transversal]] for a meantone piece or scale will be converted to a no-fives piece tempering out 1029/1024 in place of 81/80 by applying this val. | As a 5-limit temperament, the patent val for 36edo is [[Wedgies and Multivals|contorted]], meaning there are notes of it which cannot be reached from the unison using only 5-limit intervals. A curious alternative val for the 5-limit is <36 65 116| which is not contorted. It is also a meantone val, in the sense that 81/80 is tempered out. However, the "comma" |29 0 -9> is also tempered out, and the "fifth", 29\36, is actually approximately 7/4, whereas the "major third", 44\36, is actually approximately 7/3. Any 5-limit musical piece or scale which is a [[transversal]] for a meantone piece or scale will be converted to a no-fives piece tempering out 1029/1024 in place of 81/80 by applying this val. | ||
| Line 117: | Line 118: | ||
For those interested in approximations to just intonation, 36edo offers no improvement over 12edo in the 5-limit, since its nearest approximation to 5:4 is the overly-familiar 400-cent sharp third. However, it excels at approximations involving 3 and 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 <a class="wiki_link" href="/just%20intonation%20subgroup">just intonation subgroup</a>, 36edo's single degree of around 33 cents serves a double function as 49:48, the so-called <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_diesis" rel="nofollow">Slendro diesis</a> of around 36 cents, and as 64:63, the so-called <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_comma" rel="nofollow">septimal comma</a> of around 27 cents. Meanwhile, its second degree functions as 28:27, the so-called <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_third-tone" rel="nofollow">Septimal third-tone</a> (which = 49:48 x 64:63). The 2.3.7 subgroup can be extended to the <a class="wiki_link" href="/k%2AN%20subgroups">2*36 subgroup</a> 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as <a class="wiki_link" href="/72edo">72edo</a> does in the full <a class="wiki_link" href="/17-limit">17-limit</a>.<br /> | For those interested in approximations to just intonation, 36edo offers no improvement over 12edo in the 5-limit, since its nearest approximation to 5:4 is the overly-familiar 400-cent sharp third. However, it excels at approximations involving 3 and 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 <a class="wiki_link" href="/just%20intonation%20subgroup">just intonation subgroup</a>, 36edo's single degree of around 33 cents serves a double function as 49:48, the so-called <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_diesis" rel="nofollow">Slendro diesis</a> of around 36 cents, and as 64:63, the so-called <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_comma" rel="nofollow">septimal comma</a> of around 27 cents. Meanwhile, its second degree functions as 28:27, the so-called <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_third-tone" rel="nofollow">Septimal third-tone</a> (which = 49:48 x 64:63). The 2.3.7 subgroup can be extended to the <a class="wiki_link" href="/k%2AN%20subgroups">2*36 subgroup</a> 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as <a class="wiki_link" href="/72edo">72edo</a> does in the full <a class="wiki_link" href="/17-limit">17-limit</a>.<br /> | ||
<br /> | <br /> | ||
36 tempers out, like 12, 81/80, 128/125 and 648/625 in the 5-limit. It departs from 12 in the 7-limit, tempering out 686/675, and as a no-fives temperament, 1029/1024 and 118098/117649. In the 11-limit, it tempers out 56/55, 245/242 and 540/539, and is the <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a> for the rank four temperament tempering out 56/55, as well as the rank three temperament <a class="wiki_link" href="/Didymus%20rank%20three%20family">melpomene</a> tempering out 81/80 and 56/55. In the 13-limit, it tempers out 78/77, in the 17-limit 51/50, and in the 19-limit 76/75, 91/90 and 96/95.<br /> | 36 tempers out, like 12, 81/80, 128/125 and 648/625 in the 5-limit. It departs from 12 in the 7-limit, tempering out 686/675, and as a no-fives temperament, 1029/1024 and 118098/117649. The no-fives temperament tempering out 1029/1024, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/slendric">slendric</a>, is well supported by 36edo, it's generator of ~8/7 represented by 7 steps of 36edo.<br /> | ||
In the 11-limit, it tempers out 56/55, 245/242 and 540/539, and is the <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a> for the rank four temperament tempering out 56/55, as well as the rank three temperament <a class="wiki_link" href="/Didymus%20rank%20three%20family">melpomene</a> tempering out 81/80 and 56/55. In the 13-limit, it tempers out 78/77, in the 17-limit 51/50, and in the 19-limit 76/75, 91/90 and 96/95.<br /> | |||
<br /> | <br /> | ||
As a 5-limit temperament, the patent val for 36edo is <a class="wiki_link" href="/Wedgies%20and%20Multivals">contorted</a>, meaning there are notes of it which cannot be reached from the unison using only 5-limit intervals. A curious alternative val for the 5-limit is &lt;36 65 116| which is not contorted. It is also a meantone val, in the sense that 81/80 is tempered out. However, the &quot;comma&quot; |29 0 -9&gt; is also tempered out, and the &quot;fifth&quot;, 29\36, is actually approximately 7/4, whereas the &quot;major third&quot;, 44\36, is actually approximately 7/3. Any 5-limit musical piece or scale which is a <a class="wiki_link" href="/transversal">transversal</a> for a meantone piece or scale will be converted to a no-fives piece tempering out 1029/1024 in place of 81/80 by applying this val.<br /> | As a 5-limit temperament, the patent val for 36edo is <a class="wiki_link" href="/Wedgies%20and%20Multivals">contorted</a>, meaning there are notes of it which cannot be reached from the unison using only 5-limit intervals. A curious alternative val for the 5-limit is &lt;36 65 116| which is not contorted. It is also a meantone val, in the sense that 81/80 is tempered out. However, the &quot;comma&quot; |29 0 -9&gt; is also tempered out, and the &quot;fifth&quot;, 29\36, is actually approximately 7/4, whereas the &quot;major third&quot;, 44\36, is actually approximately 7/3. Any 5-limit musical piece or scale which is a <a class="wiki_link" href="/transversal">transversal</a> for a meantone piece or scale will be converted to a no-fives piece tempering out 1029/1024 in place of 81/80 by applying this val.<br /> | ||