Pepper ambiguity: Difference between revisions

(8 intermediate revisions by 5 users not shown)

Line 1:

~~<h2>IMPORTED REVISION FROM WIKISPACES</h2>~~

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.

~~This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>~~

~~: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2018-01-10 14:53:50 UTC</tt>.<br>~~

~~: The original revision id was <tt>624694375</tt>.<br>~~

~~: The revision comment was: <tt></tt><br>~~

~~The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>~~

~~<h4>Original Wikitext content:</h4>~~

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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.

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]]:

~~</pre></div>~~

~~<h4>Original HTML content:</h4>~~

* [[OEIS: A117554]] — 5-odd-limit

~~<div style~~="~~width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em~~"~~><pre style="margin:0px;border:none;background:none;word~~-~~wrap:break-word;width:200%;white-space: pre-wrap~~ ! ~~important" class="old~~-~~revision~~-~~html"><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 />~~

* [[OEIS: A117555]] — 7-odd-limit

~~<br />~~

* [[OEIS: A117556]] — 9-odd-limit

~~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~~, ~~<a class="wiki_link_ext" href="https://oeis.org/A117555" rel="nofollow">https://oeis.org/A117555</a>, ~~<a class~~=~~"wiki_link_ext" href~~=~~"https://oeis.org/A117558" rel~~=~~"nofollow">https~~:~~//oeis.org/A117558</a> and <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></pre></div>

* [[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}}

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+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.
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2018-01-10 14:53:50 UTC</tt>.<br>
-: The original revision id was <tt>624694375</tt>.<br>
-: The revision comment was: <tt></tt><br>
-The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
-<h4>Original Wikitext content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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.
+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]]:
-	</pre></div>
-<h4>Original HTML content:</h4>
+* [[OEIS: A117554]] — 5-odd-limit
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Pepper ambiguity&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Given an &lt;a class="wiki_link" href="/edo"&gt;edo&lt;/a&gt; N and a positive rational number q, we may define the &lt;em&gt;ambiguity&lt;/em&gt; 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 &lt;a class="wiki_link" href="/relative%20cent"&gt;relative cent&lt;/a&gt;s instead to measure relative error, we would get the same result.&lt;br /&gt;
+* [[OEIS: A117555]] — 7-odd-limit
-&lt;br /&gt;
+* [[OEIS: A117556]] — 9-odd-limit
-Given a finite set s of positive rational numbers, the maximum value of ambig(N, q) for all q∈s is the &lt;em&gt;Pepper ambiguity&lt;/em&gt; of N with respect to s. If the set s is the L odd limit &lt;a class="wiki_link" href="/tonality%20diamond"&gt;tonality diamond&lt;/a&gt;, 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, &lt;!-- ws:start:WikiTextUrlRule:5:https://oeis.org/A117554 --&gt;&lt;a class="wiki_link_ext" href="https://oeis.org/A117554" rel="nofollow"&gt;https://oeis.org/A117554&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:5 --&gt;, &lt;!-- ws:start:WikiTextUrlRule:6:https://oeis.org/A117555 --&gt;&lt;a class="wiki_link_ext" href="https://oeis.org/A117555" rel="nofollow"&gt;https://oeis.org/A117555&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:6 --&gt;, &lt;!-- ws:start:WikiTextUrlRule:7:https://oeis.org/A117556 --&gt;&lt;a class="wiki_link_ext" href="https://oeis.org/A117556" rel="nofollow"&gt;https://oeis.org/A117556&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7 --&gt;, &lt;!-- ws:start:WikiTextUrlRule:8:https://oeis.org/A117557 --&gt;&lt;a class="wiki_link_ext" href="https://oeis.org/A117557" rel="nofollow"&gt;https://oeis.org/A117557&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:8 --&gt;, &lt;!-- ws:start:WikiTextUrlRule:9:https://oeis.org/A117558 --&gt;&lt;a class="wiki_link_ext" href="https://oeis.org/A117558" rel="nofollow"&gt;https://oeis.org/A117558&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:9 --&gt; and &lt;!-- ws:start:WikiTextUrlRule:10:https://oeis.org/A117559 --&gt;&lt;a class="wiki_link_ext" href="https://oeis.org/A117559" rel="nofollow"&gt;https://oeis.org/A117559&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:10 --&gt;. 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.&lt;/body&gt;&lt;/html&gt;</pre></div>
+* [[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}}