Consistency: Difference between revisions
Wikispaces>hstraub **Imported revision 238923473 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 239202843 - 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:genewardsmith|genewardsmith]] and made on <tt>2011-06-28 16:42:21 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>239202843</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">[[ | <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 [[equal division of the octave|edo]], 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, N(ab) = N(a) + N(b). Normally this is considered when S is the set of q [[Odd limit|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//. | ||
An example for a system that is not consistent is [[25edo]]: | An example for a system that is not consistent 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]].</pre></div> | 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]]. | ||
[[http://www.tonalsoft.com/enc/c/consistent.aspx|consistent (TonalSoft encyclopedia)]] | |||
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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><a class=" | <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="/equal%20division%20of%20the%20octave">edo</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, N(ab) = N(a) + N(b). Normally this is considered when S is the set of q <a class="wiki_link" href="/Odd%20limit">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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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 not consistent is <a class="wiki_link" href="/25edo">25edo</a>:<br /> | ||
<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 /> | 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>.</body></html></pre></div> | 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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<a class="wiki_link_ext" href="http://www.tonalsoft.com/enc/c/consistent.aspx" rel="nofollow">consistent (TonalSoft encyclopedia)</a></body></html></pre></div> | |||