Consistency: Difference between revisions
Wikispaces>xenwolf **Imported revision 239284579 - Original comment: ** |
Wikispaces>hstraub **Imported revision 555943795 - 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:hstraub|hstraub]] and made on <tt>2015-07-30 03:29:15 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>555943795</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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
<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">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//. | <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"><span style="display: block; text-align: right;">[[一貫性|日本語]] | ||
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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 == | ==Examples== | ||
An example for a system that is //not// consistent in a particular odd limit is [[25edo]]: | An example for a system that is //not// consistent in a particular odd limit is [[25edo]]: | ||
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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. | ||
== Links == | ==Links== | ||
[[http://www.tonalsoft.com/enc/c/consistent.aspx|consistent (TonalSoft encyclopedia)]] | [[http://www.tonalsoft.com/enc/c/consistent.aspx|consistent (TonalSoft encyclopedia)]]</pre></div> | ||
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<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<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>consistent</title></head><body>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 /> | <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>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 /> | ||
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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 /> | |||
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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 <em>not</em> consistent in a particular odd limit 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 /> | ||
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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: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> | ||