Height: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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

: This revision was by author [[User:~~guest~~|~~guest~~]] and made on <tt>2012-09-~~07 03~~:49:45 UTC</tt>.<br>

: This revision was by author [[User:Sarzadoce|Sarzadoce]] and made on <tt>2012-09-08 23:32:31 UTC</tt>.<br>

: The original revision id was <tt>~~362749812~~</tt>.<br>

: The original revision id was <tt>363105768</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">=Definition:=

A **height** is a function on an abelian group which maps elements to real numbers, yielding a type of complexity measurement. Since the ~~(non-zero)~~ rationals form an abelian group under multiplication, we can assign each element a height, and hence a complexity. While there is no concensus on the restrictions of a height, we will attempt to create a definition which is practical for musical purposes.

A **height** is a function on an abelian group which maps elements to real numbers, yielding a type of complexity measurement. Since the positive rationals form an abelian group under multiplication, we can assign each element a height, and hence a complexity. While there is no concensus on the restrictions of a height, we will attempt to create a definition which is practical for musical purposes.

A height function H(q) on the rationals q should fulfill the following criteria:

A height function H(q) on the positive rationals q should fulfill the following criteria:

# Given any constant C, there are finitely many elements q such that H(q) <= C.

# There is a unique constant K such that H(q) >= K, for all q.

# H(q) = H(1/q)

# H(q^n) >= H(q) for any non-negative integer n

Since any rational q can be rewritten as a fraction n/d, we may sub this into the above equation to get H(n/d) = H(d/n). This relation is extremely useful - it tells us that we can switch n and d without any consequences on the outcome of the height.

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<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"><html><head><title>Height</title></head><body><h1 id="toc0"><a name="Definition:"></a>Definition:</h1>

A <strong>height</strong> is a function on an abelian group which maps elements to real numbers, yielding a type of complexity measurement. Since the ~~(non-zero)~~ rationals form an abelian group under multiplication, we can assign each element a height, and hence a complexity. While there is no concensus on the restrictions of a height, we will attempt to create a definition which is practical for musical purposes.<br />

A <strong>height</strong> is a function on an abelian group which maps elements to real numbers, yielding a type of complexity measurement. Since the positive rationals form an abelian group under multiplication, we can assign each element a height, and hence a complexity. While there is no concensus on the restrictions of a height, we will attempt to create a definition which is practical for musical purposes.<br />

<br />

A height function H(q) on the rationals q should fulfill the following criteria:<br />

A height function H(q) on the positive rationals q should fulfill the following criteria:<br />

<ol><li>Given any constant C, there are finitely many elements q such that H(q) &lt;= C.</li><li>There is a unique constant K such that H(q) &gt;= K, for all q.</li><li>H(q) = H(1/q)</li></ol><br />

<ol><li>Given any constant C, there are finitely many elements q such that H(q) &lt;= C.</li><li>There is a unique constant K such that H(q) &gt;= K, for all q.</li><li>H(q) = H(1/q)</li><li>H(q^n) &gt;= H(q) for any non-negative integer n</li></ol><br />

Since any rational q can be rewritten as a fraction n/d, we may sub this into the above equation to get H(n/d) = H(d/n). This relation is extremely useful - it tells us that we can switch n and d without any consequences on the outcome of the height.<br />

<br />

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:guest|guest]] and made on <tt>2012-09-07 03:49:45 UTC</tt>.<br>
+: This revision was by author [[User:Sarzadoce|Sarzadoce]] and made on <tt>2012-09-08 23:32:31 UTC</tt>.<br>
-: The original revision id was <tt>362749812</tt>.<br>
+: The original revision id was <tt>363105768</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">=Definition:=
-A **height** is a function on an abelian group which maps elements to real numbers, yielding a type of complexity measurement. Since the (non-zero) rationals form an abelian group under multiplication, we can assign each element a height, and hence a complexity. While there is no concensus on the restrictions of a height, we will attempt to create a definition which is practical for musical purposes.
+A **height** is a function on an abelian group which maps elements to real numbers, yielding a type of complexity measurement. Since the positive rationals form an abelian group under multiplication, we can assign each element a height, and hence a complexity. While there is no concensus on the restrictions of a height, we will attempt to create a definition which is practical for musical purposes.
-A height function H(q) on the rationals q should fulfill the following criteria:
+A height function H(q) on the positive rationals q should fulfill the following criteria:
 # Given any constant C, there are finitely many elements q such that H(q) &lt;= C.
 # There is a unique constant K such that H(q) &gt;= K, for all q.
 # H(q) = H(1/q)
+# H(q^n) &gt;= H(q) for any non-negative integer n
 Since any rational q can be rewritten as a fraction n/d, we may sub this into the above equation to get H(n/d) = H(d/n). This relation is extremely useful - it tells us that we can switch n and d without any consequences on the outcome of the height.
@@ Line 68: / Line 69: @@
 <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;Height&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Definition:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Definition:&lt;/h1&gt;
-  A &lt;strong&gt;height&lt;/strong&gt; is a function on an abelian group which maps elements to real numbers, yielding a type of complexity measurement. Since the (non-zero) rationals form an abelian group under multiplication, we can assign each element a height, and hence a complexity. While there is no concensus on the restrictions of a height, we will attempt to create a definition which is practical for musical purposes.&lt;br /&gt;
+  A &lt;strong&gt;height&lt;/strong&gt; is a function on an abelian group which maps elements to real numbers, yielding a type of complexity measurement. Since the positive rationals form an abelian group under multiplication, we can assign each element a height, and hence a complexity. While there is no concensus on the restrictions of a height, we will attempt to create a definition which is practical for musical purposes.&lt;br /&gt;
 &lt;br /&gt;
-A height function H(q) on the rationals q should fulfill the following criteria:&lt;br /&gt;
+A height function H(q) on the positive rationals q should fulfill the following criteria:&lt;br /&gt;
-&lt;ol&gt;&lt;li&gt;Given any constant C, there are finitely many elements q such that H(q) &amp;lt;= C.&lt;/li&gt;&lt;li&gt;There is a unique constant K such that H(q) &amp;gt;= K, for all q.&lt;/li&gt;&lt;li&gt;H(q) = H(1/q)&lt;/li&gt;&lt;/ol&gt;&lt;br /&gt;
+&lt;ol&gt;&lt;li&gt;Given any constant C, there are finitely many elements q such that H(q) &amp;lt;= C.&lt;/li&gt;&lt;li&gt;There is a unique constant K such that H(q) &amp;gt;= K, for all q.&lt;/li&gt;&lt;li&gt;H(q) = H(1/q)&lt;/li&gt;&lt;li&gt;H(q^n) &amp;gt;= H(q) for any non-negative integer n&lt;/li&gt;&lt;/ol&gt;&lt;br /&gt;
 Since any rational q can be rewritten as a fraction n/d, we may sub this into the above equation to get H(n/d) = H(d/n). This relation is extremely useful - it tells us that we can switch n and d without any consequences on the outcome of the height.&lt;br /&gt;
 &lt;br /&gt;