# Pepper ambiguity

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* log_{2}(*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 cents 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:

- OEIS: A117554 — 5-odd-limit
- OEIS: A117555 — 7-odd-limit
- OEIS: A117556 — 9-odd-limit
- OEIS: A117557 — 11-odd-limit
- OEIS: A117558 — 13-odd-limit
- OEIS: A117559 — 15-odd-limit

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*.