240edo: Difference between revisions
Wikispaces>genewardsmith **Imported revision 153944567 - Original comment: ** |
Wikispaces>JosephRuhf **Imported revision 601655298 - 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:JosephRuhf|JosephRuhf]] and made on <tt>2016-12-07 16:25:45 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>601655298</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> | ||
<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">The 240edo divides the octave into 240 steps of exactly five cents each. One important use for is in tuning marvel temperament and marvel's extension to spectacle temperament. | <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">The 240edo divides the octave into 240 steps of exactly five cents each. One important use for it is in tuning marvel temperament and marvel's extension to spectacle temperament. | ||
If we round off to the nearest five cents, we end up with a | If we round off to the nearest five cents, we end up with a [[Vals and Tuning Space|val]] (mapping to primes) for 240edo of <[[tel:240 380 557|240 380 557]] 674|. This tempers out the [[http://en.wikipedia.org/wiki/Septimal_kleisma|septimal kleisma]] of 225/224, with low resultant errors (two cents flat for the fifth, a little over a cent flat and sharp, respectively, for the major third and the 7/4.) Retuning 5-limit scales to 240edo is a simple way to to make them function as 7-limit scales while retaining very accurate tuning. However [[197edo]], despite a flatter third, gives generally better results and may be preferred. As a compromise between good results and an accurate 5 may therefore be worked out by means of retuning 5-limit scales to the 197&240 temperament. | ||
For higher limits, 240edo tempers out 243/242 in the 11-limit, 351/350 in the 13-limit, and 375/374 in the 17-limit, and adding these to the mix converts marvel temperament into spectacle temperament. This is still a planar temperament, but more complex as two unidecimal neutral thirds of 11/9 make up a fifth (which is in fact the same fifth as that of 12edo, and the 11/9 the 350 cent interval often employed in 24edo versions of Arabic music.) Musical intervals are therefore generated by octaves, major thirds, and neutral thirds in spectacle. We have: | For higher limits, 240edo tempers out 243/242 in the 11-limit, 351/350 in the 13-limit, and 375/374 in the 17-limit, and adding these to the mix converts marvel temperament into spectacle temperament. This is still a planar temperament, but more complex as two unidecimal neutral thirds of 11/9 make up a fifth (which is in fact the same fifth as that of 12edo, and the 11/9 is the 350 cent interval often employed in 24edo versions of Arabic music.) Musical intervals are therefore generated by octaves, major thirds, and neutral thirds in spectacle. We have: | ||
3 ~ 2 (11/9)^2 | 3 ~ 2 (11/9)^2 | ||
| Line 21: | Line 21: | ||
It should be noted that the exponents of 5/4 above are all positive and go no higher than 4. | It should be noted that the exponents of 5/4 above are all positive and go no higher than 4. | ||
==Scales== | ==Scales== | ||
Here are some examples of scales retuned to 240edo and hence exhibiting marvel temperament. | Here are some examples of scales retuned to 240edo and hence exhibiting marvel temperament. | ||
| Line 28: | Line 28: | ||
! | ! | ||
Ellis's Duodene : genus [33355] | Ellis's Duodene : genus [33355] | ||
12 | |||
! | ! | ||
16/15 | |||
9/8 | |||
6/5 | |||
5/4 | |||
4/3 | |||
45/32 | |||
3/2 | |||
8/5 | |||
5/3 | |||
9/5 | |||
15/8 | |||
2/1 | |||
! duodene240.scl | ! duodene240.scl | ||
! | ! | ||
Ellis's Duodene : genus [33355] retuned to 240edo | Ellis's Duodene : genus [33355] retuned to 240edo | ||
12 | |||
! | ! | ||
115. | 115. | ||
| Line 65: | Line 65: | ||
! | ! | ||
Carl Lumma's scale, 5-limit just version, TL 19-2-99 | Carl Lumma's scale, 5-limit just version, TL 19-2-99 | ||
! Also diadie1, prism, Fokker 12-tone just | ! Also diadie1, prism, Fokker 12-tone just | ||
12 | |||
! | ! | ||
16/15 | |||
9/8 | |||
75/64 | |||
5/4 | |||
4/3 | |||
45/32 | |||
3/2 | |||
8/5 | |||
5/3 | |||
225/128 | |||
15/8 | |||
2/1 | |||
| Line 139: | Line 139: | ||
1085. | 1085. | ||
1200. | 1200. | ||
! tetrads [[0, -1, 0], [0, -1, 1], [1, -1, 1], [1, -1, 2], | ! tetrads [[0, -1, 0], [0, -1, 1], [1, -1, 1], [1, -1, 2], ! [0, 0, 2], [0, -1, -2], [0, 0, 1], [0, -1, -1]] | ||
! [0, 0, 2], [0, -1, -2], [0, 0, 1], [0, -1, -1]] | |||
! doubleduo.scl | ! doubleduo.scl | ||
| Line 171: | Line 170: | ||
1200. | 1200. | ||
==Links== | ==Links== | ||
[[Shaahin Mohajeri]], an Iranian Tombak player and composer, calls his personal [[http://sites.google.com/site/240edo/|Google site]] "240edo", where he makes the point that five cents is a size close to the [[Just noticeable difference|just noticeable difference]] between pitches. | [[Shaahin Mohajeri]], an Iranian Tombak player and composer, calls his personal [[http://sites.google.com/site/240edo/|Google site]] "240edo", where he makes the point that five cents is a size close to the [[Just noticeable difference|just noticeable difference]] between pitches.</pre></div> | ||
</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>240edo</title></head><body>The 240edo divides the octave into 240 steps of exactly five cents each. One important use for is in tuning marvel temperament and marvel's extension to spectacle temperament.<br /> | <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>240edo</title></head><body>The 240edo divides the octave into 240 steps of exactly five cents each. One important use for it is in tuning marvel temperament and marvel's extension to spectacle temperament.<br /> | ||
<br /> | <br /> | ||
If we round off to the nearest five cents, we end up with a | If we round off to the nearest five cents, we end up with a <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">val</a> (mapping to primes) for 240edo of &lt;<a class="wiki_link" href="http://tel.wikispaces.com/240%20380%20557">240 380 557</a> 674|. This tempers out the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_kleisma" rel="nofollow">septimal kleisma</a> of 225/224, with low resultant errors (two cents flat for the fifth, a little over a cent flat and sharp, respectively, for the major third and the 7/4.) Retuning 5-limit scales to 240edo is a simple way to to make them function as 7-limit scales while retaining very accurate tuning. However <a class="wiki_link" href="/197edo">197edo</a>, despite a flatter third, gives generally better results and may be preferred. As a compromise between good results and an accurate 5 may therefore be worked out by means of retuning 5-limit scales to the 197&amp;240 temperament.<br /> | ||
<br /> | <br /> | ||
For higher limits, 240edo tempers out 243/242 in the 11-limit, 351/350 in the 13-limit, and 375/374 in the 17-limit, and adding these to the mix converts marvel temperament into spectacle temperament. This is still a planar temperament, but more complex as two unidecimal neutral thirds of 11/9 make up a fifth (which is in fact the same fifth as that of 12edo, and the 11/9 the 350 cent interval often employed in 24edo versions of Arabic music.) Musical intervals are therefore generated by octaves, major thirds, and neutral thirds in spectacle. We have:<br /> | For higher limits, 240edo tempers out 243/242 in the 11-limit, 351/350 in the 13-limit, and 375/374 in the 17-limit, and adding these to the mix converts marvel temperament into spectacle temperament. This is still a planar temperament, but more complex as two unidecimal neutral thirds of 11/9 make up a fifth (which is in fact the same fifth as that of 12edo, and the 11/9 is the 350 cent interval often employed in 24edo versions of Arabic music.) Musical intervals are therefore generated by octaves, major thirds, and neutral thirds in spectacle. We have:<br /> | ||
<br /> | <br /> | ||
3 ~ 2 (11/9)^2<br /> | 3 ~ 2 (11/9)^2<br /> | ||
| Line 191: | Line 189: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Scales"></a><!-- ws:end:WikiTextHeadingRule:0 -->Scales</h2> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Scales"></a><!-- ws:end:WikiTextHeadingRule:0 -->Scales</h2> | ||
<br /> | <br /> | ||
Here are some examples of scales retuned to 240edo and hence exhibiting marvel temperament.<br /> | Here are some examples of scales retuned to 240edo and hence exhibiting marvel temperament.<br /> | ||
<br /> | <br /> | ||
| Line 197: | Line 195: | ||
!<br /> | !<br /> | ||
Ellis's Duodene : genus [33355]<br /> | Ellis's Duodene : genus [33355]<br /> | ||
12<br /> | |||
!<br /> | !<br /> | ||
16/15<br /> | |||
9/8<br /> | |||
6/5<br /> | |||
5/4<br /> | |||
4/3<br /> | |||
45/32<br /> | |||
3/2<br /> | |||
8/5<br /> | |||
5/3<br /> | |||
9/5<br /> | |||
15/8<br /> | |||
2/1<br /> | |||
<br /> | <br /> | ||
! duodene240.scl<br /> | ! duodene240.scl<br /> | ||
!<br /> | !<br /> | ||
Ellis's Duodene : genus [33355] retuned to 240edo<br /> | Ellis's Duodene : genus [33355] retuned to 240edo<br /> | ||
12<br /> | |||
!<br /> | !<br /> | ||
115.<br /> | 115.<br /> | ||
| Line 234: | Line 232: | ||
!<br /> | !<br /> | ||
Carl Lumma's scale, 5-limit just version, TL 19-2-99<br /> | Carl Lumma's scale, 5-limit just version, TL 19-2-99<br /> | ||
! Also diadie1, prism, Fokker 12-tone just | ! Also diadie1, prism, Fokker 12-tone just<br /> | ||
12<br /> | |||
!<br /> | !<br /> | ||
16/15<br /> | |||
9/8<br /> | |||
75/64<br /> | |||
5/4<br /> | |||
4/3<br /> | |||
45/32<br /> | |||
3/2<br /> | |||
8/5<br /> | |||
5/3<br /> | |||
225/128<br /> | |||
15/8<br /> | |||
2/1<br /> | |||
<br /> | <br /> | ||
<br /> | <br /> | ||
| Line 340: | Line 338: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-Links"></a><!-- ws:end:WikiTextHeadingRule:2 -->Links</h2> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-Links"></a><!-- ws:end:WikiTextHeadingRule:2 -->Links</h2> | ||
<a class="wiki_link" href="/Shaahin%20Mohajeri">Shaahin Mohajeri</a>, an Iranian Tombak player and composer, calls his personal <a class="wiki_link_ext" href="http://sites.google.com/site/240edo/" rel="nofollow">Google site</a> &quot;240edo&quot;, where he makes the point that five cents is a size close to the <a class="wiki_link" href="/Just%20noticeable%20difference">just noticeable difference</a> between pitches.</body></html></pre></div> | <a class="wiki_link" href="/Shaahin%20Mohajeri">Shaahin Mohajeri</a>, an Iranian Tombak player and composer, calls his personal <a class="wiki_link_ext" href="http://sites.google.com/site/240edo/" rel="nofollow">Google site</a> &quot;240edo&quot;, where he makes the point that five cents is a size close to the <a class="wiki_link" href="/Just%20noticeable%20difference">just noticeable difference</a> between pitches.</body></html></pre></div> | ||