Height: Difference between revisions
Wikispaces>Sarzadoce **Imported revision 362649630 - Original comment: ** |
Wikispaces>Sarzadoce **Imported revision 362660064 - 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 | : This revision was by author [[User:Sarzadoce|Sarzadoce]] and made on <tt>2012-09-06 18:16:41 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>362660064</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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nd | nd | ||
[[math]] || [[math]] | [[math]] || [[math]] | ||
2^ | 2^{T1(q)} | ||
[[math]] || [[math]] | [[math]] || [[math]] | ||
T1(q) | T1(q) | ||
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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.</pre></div> | 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: | |||
[[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]]</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:13:&lt;h1&gt; --><h1 id="toc0"><a name="Definition:"></a><!-- ws:end:WikiTextHeadingRule:13 -->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:15:&lt;h1&gt; --><h1 id="toc1"><a name="Examples:"></a><!-- ws:end:WikiTextHeadingRule:15 -->Examples:</h1> | ||
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<td><!-- ws:start:WikiTextMathRule:2: | <td><!-- ws:start:WikiTextMathRule:2: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
2^ | 2^{T1(q)}&lt;br/&gt;[[math]] | ||
--><script type="math/tex">2^ | --><script type="math/tex">2^{T1(q)}</script><!-- ws:end:WikiTextMathRule:2 --><br /> | ||
</td> | </td> | ||
<td><!-- ws:start:WikiTextMathRule:3: | <td><!-- ws:start:WikiTextMathRule:3: | ||
| Line 154: | Line 165: | ||
</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, in monzo form.</body></html></pre></div> | 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.<br /> | ||
<br /> | |||
Some useful identities:<br /> | |||
<!-- ws:start:WikiTextMathRule:10: | |||
[[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:10 --><br /> | |||
<!-- ws:start:WikiTextMathRule:11: | |||
[[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:11 --><br /> | |||
<!-- ws:start:WikiTextMathRule:12: | |||
[[math]]&lt;br/&gt; | |||
nd=2^{T1(q)}&lt;br/&gt;[[math]] | |||
--><script type="math/tex">nd=2^{T1(q)}</script><!-- ws:end:WikiTextMathRule:12 --></body></html></pre></div> | |||