Height: Difference between revisions
Wikispaces>Sarzadoce **Imported revision 363887748 - Original comment: ** |
Wikispaces>Sarzadoce **Imported revision 363957564 - 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>2012-09-11 | : This revision was by author [[User:Sarzadoce|Sarzadoce]] and made on <tt>2012-09-11 21:25:02 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>363957564</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]] | ||
A **semi-height** is a function which does not obey criteria #1 above in the strictest sense, so that there is a rational number q ≠ 1 such that H(q) = H(1), resulting in an equivalence relation on its elements. An example would be octave-equivalence, where two ratios p and q are considered equivalent if the following is true: | |||
[[math]] | [[math]] | ||
2^{-v_2 \left( {p} \right)} p = 2^{-v_2 \left( {q} \right)} q | 2^{-v_2 \left( {p} \right)} p = 2^{-v_2 \left( {q} \right)} q | ||
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|| __Name:__ || __Type:__ || __H(n/d):__ || __H(q):__ || __H(q) simplified by equivalence relation:__ || | || __Name:__ || __Type:__ || __H(n/d):__ || __H(q):__ || __H(q) simplified by equivalence relation:__ || | ||
|| [[Benedetti Height|Benedetti height]] | || [[Benedetti Height|Benedetti height]] | ||
(or [[Tenney Height]]) || | (or [[Tenney Height]]) || Height || [[math]] | ||
n d | n d | ||
[[math]] || [[math]] | [[math]] || [[math]] | ||
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T1 \left( {q} \right) | T1 \left( {q} \right) | ||
[[math]] || | [[math]] || | ||
|| Weil Height || | || Weil Height || Height || [[math]] | ||
\max \left( {n , d} \right) | \max \left( {n , d} \right) | ||
[[math]] || [[math]] | [[math]] || [[math]] | ||
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T1 \left( {q} \right) + | \log_2 \left( {q} \right) | | T1 \left( {q} \right) + | \log_2 \left( {q} \right) | | ||
[[math]] || | [[math]] || | ||
|| Arithmetic Height || | || Arithmetic Height || Height || [[math]] | ||
n + d | n + d | ||
[[math]] || [[math]] | [[math]] || [[math]] | ||
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T1 \left( {q} \right) + 2 \log_2 \left( {q + 1} \right) - \log_2 \left( {q} \right) | T1 \left( {q} \right) + 2 \log_2 \left( {q + 1} \right) - \log_2 \left( {q} \right) | ||
[[math]] || | [[math]] || | ||
|| Harmonic Height || | || Harmonic Height || Semi-Height || [[math]] | ||
\dfrac {n d} {n + d} | \dfrac {n d} {n + d} | ||
[[math]] || [[math]] | [[math]] || [[math]] | ||
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T1 \left( {q} \right) - 2 \log_2 \left( {q + 1} \right) + \log_2 \left( {q} \right) | T1 \left( {q} \right) - 2 \log_2 \left( {q + 1} \right) + \log_2 \left( {q} \right) | ||
[[math]] || | [[math]] || | ||
|| [[Kees Height]] || | || [[Kees Height]] || Semi-Height || [[math]] | ||
\max \left( {2^{-v_2 \left( {n} \right)} n , | \max \left( {2^{-v_2 \left( {n} \right)} n , | ||
2^{-v_2 \left( {d} \right)} d} \right) | 2^{-v_2 \left( {d} \right)} d} \right) | ||
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--><script type="math/tex">H \left( {q} \right) \equiv F \left( {H} \left( {q} \right) \right)</script><!-- ws:end:WikiTextMathRule:0 --><br /> | --><script type="math/tex">H \left( {q} \right) \equiv F \left( {H} \left( {q} \right) \right)</script><!-- ws:end:WikiTextMathRule:0 --><br /> | ||
<br /> | <br /> | ||
A <strong>semi-height</strong> is a function which does not obey criteria #1 above in the strictest sense, so that there is a rational number q ≠ 1 such that H(q) = H(1), resulting in an equivalence relation on its elements. An example would be octave-equivalence, where two ratios p and q are considered equivalent if the following is true:<br /> | |||
<!-- ws:start:WikiTextMathRule:1: | <!-- ws:start:WikiTextMathRule:1: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
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(or <a class="wiki_link" href="/Tenney%20Height">Tenney Height</a>)<br /> | (or <a class="wiki_link" href="/Tenney%20Height">Tenney Height</a>)<br /> | ||
</td> | </td> | ||
<td> | <td>Height<br /> | ||
</td> | </td> | ||
<td><!-- ws:start:WikiTextMathRule:4: | <td><!-- ws:start:WikiTextMathRule:4: | ||
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<td>Weil Height<br /> | <td>Weil Height<br /> | ||
</td> | </td> | ||
<td> | <td>Height<br /> | ||
</td> | </td> | ||
<td><!-- ws:start:WikiTextMathRule:7: | <td><!-- ws:start:WikiTextMathRule:7: | ||
| Line 184: | Line 184: | ||
<td>Arithmetic Height<br /> | <td>Arithmetic Height<br /> | ||
</td> | </td> | ||
<td> | <td>Height<br /> | ||
</td> | </td> | ||
<td><!-- ws:start:WikiTextMathRule:10: | <td><!-- ws:start:WikiTextMathRule:10: | ||
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<td>Harmonic Height<br /> | <td>Harmonic Height<br /> | ||
</td> | </td> | ||
<td> | <td>Semi-Height<br /> | ||
</td> | </td> | ||
<td><!-- ws:start:WikiTextMathRule:13: | <td><!-- ws:start:WikiTextMathRule:13: | ||
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<td><a class="wiki_link" href="/Kees%20Height">Kees Height</a><br /> | <td><a class="wiki_link" href="/Kees%20Height">Kees Height</a><br /> | ||
</td> | </td> | ||
<td> | <td>Semi-Height<br /> | ||
</td> | </td> | ||
<td><!-- ws:start:WikiTextMathRule:16: | <td><!-- ws:start:WikiTextMathRule:16: | ||