Consistency
IMPORTED REVISION FROM WIKISPACES
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- This revision was by author genewardsmith and made on 2011-06-28 16:51:26 UTC.
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Original Wikitext content:
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//. 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. 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)]]
Original HTML content:
<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 /> <br /> An example for a system that is not consistent 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 /> <a class="wiki_link_ext" href="http://www.tonalsoft.com/enc/c/consistent.aspx" rel="nofollow">consistent (TonalSoft encyclopedia)</a></body></html>