Consistency: Difference between revisions
Wikispaces>MasonGreen1 **Imported revision 570782225 - Original comment: ** |
Wikispaces>MasonGreen1 **Imported revision 570782573 - 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:MasonGreen1|MasonGreen1]] and made on <tt>2015-12-26 | : This revision was by author [[User:MasonGreen1|MasonGreen1]] and made on <tt>2015-12-26 02:20:16 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>570782573</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> | ||
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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 <= q >= v. | 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. | 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. | ||
One notable example: [[46edo]] is not consistent in the 15 odd limit. The 15:13 interval is very closer slightly closer to 9 degrees of 46edo than to 10 degrees, but the //functional// 15:13 (the difference between 46edo's versions of 15:8 and 13:8) is 10 degrees. However, if we compress the octave slightly (by about a cent), this discrepancy no longer occurs, and we end up with an 18-//integer//-limit consistent system, which makes it ideal for approximating mode 8 of the harmonic series. | |||
==Links== | ==Links== | ||
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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 /> | 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 /> | <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 /> | 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 /> | ||
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One notable example: <a class="wiki_link" href="/46edo">46edo</a> is not consistent in the 15 odd limit. The 15:13 interval is very closer slightly closer to 9 degrees of 46edo than to 10 degrees, but the <em>functional</em> 15:13 (the difference between 46edo's versions of 15:8 and 13:8) is 10 degrees. However, if we compress the octave slightly (by about a cent), this discrepancy no longer occurs, and we end up with an 18-<em>integer</em>-limit consistent system, which makes it ideal for approximating mode 8 of the harmonic series.<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-Links"></a><!-- ws:end:WikiTextHeadingRule:4 -->Links</h2> | <!-- 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> | ||