Consistency: Difference between revisions
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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> | ||
: 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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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> | ||
<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 /> | |||
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> | ||
Revision as of 00:45, 26 December 2015
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author MasonGreen1 and made on 2015-12-26 00:45:35 UTC.
- The original revision id was 570782225.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
<span style="display: block; text-align: right;">[[一貫性|日本語]] </span> If N-edo is an [[edo|equal division of the octave]], and if for any interval r, N(r) is the best N-edo approximation to r, then N is //consistent// with respect to a set of intervals S if for any two intervals a and b in S where ab is also in S, N(ab) = N(a) + N(b). Normally this is considered when S is the set of [[Odd limit|q odd limit intervals]], consisting of everything of the form 2^n u/v, where u and v are odd integers less than or equal to q. N is then said to be //q limit consistent//. If each interval in the q-limit is mapped to a unique value by N, then it said to be //uniquely q limit consistent//. ==Examples== 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. 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. ==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== [[http://www.tonalsoft.com/enc/c/consistent.aspx|consistent (TonalSoft encyclopedia)]]
Original HTML content:
<html><head><title>consistent</title></head><body><span style="display: block; text-align: right;"><a class="wiki_link" href="/%E4%B8%80%E8%B2%AB%E6%80%A7">日本語</a><br /> </span><br /> If N-edo is an <a class="wiki_link" href="/edo">equal division of the octave</a>, and if for any interval r, N(r) is the best N-edo approximation to r, then N is <em>consistent</em> with respect to a set of intervals S if for any two intervals a and b in S where ab is also in S, N(ab) = N(a) + N(b). Normally this is considered when S is the set of <a class="wiki_link" href="/Odd%20limit">q odd limit intervals</a>, consisting of everything of the form 2^n u/v, where u and v are odd integers less than or equal to q. N is then said to be <em>q limit consistent</em>. If each interval in the q-limit is mapped to a unique value by N, then it said to be <em>uniquely q limit consistent</em>.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-Examples"></a><!-- ws:end:WikiTextHeadingRule:0 -->Examples</h2> 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 /> <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 /> <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 /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="x-Generalization"></a><!-- ws:end:WikiTextHeadingRule:2 -->Generalization</h2> <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 <= q >= 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:<h2> --><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>