POTE tuning: Difference between revisions
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Wikispaces>genewardsmith **Imported revision 350848958 - Original comment: ** |
Wikispaces>clumma **Imported revision 588204701 - 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: | : This revision was by author [[User:clumma|clumma]] and made on <tt>2016-07-27 14:07:06 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>588204701</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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<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;white-space: pre-wrap ! important" class="old-revision-html">**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. | <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;white-space: pre-wrap ! important" class="old-revision-html">**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 [[val]] | 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 [[val|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)] | #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)] | ||
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<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>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 /> | <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>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 /> | <br /> | ||
The POTE tuning for a <a class="wiki_link" href="/map%20matrix">map matrix</a> such as M = [&lt;1 0 2 -1|, &lt;0 5 1 12|] (the <a class="wiki_link" href="/map">map</a> for 7-limit <a class="wiki_link" href="/Magic%20family">magic</a>, which consists of a linearly independent list of <a class="wiki_link" href="/val"> | The POTE tuning for a <a class="wiki_link" href="/map%20matrix">map matrix</a> such as M = [&lt;1 0 2 -1|, &lt;0 5 1 12|] (the <a class="wiki_link" href="/map">map</a> for 7-limit <a class="wiki_link" href="/Magic%20family">magic</a>, which consists of a linearly independent list of <a class="wiki_link" href="/val">vals</a> 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 /> | #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 /> | ||
Revision as of 14:07, 27 July 2016
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author clumma and made on 2016-07-27 14:07:06 UTC.
- The original revision id was 588204701.
- 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 [[val|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 can be generalized to subgroup temperaments so long as the group contains 2 by [[Lp tuning|POL2 tuning]].
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 <a class="wiki_link" href="/map%20matrix">map matrix</a> such as M = [<1 0 2 -1|, <0 5 1 12|] (the <a class="wiki_link" href="/map">map</a> for 7-limit <a class="wiki_link" href="/Magic%20family">magic</a>, which consists of a linearly independent list of <a class="wiki_link" href="/val">vals</a> 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 <a class="wiki_link" href="/generator">generator</a> 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 can be generalized to subgroup temperaments so long as the group contains 2 by <a class="wiki_link" href="/Lp%20tuning">POL2 tuning</a>.</body></html>