Pepper ambiguity
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- This revision was by author genewardsmith and made on 2018-01-10 14:53:50 UTC.
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Original Wikitext content:
Given an [[edo]] N and a positive rational number q, we may define the //ambiguity// ambig(N, q) of q in N edo by first computing u = N log2(q), and from there v = abs(u - round(u)). Then ambig(N, q) = v/(1-v). Since v is a measure of the relative error of q in is best approximation in N edo, and 1-v of its second best approximation, ambig(N, q) is the ratio of the best approximation to the second best. If we used [[relative cent]]s instead to measure relative error, we would get the same result. Given a finite set s of positive rational numbers, the maximum value of ambig(N, q) for all q∈s is the //Pepper ambiguity// of N with respect to s. If the set s is the L odd limit [[tonality diamond]], this is the L-limit Pepper ambiguity of N. Lists of N of decreasing Pepper ambiguity can be found on the On-Line Encyclopedia of Integer Sequences, https://oeis.org/A117554, https://oeis.org/A117555, https://oeis.org/A117556, https://oeis.org/A117557, https://oeis.org/A117558 and https://oeis.org/A117559. We may also define the mean ambiguity for N with respect to s by taking the mean of ambig(N, q) for all members q of s.
Original HTML content:
<html><head><title>Pepper ambiguity</title></head><body>Given an <a class="wiki_link" href="/edo">edo</a> N and a positive rational number q, we may define the <em>ambiguity</em> ambig(N, q) of q in N edo by first computing u = N log2(q), and from there v = abs(u - round(u)). Then ambig(N, q) = v/(1-v). Since v is a measure of the relative error of q in is best approximation in N edo, and 1-v of its second best approximation, ambig(N, q) is the ratio of the best approximation to the second best. If we used <a class="wiki_link" href="/relative%20cent">relative cent</a>s instead to measure relative error, we would get the same result.<br /> <br /> Given a finite set s of positive rational numbers, the maximum value of ambig(N, q) for all q∈s is the <em>Pepper ambiguity</em> of N with respect to s. If the set s is the L odd limit <a class="wiki_link" href="/tonality%20diamond">tonality diamond</a>, this is the L-limit Pepper ambiguity of N. Lists of N of decreasing Pepper ambiguity can be found on the On-Line Encyclopedia of Integer Sequences, <!-- ws:start:WikiTextUrlRule:5:https://oeis.org/A117554 --><a class="wiki_link_ext" href="https://oeis.org/A117554" rel="nofollow">https://oeis.org/A117554</a><!-- ws:end:WikiTextUrlRule:5 -->, <!-- ws:start:WikiTextUrlRule:6:https://oeis.org/A117555 --><a class="wiki_link_ext" href="https://oeis.org/A117555" rel="nofollow">https://oeis.org/A117555</a><!-- ws:end:WikiTextUrlRule:6 -->, <!-- ws:start:WikiTextUrlRule:7:https://oeis.org/A117556 --><a class="wiki_link_ext" href="https://oeis.org/A117556" rel="nofollow">https://oeis.org/A117556</a><!-- ws:end:WikiTextUrlRule:7 -->, <!-- ws:start:WikiTextUrlRule:8:https://oeis.org/A117557 --><a class="wiki_link_ext" href="https://oeis.org/A117557" rel="nofollow">https://oeis.org/A117557</a><!-- ws:end:WikiTextUrlRule:8 -->, <!-- ws:start:WikiTextUrlRule:9:https://oeis.org/A117558 --><a class="wiki_link_ext" href="https://oeis.org/A117558" rel="nofollow">https://oeis.org/A117558</a><!-- ws:end:WikiTextUrlRule:9 --> and <!-- ws:start:WikiTextUrlRule:10:https://oeis.org/A117559 --><a class="wiki_link_ext" href="https://oeis.org/A117559" rel="nofollow">https://oeis.org/A117559</a><!-- ws:end:WikiTextUrlRule:10 -->. We may also define the mean ambiguity for N with respect to s by taking the mean of ambig(N, q) for all members q of s.</body></html>