Consistency: Difference between revisions
Wikispaces>hstraub **Imported revision 555943795 - Original comment: ** |
Wikispaces>MasonGreen1 **Imported revision 570782225 - 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:MasonGreen1|MasonGreen1]] and made on <tt>2015-12-26 00:45:35 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>570782225</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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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. | ||
==Generalization== | |||
It is possible to generalize the concept of consistency to non-edo equal temperaments. Because octaves are no longer equivalent, instead of an odd limit we must use an integer limit, and the term 2^n in the above equation is no longer present. Instead, the set S consists of all intervals u/v where u <= q >= v. | |||
This also means that the concept of octave inversion no longer applies: in this example, 13:9 is in S, but 18:13 is not. | |||
==Links== | ==Links== | ||
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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 /> | 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: | <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-Generalization"></a><!-- ws:end:WikiTextHeadingRule:2 -->Generalization</h2> | ||
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It is possible to generalize the concept of consistency to non-edo equal temperaments. Because octaves are no longer equivalent, instead of an odd limit we must use an integer limit, and the term 2^n in the above equation is no longer present. Instead, the set S consists of all intervals u/v where u &lt;= q &gt;= v.<br /> | |||
<br /> | |||
This also means that the concept of octave inversion no longer applies: in this example, 13:9 is in S, but 18:13 is not. <br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-Links"></a><!-- ws:end:WikiTextHeadingRule:4 -->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> | ||