Height: Difference between revisions

Revision as of 18:34, 6 September 2012

IMPORTED REVISION FROM WIKISPACES

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

This revision was by author Sarzadoce and made on 2012-09-06 18:34:21 UTC.

The original revision id was 362662950.

The revision comment was:

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

Original Wikitext content:

=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 function H(q) on the 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)

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.

If we have a function F(x) which is strictly increasing on the positive reals, then F(H(q)) will rank elements in the same order as H(q). We can therefore establish the following equivalence relation:
[[math]]
H(q) \equiv F(H(q))
[[math]]

=Examples:= 
|| Name: || H(n/d) || H(q) || H(q) simplified by equivalence relation ||
|| Benedetti Height || [[math]]
nd
[[math]] || [[math]]
2^{T1(q)}
[[math]] || [[math]]
T1(q)
[[math]] ||
|| Weil Height || [[math]]
max(n,d)
[[math]] || [[math]]
2^{(T1(q)+|log_2(q)|)/2}
[[math]] || [[math]]
T1(q)+|log_2(q)|
[[math]] ||
|| ?? || [[math]]
n+d
[[math]] || [[math]]
2^{T1(q)/2} (q+1)/q^{1/2}
[[math]] || [[math]]
T1(q)+2log_2(q+1)-log_2(q)
[[math]] ||
|| Kees Height || [[math]]
max(2^{-v_2(n)}n,
2^{-v_2(d)}d)
[[math]] || [[math]]
2^{(T1(2^{-v_2(q)}q)+|log_2(q)-v_2(q)|)/2}
[[math]] || [[math]]

T1(2^{-v_2(q)}q)+|log_2(q)-v_2(q)|
[[math]] ||
||   ||   ||   ||   ||
Where T1(q) is the [[xenharmonic/Generalized Tenney Norms and Tp Interval Space#The%20Tenney%20Norm%20(T1%20norm)|tenney norm]] of q in monzo form, and vp(x) is the [[http://en.wikipedia.org/wiki/P-adic_order|p-adic valuation]] of x.

Some useful identities:
[[math]]
n=2^{(T1(q)\pm|log_2(q)|)/2}
[[math]]
[[math]]
d=2^{(T1(q)\mp|log_2(q)|)/2}
[[math]]
[[math]]
nd=2^{T1(q)}
[[math]]

Original HTML content:

<html><head><title>Height</title></head><body><!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc0"><a name="Definition:"></a><!-- ws:end:WikiTextHeadingRule:16 -->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 />
<br />
A height function H(q) on the 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 />
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 />
If we have a function F(x) which is strictly increasing on the positive reals, then F(H(q)) will rank elements in the same order as H(q). We can therefore establish the following equivalence relation:<br />
<!-- ws:start:WikiTextMathRule:0:
[[math]]&lt;br/&gt;
H(q) \equiv F(H(q))&lt;br/&gt;[[math]]
 --><script type="math/tex">H(q) \equiv F(H(q))</script><!-- ws:end:WikiTextMathRule:0 --><br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h1&gt; --><h1 id="toc1"><a name="Examples:"></a><!-- ws:end:WikiTextHeadingRule:18 -->Examples:</h1>
 

<table class="wiki_table">
    <tr>
        <td>Name:<br />
</td>
        <td>H(n/d)<br />
</td>
        <td>H(q)<br />
</td>
        <td>H(q) simplified by equivalence relation<br />
</td>
    </tr>
    <tr>
        <td>Benedetti Height<br />
</td>
        <td><!-- ws:start:WikiTextMathRule:1:
[[math]]&lt;br/&gt;
nd&lt;br/&gt;[[math]]
 --><script type="math/tex">nd</script><!-- ws:end:WikiTextMathRule:1 --><br />
</td>
        <td><!-- ws:start:WikiTextMathRule:2:
[[math]]&lt;br/&gt;
2^{T1(q)}&lt;br/&gt;[[math]]
 --><script type="math/tex">2^{T1(q)}</script><!-- ws:end:WikiTextMathRule:2 --><br />
</td>
        <td><!-- ws:start:WikiTextMathRule:3:
[[math]]&lt;br/&gt;
T1(q)&lt;br/&gt;[[math]]
 --><script type="math/tex">T1(q)</script><!-- ws:end:WikiTextMathRule:3 --><br />
</td>
    </tr>
    <tr>
        <td>Weil Height<br />
</td>
        <td><!-- ws:start:WikiTextMathRule:4:
[[math]]&lt;br/&gt;
max(n,d)&lt;br/&gt;[[math]]
 --><script type="math/tex">max(n,d)</script><!-- ws:end:WikiTextMathRule:4 --><br />
</td>
        <td><!-- ws:start:WikiTextMathRule:5:
[[math]]&lt;br/&gt;
2^{(T1(q)+|log_2(q)|)/2}&lt;br/&gt;[[math]]
 --><script type="math/tex">2^{(T1(q)+|log_2(q)|)/2}</script><!-- ws:end:WikiTextMathRule:5 --><br />
</td>
        <td><!-- ws:start:WikiTextMathRule:6:
[[math]]&lt;br/&gt;
T1(q)+|log_2(q)|&lt;br/&gt;[[math]]
 --><script type="math/tex">T1(q)+|log_2(q)|</script><!-- ws:end:WikiTextMathRule:6 --><br />
</td>
    </tr>
    <tr>
        <td>??<br />
</td>
        <td><!-- ws:start:WikiTextMathRule:7:
[[math]]&lt;br/&gt;
n+d&lt;br/&gt;[[math]]
 --><script type="math/tex">n+d</script><!-- ws:end:WikiTextMathRule:7 --><br />
</td>
        <td><!-- ws:start:WikiTextMathRule:8:
[[math]]&lt;br/&gt;
2^{T1(q)/2} (q+1)/q^{1/2}&lt;br/&gt;[[math]]
 --><script type="math/tex">2^{T1(q)/2} (q+1)/q^{1/2}</script><!-- ws:end:WikiTextMathRule:8 --><br />
</td>
        <td><!-- ws:start:WikiTextMathRule:9:
[[math]]&lt;br/&gt;
T1(q)+2log_2(q+1)-log_2(q)&lt;br/&gt;[[math]]
 --><script type="math/tex">T1(q)+2log_2(q+1)-log_2(q)</script><!-- ws:end:WikiTextMathRule:9 --><br />
</td>
    </tr>
    <tr>
        <td>Kees Height<br />
</td>
        <td><!-- ws:start:WikiTextMathRule:10:
[[math]]&lt;br/&gt;
max(2^{-v_2(n)}n,&lt;br /&gt;
2^{-v_2(d)}d)&lt;br/&gt;[[math]]
 --><script type="math/tex">max(2^{-v_2(n)}n,
2^{-v_2(d)}d)</script><!-- ws:end:WikiTextMathRule:10 --><br />
</td>
        <td><!-- ws:start:WikiTextMathRule:11:
[[math]]&lt;br/&gt;
2^{(T1(2^{-v_2(q)}q)+|log_2(q)-v_2(q)|)/2}&lt;br/&gt;[[math]]
 --><script type="math/tex">2^{(T1(2^{-v_2(q)}q)+|log_2(q)-v_2(q)|)/2}</script><!-- ws:end:WikiTextMathRule:11 --><br />
</td>
        <td><!-- ws:start:WikiTextMathRule:12:
[[math]]&lt;br/&gt;
T1(2^{-v_2(q)}q)+|log_2(q)-v_2(q)|&lt;br/&gt;[[math]]
 --><script type="math/tex">T1(2^{-v_2(q)}q)+|log_2(q)-v_2(q)|</script><!-- ws:end:WikiTextMathRule:12 --><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
</table>

Where T1(q) is the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Generalized%20Tenney%20Norms%20and%20Tp%20Interval%20Space#The%20Tenney%20Norm%20(T1%20norm)">tenney norm</a> of q in monzo form, and vp(x) is the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/P-adic_order" rel="nofollow">p-adic valuation</a> of x.<br />
<br />
Some useful identities:<br />
<!-- ws:start:WikiTextMathRule:13:
[[math]]&lt;br/&gt;
n=2^{(T1(q)\pm|log_2(q)|)/2}&lt;br/&gt;[[math]]
 --><script type="math/tex">n=2^{(T1(q)\pm|log_2(q)|)/2}</script><!-- ws:end:WikiTextMathRule:13 --><br />
<!-- ws:start:WikiTextMathRule:14:
[[math]]&lt;br/&gt;
d=2^{(T1(q)\mp|log_2(q)|)/2}&lt;br/&gt;[[math]]
 --><script type="math/tex">d=2^{(T1(q)\mp|log_2(q)|)/2}</script><!-- ws:end:WikiTextMathRule:14 --><br />
<!-- ws:start:WikiTextMathRule:15:
[[math]]&lt;br/&gt;
nd=2^{T1(q)}&lt;br/&gt;[[math]]
 --><script type="math/tex">nd=2^{T1(q)}</script><!-- ws:end:WikiTextMathRule:15 --></body></html>

@@ 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:Sarzadoce|Sarzadoce]] and made on <tt>2012-09-06 18:16:41 UTC</tt>.<br>
+: This revision was by author [[User:Sarzadoce|Sarzadoce]] and made on <tt>2012-09-06 18:34:21 UTC</tt>.<br>
-: The original revision id was <tt>362660064</tt>.<br>
+: The original revision id was <tt>362662950</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>
@@ Line 43: / Line 43: @@
 [[math]] || [[math]]
 T1(q)+2log_2(q+1)-log_2(q)
+[[math]] ||
+|| Kees Height || [[math]]
+max(2^{-v_2(n)}n,
+^{-v_2(d)}d)
+[[math]] || [[math]]
+^{(T1(2^{-v_2(q)}q)+|log_2(q)-v_2(q)|)/2}
+[[math]] || [[math]]
+T1(2^{-v_2(q)}q)+|log_2(q)-v_2(q)|
 [[math]] ||
 ||   ||   ||   ||   ||
-||   ||   ||   ||   ||
+Where T1(q) is the [[xenharmonic/Generalized Tenney Norms and Tp Interval Space#The%20Tenney%20Norm%20(T1%20norm)|tenney norm]] of q in monzo form, and vp(x) is the [[http://en.wikipedia.org/wiki/P-adic_order|p-adic valuation]] of x.
-Where T1(q) is the [[xenharmonic/Generalized Tenney Norms and Tp Interval Space#The%20Tenney%20Norm%20(T1%20norm)|tenney norm]] of q, in monzo form.
 Some useful identities:
@@ Line 59: / Line 67: @@
 [[math]]</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;Height&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:13:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Definition:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:13 --&gt;Definition:&lt;/h1&gt;
+<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;
 &lt;br /&gt;
@@ Line 72: / Line 80: @@
   --&gt;&lt;script type="math/tex"&gt;H(q) \equiv F(H(q))&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:0 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:15:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Examples:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:15 --&gt;Examples:&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Examples:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Examples:&lt;/h1&gt;
@@ Line 144: / Line 152: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Kees Height&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;!-- ws:start:WikiTextMathRule:10:
+[[math]]&amp;lt;br/&amp;gt;
+max(2^{-v_2(n)}n,&amp;lt;br /&amp;gt;
+^{-v_2(d)}d)&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;max(2^{-v_2(n)}n,
+^{-v_2(d)}d)&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:10 --&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;!-- ws:start:WikiTextMathRule:11:
+[[math]]&amp;lt;br/&amp;gt;
+^{(T1(2^{-v_2(q)}q)+|log_2(q)-v_2(q)|)/2}&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;2^{(T1(2^{-v_2(q)}q)+|log_2(q)-v_2(q)|)/2}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:11 --&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;!-- ws:start:WikiTextMathRule:12:
+[[math]]&amp;lt;br/&amp;gt;
+T1(2^{-v_2(q)}q)+|log_2(q)-v_2(q)|&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;T1(2^{-v_2(q)}q)+|log_2(q)-v_2(q)|&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:12 --&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 165: / Line 184: @@
 &lt;/table&gt;
-Where T1(q) is the &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Generalized%20Tenney%20Norms%20and%20Tp%20Interval%20Space#The%20Tenney%20Norm%20(T1%20norm)"&gt;tenney norm&lt;/a&gt; of q, in monzo form.&lt;br /&gt;
+Where T1(q) is the &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Generalized%20Tenney%20Norms%20and%20Tp%20Interval%20Space#The%20Tenney%20Norm%20(T1%20norm)"&gt;tenney norm&lt;/a&gt; of q in monzo form, and vp(x) is the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/P-adic_order" rel="nofollow"&gt;p-adic valuation&lt;/a&gt; of x.&lt;br /&gt;
 &lt;br /&gt;
 Some useful identities:&lt;br /&gt;
-&lt;!-- ws:start:WikiTextMathRule:10:
+&lt;!-- ws:start:WikiTextMathRule:13:
 [[math]]&amp;lt;br/&amp;gt;
 n=2^{(T1(q)\pm|log_2(q)|)/2}&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;n=2^{(T1(q)\pm|log_2(q)|)/2}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:10 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt;n=2^{(T1(q)\pm|log_2(q)|)/2}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:13 --&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextMathRule:11:
+&lt;!-- ws:start:WikiTextMathRule:14:
 [[math]]&amp;lt;br/&amp;gt;
 d=2^{(T1(q)\mp|log_2(q)|)/2}&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;d=2^{(T1(q)\mp|log_2(q)|)/2}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:11 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt;d=2^{(T1(q)\mp|log_2(q)|)/2}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:14 --&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextMathRule:12:
+&lt;!-- ws:start:WikiTextMathRule:15:
 [[math]]&amp;lt;br/&amp;gt;
 nd=2^{T1(q)}&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;nd=2^{T1(q)}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:12 --&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
+  --&gt;&lt;script type="math/tex"&gt;nd=2^{T1(q)}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:15 --&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Height: Difference between revisions

Revision as of 18:34, 6 September 2012

IMPORTED REVISION FROM WIKISPACES

Original Wikitext content:

Original HTML content:

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