Kees semi-height: Difference between revisions

Line 1:

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

Given a ratio of positive integers p/q, the ''Kees [[Height|height]]'' is found by first removing factors of two and all common factors from p/q, producing a ratio a/b of relatively prime odd positive integers. Then kees(p/q) = kees(a/b) = max(a, b). The Kees "expressibility" is then the logarithm base two of the Kees height.

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

~~: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2016-03-06 08:48:32 UTC</tt>.<br>~~

~~: The original revision id was <tt>576689845</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 a ratio of positive integers p/q, the //Kees [[height]]// is found by first removing factors of two and all common factors from p/q, producing a ratio a/b of relatively prime odd positive integers. Then kees(p/q) = kees(a/b) = max(a, b). The Kees "expressibility" is then the logarithm base two of the Kees height.

Expressibility can be extended to all vectors in [[Monzos_and_Interval_Space|interval space]], by means of the formula KE(|m2 m3 m5... mp>) = (|m3 + m5+ ... +mp| + |m3| + |m5| + ... + |mp|)/2, where "KE" denotes Kees expressibility and |m2 m3 m5 ... mp> is a vector with weighted coordinates in interval space. It can also be thought of as the quotient norm of Weil height, mod 2/1. Additionally, it can<span style="line-height: 1.5;"> be extended to tempered intervals using the quotient norm.</span>

~~Expressibility can be extended~~ to ~~all vectors in~~ [[~~Monzos and Interval Space~~|~~interval space~~]], by means of the formula KE(|m2 m3 m5... mp>) = (|m3 + m5+ ... +mp| + |m3| + |m5| + ... + |mp|)/2, where "KE" denotes Kees expressibility and |m2 m3 m5 ... mp> is a vector with weighted coordinates in interval space. It can also be thought of as the quotient norm of Weil height, mod 2/1. Additionally, it can<span style="line-height: 1.5;"> be extended to tempered intervals using the quotient norm.~~</span>~~

The set of JI intervals with Kees height less than or equal to an odd integer q comprises the [[Odd_limit|q odd limit]].

The ~~set~~ of ~~JI intervals with~~ Kees height ~~less than or equal~~ to ~~an odd integer q comprises the~~ [[~~Odd limit~~|~~q odd limit~~]].

The point of Kees height is to serve as a metric/height on [[Pitch_class|JI pitch classes]] corresponding to [[Benedetti_height|Benedetti height]] on pitches. The measure was proposed by [[Kees_van_Prooijen|Kees van Prooijen]].

~~The point of Kees height is to serve as a metric~~/~~height on [[Pitch class|JI pitch classes]] corresponding to [[Benedetti height]] on pitches~~. ~~The measure was proposed by [[~~Kees ~~van Prooijen]~~].

[http://www.kees.cc/tuning/perbl.html Kees tuning pages]

~~[[http://www.kees.cc/tuning/perbl.html|Kees tuning pages]]~~

==Examples==

~~==Examples==~~

{| class="wikitable"

~~||= **intervals** ||= **kees height** ||~~

|-

~~||= 7/4, 7/5, 7/6, 8/7 ||= 7 ||~~

| style="text-align:center;" | '''intervals'''

~~||= 5/3, 8/5, 5/4, 6/5 ||= 5~~ ||

| style="text-align:center;" | '''kees height'''

~~||= 4/3, 3/2 ||= 3 ||~~

|-

~~||= 2/1 ||= 1 ||</pre></div>~~

| style="text-align:center;" | 7/4, 7/5, 7/6, 8/7

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

| style="text-align:center;" | 7

<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>Kees Height</title></head><body>Given a ratio of positive integers p/q, the <em>Kees <a class="~~wiki_link~~" href="/height">height</a></em> is found by first removing factors of two and all common factors from p/q, producing a ratio a/b of relatively prime odd positive integers. Then kees(p/q) = kees(a/b) = max(a, b). The Kees &quot;expressibility&quot; is then the logarithm base two of the Kees height.<br />

|-

~~<br />~~

| style="text-align:center;" | 5/3, 8/5, 5/4, 6/5

~~<br />~~

| style="text-align:center;" | 5

~~Expressibility can be extended to all vectors in <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">interval space</a>, by means of the formula KE(~~|m2 m3 m5... mp&gt;) = (|m3 + m5+ ... +mp| + |m3| + |m5| + ... + |mp|)/2, where &quot;KE&quot; denotes Kees expressibility and |m2 m3 m5 ... mp&gt; is a vector with weighted coordinates in interval space. It can also be thought of as the quotient norm of Weil height, mod 2/1. Additionally, it can<span style="line-~~height: 1.5;"> be extended to tempered intervals using the quotient norm.</span><br />~~

|-

~~<br />~~

| style="text-align:center;" | 4/3, 3/2

~~The set of JI intervals with Kees height less than or equal to an odd integer q comprises the <a class="wiki_link" href="/Odd%20limit">q odd limit</a>.<br />~~

| style="text-align:center;" | 3

~~<br />~~

|-

The point of Kees height is to serve as a metric/height on <a class="wiki_link" href="/Pitch%20class">JI pitch classes</a> corresponding to <a class="wiki_link" href="/Benedetti%20height">Benedetti height</a> on pitches. The measure was proposed by <a class="wiki_link" href="/Kees%20van%20Prooijen">Kees van Prooijen</a>.<br />

| style="text-align:center;" | 2/1

~~<br />~~

| style="text-align:center;" | 1

~~<a class="wiki_link_ext" href="http://www.kees.cc/tuning/perbl.html" rel="nofollow">Kees tuning pages</a><br />~~

|}

~~<br />~~

[[Category:definition]]

~~<h2 id="toc0"><a name="x-Examples"></a>Examples</h2>~~

[[Category:height]]

[[Category:ji]]

[[Category:measure]]

~~<table class="wiki_table">~~

[[Category:psychoacoustics]]

~~<tr>~~

[[Category:sonance]]

~~<td~~ style="text-align: center;"~~><strong>~~intervals~~</strong><br />~~

[[Category:sonance-measure]]

~~</td>~~

[[Category:theory]]

~~<td~~ style="text-align: center;"~~><strong>~~kees height~~</strong><br />~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td~~ style="text-align: center;"~~>~~7/4, 7/5, 7/6, 8/7~~<br />~~

~~</td>~~

~~<td~~ style="text-align: center;"~~>~~7~~<br />~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td~~ style="text-align: center;"~~>~~5/3, 8/5, 5/4, 6/5~~<br />~~

~~</td>~~

~~<td~~ style="text-align: center;"~~>~~5~~<br />~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td~~ style="text-align: center;"~~>~~4/3, 3/2~~<br />~~

~~</td>~~

~~<td~~ style="text-align: center;"~~>~~3~~<br />~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td~~ style="text-align: center;"~~>~~2/1~~<br />~~

~~</td>~~

~~<td~~ style="text-align: center;"~~>~~1~~<br />~~

~~</td>~~

~~</tr>~~

~~</table>~~

~~</body></html></pre></div>~~

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+Given a ratio of positive integers p/q, the ''Kees [[Height|height]]''  is found by first removing factors of two and all common factors from p/q, producing a ratio a/b of relatively prime odd positive integers. Then kees(p/q) = kees(a/b) = max(a, b). The Kees "expressibility" is then the logarithm base two of the Kees height.
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2016-03-06 08:48:32 UTC</tt>.<br>
-: The original revision id was <tt>576689845</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 a ratio of positive integers p/q, the //Kees [[height]]//  is found by first removing factors of two and all common factors from p/q, producing a ratio a/b of relatively prime odd positive integers. Then kees(p/q) = kees(a/b) = max(a, b). The Kees "expressibility" is then the logarithm base two of the Kees height.
+Expressibility can be extended to all vectors in [[Monzos_and_Interval_Space|interval space]], by means of the formula KE(|m2 m3 m5... mp&gt;) = (|m3 + m5+ ... +mp| + |m3| + |m5| + ... + |mp|)/2, where "KE" denotes Kees expressibility and |m2 m3 m5 ... mp&gt; is a vector with weighted coordinates in interval space. It can also be thought of as the quotient norm of Weil height, mod 2/1. Additionally, it can<span style="line-height: 1.5;"> be extended to tempered intervals using the quotient norm.</span>
-Expressibility can be extended to all vectors in [[Monzos and Interval Space|interval space]], by means of the formula KE(|m2 m3 m5... mp&gt;) = (|m3 + m5+ ... +mp| + |m3| + |m5| + ... + |mp|)/2, where "KE" denotes Kees expressibility and |m2 m3 m5 ... mp&gt; is a vector with weighted coordinates in interval space. It can also be thought of as the quotient norm of Weil height, mod 2/1. Additionally, it can&lt;span style="line-height: 1.5;"&gt; be extended to tempered intervals using the quotient norm.&lt;/span&gt;
+The set of JI intervals with Kees height less than or equal to an odd integer q comprises the [[Odd_limit|q odd limit]].
-The set of JI intervals with Kees height less than or equal to an odd integer q comprises the [[Odd limit|q odd limit]].
+The point of Kees height is to serve as a metric/height on [[Pitch_class|JI pitch classes]] corresponding to [[Benedetti_height|Benedetti height]] on pitches. The measure was proposed by [[Kees_van_Prooijen|Kees van Prooijen]].
-The point of Kees height is to serve as a metric/height on [[Pitch class|JI pitch classes]] corresponding to [[Benedetti height]] on pitches. The measure was proposed by [[Kees van Prooijen]].
+[http://www.kees.cc/tuning/perbl.html Kees tuning pages]
-[[http://www.kees.cc/tuning/perbl.html|Kees tuning pages]]
+==Examples==
-==Examples==
+{| class="wikitable"
-||= **intervals** ||= **kees height** ||
+|-
-||= 7/4, 7/5, 7/6, 8/7 ||= 7 ||
+| style="text-align:center;" | '''intervals'''
-||= 5/3, 8/5, 5/4, 6/5 ||= 5 ||
+| style="text-align:center;" | '''kees height'''
-||= 4/3, 3/2 ||= 3 ||
+|-
-||= 2/1 ||= 1 ||</pre></div>
+| style="text-align:center;" | 7/4, 7/5, 7/6, 8/7
-<h4>Original HTML content:</h4>
+| style="text-align:center;" | 7
-<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;Kees Height&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Given a ratio of positive integers p/q, the &lt;em&gt;Kees &lt;a class="wiki_link" href="/height"&gt;height&lt;/a&gt;&lt;/em&gt;  is found by first removing factors of two and all common factors from p/q, producing a ratio a/b of relatively prime odd positive integers. Then kees(p/q) = kees(a/b) = max(a, b). The Kees &amp;quot;expressibility&amp;quot; is then the logarithm base two of the Kees height.&lt;br /&gt;
+|-
-&lt;br /&gt;
+| style="text-align:center;" | 5/3, 8/5, 5/4, 6/5
-&lt;br /&gt;
+| style="text-align:center;" | 5
-Expressibility can be extended to all vectors in &lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;interval space&lt;/a&gt;, by means of the formula KE(|m2 m3 m5... mp&amp;gt;) = (|m3 + m5+ ... +mp| + |m3| + |m5| + ... + |mp|)/2, where &amp;quot;KE&amp;quot; denotes Kees expressibility and |m2 m3 m5 ... mp&amp;gt; is a vector with weighted coordinates in interval space. It can also be thought of as the quotient norm of Weil height, mod 2/1. Additionally, it can&lt;span style="line-height: 1.5;"&gt; be extended to tempered intervals using the quotient norm.&lt;/span&gt;&lt;br /&gt;
+|-
-&lt;br /&gt;
+| style="text-align:center;" | 4/3, 3/2
-The set of JI intervals with Kees height less than or equal to an odd integer q comprises the &lt;a class="wiki_link" href="/Odd%20limit"&gt;q odd limit&lt;/a&gt;.&lt;br /&gt;
+| style="text-align:center;" | 3
-&lt;br /&gt;
+|-
-The point of Kees height is to serve as a metric/height on &lt;a class="wiki_link" href="/Pitch%20class"&gt;JI pitch classes&lt;/a&gt; corresponding to &lt;a class="wiki_link" href="/Benedetti%20height"&gt;Benedetti height&lt;/a&gt; on pitches. The measure was proposed by &lt;a class="wiki_link" href="/Kees%20van%20Prooijen"&gt;Kees van Prooijen&lt;/a&gt;.&lt;br /&gt;
+| style="text-align:center;" | 2/1
-&lt;br /&gt;
+| style="text-align:center;" | 1
-&lt;a class="wiki_link_ext" href="http://www.kees.cc/tuning/perbl.html" rel="nofollow"&gt;Kees tuning pages&lt;/a&gt;&lt;br /&gt;
+|}
-&lt;br /&gt;
+[[Category:definition]]
-&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Examples&lt;/h2&gt;
+[[Category:height]]
+[[Category:ji]]
+[[Category:measure]]
-&lt;table class="wiki_table"&gt;
+[[Category:psychoacoustics]]
-    &lt;tr&gt;
+[[Category:sonance]]
-        &lt;td style="text-align: center;"&gt;&lt;strong&gt;intervals&lt;/strong&gt;&lt;br /&gt;
+[[Category:sonance-measure]]
-&lt;/td&gt;
+[[Category:theory]]
-        &lt;td style="text-align: center;"&gt;&lt;strong&gt;kees height&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;7/4, 7/5, 7/6, 8/7&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;7&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;5/3, 8/5, 5/4, 6/5&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;5&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;4/3, 3/2&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;3&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;2/1&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;1&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-&lt;/table&gt;
-&lt;/body&gt;&lt;/html&gt;</pre></div>