Pepper ambiguity: Difference between revisions

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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 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|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.
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]]
 
{{todo|improve synopsis|text=add a non-mathy paragraph at the start}}

Latest revision as of 10:55, 13 April 2025

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 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:

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.

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

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