POTE tuning: Difference between revisions
Wikispaces>genewardsmith **Imported revision 249241283 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 249241903 - 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:genewardsmith|genewardsmith]] and made on <tt>2011-08-29 19: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-08-29 19:25:17 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>249241903</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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The POTE tuning for a map matrix such as M = [<1 0 2 -1|, <0 5 1 12|] (the map for 7-limit [[Magic family|magic]], which consists of a linearly independent list of vals defining magic) can be found as follows: | The POTE tuning for a map matrix such as M = [<1 0 2 -1|, <0 5 1 12|] (the map for 7-limit [[Magic family|magic]], which consists of a linearly independent list of vals defining magic) can be found as follows: | ||
# Form a matrix V from M by multiplying by the diagonal matrix which is zero off the diagonal and 1/log2(p) on the diagonal; in other words the diagonal is [1 1/log2(3) 1/log2(5) 1/log2(7)]. Another way to say this is that each val is "weighted" by dividing through by the logarithms, so that V = [<1 0 2/log2(5) -1/log2(7)| <5/log2(3) 1/log2(5) 12/log2(7)] | #1 Form a matrix V from M by multiplying by the diagonal matrix which is zero off the diagonal and 1/log2(p) on the diagonal; in other words the diagonal is [1 1/log2(3) 1/log2(5) 1/log2(7)]. Another way to say this is that each val is "weighted" by dividing through by the logarithms, so that V = [<1 0 2/log2(5) -1/log2(7)| <5/log2(3) 1/log2(5) 12/log2(7)] | ||
# Find the matrix P = V*(VV*)^(-1), where V* is the transpose matrix. | #2 Find the matrix P = V*(VV*)^(-1), where V* is the transpose matrix. | ||
# Find T = <1 1 1 1|P. | #3 Find T = <1 1 1 1|P. | ||
# Find POTE = T/T[1]; in other words T scalar divided by T[1], the first element of T. | #4 Find POTE = T/T[1]; in other words T scalar divided by T[1], the first element of T. | ||
If you carry out these operations, you should find | If you carry out these operations, you should find | ||
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The POTE tuning for a map matrix such as M = [&lt;1 0 2 -1|, &lt;0 5 1 12|] (the map for 7-limit <a class="wiki_link" href="/Magic%20family">magic</a>, which consists of a linearly independent list of vals defining magic) can be found as follows:<br /> | The POTE tuning for a map matrix such as M = [&lt;1 0 2 -1|, &lt;0 5 1 12|] (the map for 7-limit <a class="wiki_link" href="/Magic%20family">magic</a>, which consists of a linearly independent list of vals defining magic) can be found as follows:<br /> | ||
<br /> | <br /> | ||
#1 Form a matrix V from M by multiplying by the diagonal matrix which is zero off the diagonal and 1/log2(p) on the diagonal; in other words the diagonal is [1 1/log2(3) 1/log2(5) 1/log2(7)]. Another way to say this is that each val is &quot;weighted&quot; by dividing through by the logarithms, so that V = [&lt;1 0 2/log2(5) -1/log2(7)| &lt;5/log2(3) 1/log2(5) 12/log2(7)]<br /> | |||
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#2 Find the matrix P = V*(VV*)^(-1), where V* is the transpose matrix.<br /> | |||
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#3 Find T = &lt;1 1 1 1|P.<br /> | |||
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#4 Find POTE = T/T[1]; in other words T scalar divided by T[1], the first element of T.<br /> | |||
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If you carry out these operations, you should find <br /> | If you carry out these operations, you should find <br /> | ||
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