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Tp tuning: Difference between revisions

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**Imported revision 354237462 - Original comment: **
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**Imported revision 354241872 - 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>2012-07-21 19:15:29 UTC</tt>.<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-21 20:39:05 UTC</tt>.<br>
: The original revision id was <tt>354237462</tt>.<br>
: The original revision id was <tt>354241872</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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=L2 tuning=
=L2 tuning=
In the special case where p = 2, the Lp norm becomes the L2 norm, which is the [[Tenney-Euclidean metrics|Tenney-Euclidean]] norm, or TE complexity. Associated to this norm is L2 tuning, or TE tuning, extended to arbitrary JI groups, and the L2 error, which is E2(S) for the temperament S, and which is approximately proportional to [[Tenney-Euclidean temperament measures#TE error|TE error]]. Starting from a [[gencom]] for the temperament, we may find the tuning and the L2 error by the following procedure:
In the special case where p = 2, the Lp norm becomes the L2 norm, which is the [[Tenney-Euclidean metrics|Tenney-Euclidean]] norm, or TE complexity. Associated to this norm is L2 tuning, or TE tuning, extended to arbitrary JI groups, and the L2 error, which is E2(S) for the temperament S, and which is approximately proportional to [[Tenney-Euclidean temperament measures#TE error|TE error]].  


# Convert the gencom into monzo form.
For an example, consider [[Chromatic pairs#Indium|indium temperament]], with group 2.5/3.7/3.11/3 and comma basis 3025/3024 and 3125/3087. The corresponding full 11-limit temperament is of rank three, and using the [[Tenney-Euclidean tuning|usual methods]], in particular the pseudoinverse, we find that the L2 tuning map is &lt;1199.552 1901.846 2783.579 3371.401 4153.996|. Applying that to 12/11 gives a generator of 146.995, and multiplying that by (1200.0/1199.552) gives a POL2 tuning, or extended POTE tuning, of 147.010. Converting the tuning map to weighted coordinates and subtracting &lt;1200 1200 1200 1200 1200| gives &lt;-0.4475 -.0685 -1.1778 0.9172 0.7741|. The ordinary Euclideam norm of this, ie the square root of the dot product, gives an error of 1.7414 cents.
# Convert the monzos to weighted coordinates.
</pre></div>
# Multiply each monzo by an indeterminate, creating a parametrized n-dimensional weighted monzo, where n is the number of primes in the prime limit; call that M.
# Take the dot product of M with a vector &lt;x1 x2 ... xn| consisting of n new indeterminates x1, x2, ..., xn; call that X.
# Maximize X^2 subject to the constraint that the M∙M, the dot product of M with itself, is equal to 1. This is maximizing a quadratic subject to a quadratic constraint, and so is easily done via Lagrange multipliers.
# The result of the maximization process is a positive definite quadratic form Q(x1, x2, ..., xn) in n indeterminates x1, x2, ..., xn.
# Find the point T on the subspace of G-tuning space spanned by the [[gencom|gencom mapping]] closest to the JIP using the distance function defined by Q. As usual the JIP is &lt;1 1 1 ... 1|. Since Q is a quadratic polynomial, this minimal distance is easily found by calculus methods.
# Since the primes may or may not be in the group G, the mapping of primes in T may or may not make sense by itself; however, now unweight T and apply it to the generators of the gencom, and obtain the TE tuning of those generators, which defines the TE tuning for the temperament defined by the gencom.
# Given the tuning T, we may find the L2 error as PE2s(T) by constrained optimization of Err(T) under the constraint M∙M = 1. It is important to bear in mind that while the tuning is found subject to the constraint that the commas are tempered out, the error is found by optimizing over the entire group G.</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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Tp tuning&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:10:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextTocRule:12: --&gt; | &lt;a href="#Dual norm"&gt;Dual norm&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt; | &lt;a href="#Applying the Hahn-Banach theorem"&gt;Applying the Hahn-Banach theorem&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt; | &lt;a href="#L2 tuning"&gt;L2 tuning&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;
<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Tp tuning&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:10:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextTocRule:12: --&gt; | &lt;a href="#Dual norm"&gt;Dual norm&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt; | &lt;a href="#Applying the Hahn-Banach theorem"&gt;Applying the Hahn-Banach theorem&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt; | &lt;a href="#L2 tuning"&gt;L2 tuning&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;
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&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="L2 tuning"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;L2 tuning&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="L2 tuning"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;L2 tuning&lt;/h1&gt;
In the special case where p = 2, the Lp norm becomes the L2 norm, which is the &lt;a class="wiki_link" href="/Tenney-Euclidean%20metrics"&gt;Tenney-Euclidean&lt;/a&gt; norm, or TE complexity. Associated to this norm is L2 tuning, or TE tuning, extended to arbitrary JI groups, and the L2 error, which is E2(S) for the temperament S, and which is approximately proportional to &lt;a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE error"&gt;TE error&lt;/a&gt;. Starting from a &lt;a class="wiki_link" href="/gencom"&gt;gencom&lt;/a&gt; for the temperament, we may find the tuning and the L2 error by the following procedure:&lt;br /&gt;
In the special case where p = 2, the Lp norm becomes the L2 norm, which is the &lt;a class="wiki_link" href="/Tenney-Euclidean%20metrics"&gt;Tenney-Euclidean&lt;/a&gt; norm, or TE complexity. Associated to this norm is L2 tuning, or TE tuning, extended to arbitrary JI groups, and the L2 error, which is E2(S) for the temperament S, and which is approximately proportional to &lt;a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE error"&gt;TE error&lt;/a&gt;. &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;ol&gt;&lt;li&gt;Convert the gencom into monzo form.&lt;/li&gt;&lt;li&gt;Convert the monzos to weighted coordinates.&lt;/li&gt;&lt;li&gt;Multiply each monzo by an indeterminate, creating a parametrized n-dimensional weighted monzo, where n is the number of primes in the prime limit; call that M.&lt;/li&gt;&lt;li&gt;Take the dot product of M with a vector &amp;lt;x1 x2 ... xn| consisting of n new indeterminates x1, x2, ..., xn; call that X.&lt;/li&gt;&lt;li&gt;Maximize X^2 subject to the constraint that the M∙M, the dot product of M with itself, is equal to 1. This is maximizing a quadratic subject to a quadratic constraint, and so is easily done via Lagrange multipliers.&lt;/li&gt;&lt;li&gt;The result of the maximization process is a positive definite quadratic form Q(x1, x2, ..., xn) in n indeterminates x1, x2, ..., xn.&lt;/li&gt;&lt;li&gt;Find the point T on the subspace of G-tuning space spanned by the &lt;a class="wiki_link" href="/gencom"&gt;gencom mapping&lt;/a&gt; closest to the JIP using the distance function defined by Q. As usual the JIP is &amp;lt;1 1 1 ... 1|. Since Q is a quadratic polynomial, this minimal distance is easily found by calculus methods.&lt;/li&gt;&lt;li&gt;Since the primes may or may not be in the group G, the mapping of primes in T may or may not make sense by itself; however, now unweight T and apply it to the generators of the gencom, and obtain the TE tuning of those generators, which defines the TE tuning for the temperament defined by the gencom.&lt;/li&gt;&lt;li&gt;Given the tuning T, we may find the L2 error as PE2s(T) by constrained optimization of Err(T) under the constraint M∙M = 1. It is important to bear in mind that while the tuning is found subject to the constraint that the commas are tempered out, the error is found by optimizing over the entire group G.&lt;/li&gt;&lt;/ol&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
For an example, consider &lt;a class="wiki_link" href="/Chromatic%20pairs#Indium"&gt;indium temperament&lt;/a&gt;, with group 2.5/3.7/3.11/3 and comma basis 3025/3024 and 3125/3087. The corresponding full 11-limit temperament is of rank three, and using the &lt;a class="wiki_link" href="/Tenney-Euclidean%20tuning"&gt;usual methods&lt;/a&gt;, in particular the pseudoinverse, we find that the L2 tuning map is &amp;lt;1199.552 1901.846 2783.579 3371.401 4153.996|. Applying that to 12/11 gives a generator of 146.995, and multiplying that by (1200.0/1199.552) gives a POL2 tuning, or extended POTE tuning, of 147.010. Converting the tuning map to weighted coordinates and subtracting &amp;lt;1200 1200 1200 1200 1200| gives &amp;lt;-0.4475 -.0685 -1.1778 0.9172 0.7741|. The ordinary Euclideam norm of this, ie the square root of the dot product, gives an error of 1.7414 cents.&lt;/body&gt;&lt;/html&gt;</pre></div>
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