240edo: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>

: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-12-07 16:25:45 UTC</tt>.<br>

: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-12-07 16:29:22 UTC</tt>.<br>

: The original revision id was <tt>~~601655298~~</tt>.<br>

: The original revision id was <tt>601655574</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>

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<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 [[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.

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, whitherfore a compromise between good results and an accurate 5 may 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 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:

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<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 />

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 />

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, whitherfore a compromise between good results and an accurate 5 may be worked out by means of retuning 5-limit scales to the 197&amp;240 temperament.<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 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 />

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-12-07 16:25:45 UTC</tt>.<br>
+: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-12-07 16:29:22 UTC</tt>.<br>
-: The original revision id was <tt>601655298</tt>.<br>
+: The original revision id was <tt>601655574</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>
@@ Line 8: / Line 8: @@
 <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 [[Vals and Tuning Space|val]] (mapping to primes) for 240edo of &lt;[[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&amp;240 temperament.
+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 &lt;[[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, whitherfore a compromise between good results and an accurate 5 may be worked out by means of retuning 5-limit scales to the 197&amp;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 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:
@@ Line 175: / Line 175: @@
 <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">&lt;html&gt;&lt;head&gt;&lt;title&gt;240edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;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.&lt;br /&gt;
 &lt;br /&gt;
-If we round off to the nearest five cents, we end up with a &lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&gt;val&lt;/a&gt; (mapping to primes) for 240edo of &amp;lt;&lt;a class="wiki_link" href="http://tel.wikispaces.com/240%20380%20557"&gt;240 380 557&lt;/a&gt; 674|. This tempers out the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_kleisma" rel="nofollow"&gt;septimal kleisma&lt;/a&gt; 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 &lt;a class="wiki_link" href="/197edo"&gt;197edo&lt;/a&gt;, 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;amp;240 temperament.&lt;br /&gt;
+If we round off to the nearest five cents, we end up with a &lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&gt;val&lt;/a&gt; (mapping to primes) for 240edo of &amp;lt;&lt;a class="wiki_link" href="http://tel.wikispaces.com/240%20380%20557"&gt;240 380 557&lt;/a&gt; 674|. This tempers out the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_kleisma" rel="nofollow"&gt;septimal kleisma&lt;/a&gt; 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 &lt;a class="wiki_link" href="/197edo"&gt;197edo&lt;/a&gt;, despite a flatter third, gives generally better results and may be preferred, whitherfore a compromise between good results and an accurate 5 may be worked out by means of retuning 5-limit scales to the 197&amp;amp;240 temperament.&lt;br /&gt;
 &lt;br /&gt;
 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:&lt;br /&gt;