Height: Difference between revisions
Wikispaces>Sarzadoce **Imported revision 362660064 - Original comment: ** |
Wikispaces>Sarzadoce **Imported revision 362662950 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | 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: | : 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> | : The original revision id was <tt>362662950</tt>.<br> | ||
: The revision comment was: <tt></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> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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[[math]] || [[math]] | [[math]] || [[math]] | ||
T1(q)+2log_2(q+1)-log_2(q) | 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]] || | [[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 | |||
Some useful identities: | Some useful identities: | ||
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[[math]]</pre></div> | [[math]]</pre></div> | ||
<h4>Original HTML content:</h4> | <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><!-- ws:start:WikiTextHeadingRule: | <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><!-- 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 /> | 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 /> | <br /> | ||
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--><script type="math/tex">H(q) \equiv F(H(q))</script><!-- ws:end:WikiTextMathRule:0 --><br /> | --><script type="math/tex">H(q) \equiv F(H(q))</script><!-- ws:end:WikiTextMathRule:0 --><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h1&gt; --><h1 id="toc1"><a name="Examples:"></a><!-- ws:end:WikiTextHeadingRule:18 -->Examples:</h1> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td><br /> | <td>Kees Height<br /> | ||
</td> | </td> | ||
<td><br /> | <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> | ||
<td><br /> | <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> | ||
<td><br /> | <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> | </td> | ||
</tr> | </tr> | ||
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</table> | </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 | 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 /> | <br /> | ||
Some useful identities:<br /> | Some useful identities:<br /> | ||
<!-- ws:start:WikiTextMathRule: | <!-- ws:start:WikiTextMathRule:13: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
n=2^{(T1(q)\pm|log_2(q)|)/2}&lt;br/&gt;[[math]] | 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: | --><script type="math/tex">n=2^{(T1(q)\pm|log_2(q)|)/2}</script><!-- ws:end:WikiTextMathRule:13 --><br /> | ||
<!-- ws:start:WikiTextMathRule: | <!-- ws:start:WikiTextMathRule:14: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
d=2^{(T1(q)\mp|log_2(q)|)/2}&lt;br/&gt;[[math]] | 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: | --><script type="math/tex">d=2^{(T1(q)\mp|log_2(q)|)/2}</script><!-- ws:end:WikiTextMathRule:14 --><br /> | ||
<!-- ws:start:WikiTextMathRule: | <!-- ws:start:WikiTextMathRule:15: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
nd=2^{T1(q)}&lt;br/&gt;[[math]] | nd=2^{T1(q)}&lt;br/&gt;[[math]] | ||
--><script type="math/tex">nd=2^{T1(q)}</script><!-- ws:end:WikiTextMathRule: | --><script type="math/tex">nd=2^{T1(q)}</script><!-- ws:end:WikiTextMathRule:15 --></body></html></pre></div> | ||