View source for 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<sub>2</sub>(''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]]: 

* [[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''.
{| class="wikitable"
|+
!odd-limit
!list of EDOs with decreasing relative error
|-
|1
|None
|-
|3
|1, 2, 5, 12, 41, 53, 306, 665, 15601, 31867, 79335, 111202, 190537
|-
|5
|1, 3, 12, 19, 34, 53, 118, 441, 612, 730, 1171, 1783, 2513, 4296, 25164, 52841, 73709, 78005, 229719
|-
|7
|1, 2, 3, 4, 12, 22, 27, 31, 99, 171, 3125, 6691, 11664, 18355, 84814, 103169
|-
|9
|1, 2, 4, 5, 12, 19, 31, 41, 99, 171, 3125, 11664, 18355, 84814, 103169
|-
|11
|1, 2, 5, 16, 22, 31, 72, 270, 342, 1848, 6421, 6691, 14618, 26894, 40006, 54624, 121524, 258008, 903475
|-
|13
|1, 2, 7, 8, 24, 37, 46, 58, 130, 198, 224, 270, 494, 1506, 2684, 5585, 6079, 14618, 20203, 81860, 87939, 96478, 161530, 258008
|-
|15
|1, 2, 7, 8, 24, 58, 111, 130, 224, 270, 494, 2190, 2684, 5585, 6079, 14618, 20203, 81860, 96478, 161530, 258008
|-
|17
|1, 2, 8, 24, 72, 94, 111, 311, 581, 764, 1506, 2460, 3395, 7033, 14348, 16808, 20203, 102557, 419538
|-
|19
|1, 2, 3, 7, 8, 24, 311, 581, 1178, 1578, 2000, 3395, 8539, 16808, 20203, 360565, 419538
|-
|21
|1, 2, 3, 8, 24, 54, 72, 118, 311, 581, 1178, 1578, 2460, 3395, 8539, 16808, 20203, 360565, 419538
|-
|23
|1, 2, 3, 8, 9, 10, 54, 175, 311, 1578, 2460, 10028, 16808, 58973, 360565, 419538, 937060
|-
|25
|1, 2, 3, 8, 9, 10, 31, 55, 68, 175, 311, 1578, 16808, 58973, 360565, 419538
|-
|27
|1, 2, 3, 8, 9, 10, 31, 55, 68, 152, 183, 422, 526, 1578, 16808, 58973, 360565, 419538
|-
|29
|1, 2, 3, 9, 11, 18, 31, 55, 94, 170, 183, 436, 526, 1578, 15112, 16808, 360565
|-
|31
|1, 2, 3, 9, 16, 18, 31, 55, 129, 147, 183, 279, 436, 3513, 4349, 6850, 9934, 15112, 16808
|}

== See also ==
* [[Relative error]]

[[Category:EDO theory pages]]
[[Category:Terms]]

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