POTE tuning: Difference between revisions
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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 /> | |||
<br /> | |||
#2 Find the matrix P = V*(VV*)^(-1), where V* is the transpose matrix.<br /> | |||
<br /> | |||
#3 Find T = &lt;1 1 1 1|P.<br /> | |||
<br /> | |||
#4 Find POTE = T/T[1]; in other words T scalar divided by T[1], the first element of T.<br /> | |||
<br /> | |||
If you carry out these operations, you should find <br /> | If you carry out these operations, you should find <br /> | ||
<br /> | <br /> | ||
Revision as of 19:25, 29 August 2011
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2011-08-29 19:25:17 UTC.
- The original revision id was 249241903.
- 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:
**POTE tuning** is the short form of **Pure-Octaves [[Tenney-Euclidean tuning#Pure octaves TE tuning]]**, a good choice for a standard tuning enforcing just 2s as octaves. 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: #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)] #2 Find the matrix P = V*(VV*)^(-1), where V* is the transpose matrix. #3 Find T = <1 1 1 1|P. #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 V ~ [<1 0 0.861 -0.356|, <0 3.155 0.431 4.274|] T ~ <1.000902 0.317246| POTE ~ <1 0.3169600| The tuning of the POTE generator corresponding to the mapping M is therefore 0.31696 octaves, or 380.252 cents. Naturally, this only gives the single POTE generator in the rank two case, and only when the map M is in period-generator form, but the POTE tuning can still be found in this way for mappings defining higher rank temperaments. The method also, of course, should be modified if subgroup temperaments are being considered.
Original HTML content:
<html><head><title>POTE tuning</title></head><body><strong>POTE tuning</strong> is the short form of <strong>Pure-Octaves <a class="wiki_link" href="/Tenney-Euclidean%20tuning#Pure octaves TE tuning">Tenney-Euclidean tuning</a></strong>, a good choice for a standard tuning enforcing just 2s as octaves.<br /> <br /> The POTE tuning for a map matrix such as M = [<1 0 2 -1|, <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 /> #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)]<br /> <br /> #2 Find the matrix P = V*(VV*)^(-1), where V* is the transpose matrix.<br /> <br /> #3 Find T = <1 1 1 1|P.<br /> <br /> #4 Find POTE = T/T[1]; in other words T scalar divided by T[1], the first element of T.<br /> <br /> If you carry out these operations, you should find <br /> <br /> V ~ [<1 0 0.861 -0.356|, <0 3.155 0.431 4.274|]<br /> <br /> T ~ <1.000902 0.317246|<br /> <br /> POTE ~ <1 0.3169600|<br /> <br /> The tuning of the POTE generator corresponding to the mapping M is therefore 0.31696 octaves, or 380.252 cents. Naturally, this only gives the single POTE generator in the rank two case, and only when the map M is in period-generator form, but the POTE tuning can still be found in this way for mappings defining higher rank temperaments. The method also, of course, should be modified if subgroup temperaments are being considered.</body></html>