Pepper ambiguity: Difference between revisions

Given an [[EDO|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|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|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/A117554], [https://oeis.org/A117555 https://oeis.org/A117555], [https://oeis.org/A117556 https://oeis.org/A117556], [https://oeis.org/A117557 https://oeis.org/A117557], [https://oeis.org/A117558 https://oeis.org/A117558] and [https://oeis.org/A117559 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.