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Kees semi-height: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
Given a [[ratio]] of positive integers ''p''/''q'', the '''Kees semi-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 semi-height.
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>2013-12-30 10:42:25 UTC</tt>.<br>
: The original revision id was <tt>479870430</tt>.<br>
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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. 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]] on pitches. The measure was proposed by [[Kees van Prooijen]].
Expressibility can be extended to all vectors in [[Monzos and interval space|interval space]], by means of the formula


[[http://www.kees.cc/tuning/perbl.html|Kees tuning pages]]
<math> \lVert |m_2 \, m_3 \, m_5 \ldots m_p \rangle \rVert_{K1} = (|m_3 + m_5 + ... + m_p| + |m_3| + |m_5| + ... + |m_p|)/2</math>
 
where "K1" denotes Kees expressibility and {{monzo| ''m''<sub>2</sub> ''m''<sub>3</sub> ''m''<sub>5</sub> … ''m''<sub>p</sub> }} is a vector with weighted coordinates in interval space.
 
The set of JI intervals with Kees semi-height less than or equal to an odd integer q comprises the [[Odd limit|''q''-odd-limit]].
 
The Kees semi-height is only a semi-height function, rather than a true [[height]] function, because the set of all ratios with less than some Kees semi-height is infinite and unbounded. Thus it is only a seminorm (or a "semimetric," sometimes called "pseudometric") on the space of JI intervals. However, if one looks at it as a function bounding sets of octave-equivalent [[Pitch class|JI pitch classes]], then there are only finitely many pitch classes with less than some specified Kees expressibility, making it sort of a height function on these "generalized rationals" which are octave equivalent.
 
In linear-algebraic terms, the Kees expressibility is a [[wikipedia: Seminorm|seminorm]] rather than a true norm; because the distance between two different intervals can be zero (if they are simply octave transpositions of one another). However, if one looks at the space of octave-equivalent intervals, which can be kind of thought of as "tempering" 2/1 as a "comma" and looking at the resulting equivalence classes, the Kees expressibility is a true norm on this space. The Kees expressibility can also be thought of as the quotient norm of Weil height mod 2/1. Additionally, it can be extended to tempered intervals using the quotient norm mod additional commas as a form of [[temperamental complexity]].
 
The Kees semi-height is often used as a "default" measure of complexity for octave-equivalent pitch classes, similarly to the use of [[Benedetti height]] on pitches (although the Kees semi-height is not the same as "octave-equivalent Benedetti height", though it is related in a different way).
 
The use of max(''a'', ''b'') as a complexity function, with or without octave equivalence, is very old; according to [[Paul Erlich]], it may date back even to the Renaissance. In the 20th century the octave-equivalent version was used by [[Harry Partch]], among others. The metric (and particularly the logarithmic version) has since become associated with [[Kees van Prooijen]], who studied extensively its properties as a norm on the space of pitch classes.


== Examples ==
== Examples ==
||= **interval** ||= **kees height** ||
||= 5/3 ||= 5 ||
||= 4/3 ||= 3 ||
||= 2/1 ||= 1 ||</pre></div>
<h4>Original HTML 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;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 height&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. 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;
&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;
&lt;br /&gt;
&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;
&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;


{| class="wikitable center-all"
|-
! Intervals
! Kees Height
! Deduction Steps
|-
| 7/4
| 7
| 7/4 &rarr;  7/1; max(7, 1) &rarr; 7
|-
| 7/5
| 7
| max(7, 5) &rarr; 7
|-
| 7/6
| 7
| 7/6 &rarr; 7/3; max(7, 3) &rarr; 7
|-
| 8/7
| 7
| 8/7 &rarr; 1/7; max(1, 7) &rarr; 7
|-
| 5/3
| 5
| max(3, 5) &rarr; 5
|-
| 8/5
| 5
| 8/5 &rarr; 1/5; max(1, 5) &rarr; 5
|-
| 5/4
| 5
| 5/4 &rarr; 5/1; max (5, 1) &rarr; 5
|-
| 6/5
| 5
| 6/5 &rarr; 3/5; max(3, 5) &rarr; 5
|-
| 4/3
| 3
| 4/3 &rarr; 1/3; max(1, 3) &rarr; 3
|-
| 3/2
| 3
| 3/2 &rarr; 3/1; max(3, 1) &rarr; 3
|-
| 2/1
| 1
| 2/1 &rarr; 1/1; max(1, 1) &rarr; 1
|-
| 9/5
| 9
| max(9, 5) &rarr; 9
|-
| 10/9
| 9
| 10/9 &rarr; 5/9; max(5, 9) &rarr; 9
|-
| 15/14
| 15
| 15/14 &rarr; 15/7; max(15, 7) &rarr; 15
|-
| 28/15
| 15
| 28/15 &rarr; 7/15; max(7, 15) &rarr; 15
|-
| 25/26
| 25
| 25/26 &rarr; 25/13; max(25, 13) &rarr; 25
|-
| 27/25
| 27
| max(27, 25) &rarr; 27
|-
| 25/24
| 25
| 25/24 &rarr; 25/3; max(25, 3) &rarr; 25
|}


&lt;table class="wiki_table"&gt;
== External links ==
    &lt;tr&gt;
* [http://www.kees.cc/tuning/perbl.html Kees tuning pages]
        &lt;td style="text-align: center;"&gt;&lt;strong&gt;interval&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &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;5/3&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&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>
[[Category:Terms]]
[[Category:Interval complexity measures]]
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