Consistency: Difference between revisions
Wikispaces>xenwolf **Imported revision 239284485 - 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:xenwolf|xenwolf]] and made on <tt>2011-06-29 04: | : This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2011-06-29 04:06:36 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>239284579</tt>.<br> | ||
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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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== Examples == | == Examples == | ||
An example for a system that is not consistent is [[25edo]]: | An example for a system that is //not// consistent in a particular odd limit is [[25edo]]: | ||
The best approximation for the interval of [[7_6|7/6]] (the septimal subminor third) in 25edo is 6 steps, the best approximation for the [[3_2|perfect fifth 3/2]] is 15 steps. | The best approximation for the interval of [[7_6|7/6]] (the septimal subminor third) in 25edo is 6 steps, the best approximation for the [[3_2|perfect fifth 3/2]] is 15 steps. | ||
Adding the two just intervals gives 3/2 * 7/6 = [[7_4|7/4]], the harmonic seventh, for which the best approximation in 25edo is 20 steps. Adding the two approximated intervals, however, gives 21 steps. This means that 25edo is not consistent in [[7-limit]]. | Adding the two just intervals gives 3/2 * 7/6 = [[7_4|7/4]], the harmonic seventh, for which the best approximation in 25edo is 20 steps. Adding the two approximated intervals, however, gives 21 steps. This means that 25edo is not consistent in [[7-limit]]. | ||
An example for a system that is consistent in the 3-limit is [[12edo]]: the (up to 12) multiples of the just fifth ([[3_2|3:2]]) are consistently approximated by the 12-edo steps. | An example for a system that //is// consistent in the 3-limit is [[12edo]]: the (up to 12) multiples of the just fifth ([[3_2|3:2]]) are consistently approximated by the 12-edo steps. | ||
== Links == | == Links == | ||
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<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Examples"></a><!-- ws:end:WikiTextHeadingRule:0 --> Examples </h2> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Examples"></a><!-- ws:end:WikiTextHeadingRule:0 --> Examples </h2> | ||
An example for a system that is not consistent is <a class="wiki_link" href="/25edo">25edo</a>:<br /> | An example for a system that is <em>not</em> consistent in a particular odd limit is <a class="wiki_link" href="/25edo">25edo</a>:<br /> | ||
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The best approximation for the interval of <a class="wiki_link" href="/7_6">7/6</a> (the septimal subminor third) in 25edo is 6 steps, the best approximation for the <a class="wiki_link" href="/3_2">perfect fifth 3/2</a> is 15 steps.<br /> | The best approximation for the interval of <a class="wiki_link" href="/7_6">7/6</a> (the septimal subminor third) in 25edo is 6 steps, the best approximation for the <a class="wiki_link" href="/3_2">perfect fifth 3/2</a> is 15 steps.<br /> | ||
Adding the two just intervals gives 3/2 * 7/6 = <a class="wiki_link" href="/7_4">7/4</a>, the harmonic seventh, for which the best approximation in 25edo is 20 steps. Adding the two approximated intervals, however, gives 21 steps. This means that 25edo is not consistent in <a class="wiki_link" href="/7-limit">7-limit</a>.<br /> | Adding the two just intervals gives 3/2 * 7/6 = <a class="wiki_link" href="/7_4">7/4</a>, the harmonic seventh, for which the best approximation in 25edo is 20 steps. Adding the two approximated intervals, however, gives 21 steps. This means that 25edo is not consistent in <a class="wiki_link" href="/7-limit">7-limit</a>.<br /> | ||
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An example for a system that is consistent in the 3-limit is <a class="wiki_link" href="/12edo">12edo</a>: the (up to 12) multiples of the just fifth (<a class="wiki_link" href="/3_2">3:2</a>) are consistently approximated by the 12-edo steps.<br /> | An example for a system that <em>is</em> consistent in the 3-limit is <a class="wiki_link" href="/12edo">12edo</a>: the (up to 12) multiples of the just fifth (<a class="wiki_link" href="/3_2">3:2</a>) are consistently approximated by the 12-edo steps.<br /> | ||
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<!-- 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_ext" href="http://www.tonalsoft.com/enc/c/consistent.aspx" rel="nofollow">consistent (TonalSoft encyclopedia)</a></body></html></pre></div> | <a class="wiki_link_ext" href="http://www.tonalsoft.com/enc/c/consistent.aspx" rel="nofollow">consistent (TonalSoft encyclopedia)</a></body></html></pre></div> | ||