Tp tuning: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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>2012-06-26 14:01:37 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-06-26 14:08:05 UTC</tt>.<br>

: The original revision id was <tt>~~348181842~~</tt>.<br>

: The original revision id was <tt>348184558</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>

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# Convert the monzos to weighted coordinates.

# 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 consisting of n new indeterminates x1, x2, ..., xn~~, and take the dot product of that with M~~; call that X.

# Take the dot product of M with a vector <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 maximinzing a quadric subject to a quadratic constraint, and so ~~this~~ is easily done via Lagrange multipliers.

# Maximize X^2 subject to the constraint that the M.M, the dot product of M with itself, is equal to 1. This is maximinzing a quadric 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~~, which as~~ usual is <1 1 1 ... 1|.

# 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 <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.</pre></div>

<h4>Original HTML content:</h4>

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In the special case where p = 2, the Lp norm becomes the L2 norm, which is the <a class="wiki_link" href="/Tenney-Euclidean%20metrics">Tenney-Euclidean</a> norm, or TE complexity. Associated to this norm is L2 tuning, or TE tuning, extended to arbitrary JI groups. Starting from a <a class="wiki_link" href="/gencom">gencom</a> for the temperament, we may find the tuning by the following proceedure:<br />

<br />

<ol><li>Convert the gencom into monzo form.</li><li>Convert the monzos to weighted coordinates.</li><li>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.</li><li>Take the dot product of M with a vector consisting of n new indeterminates x1, x2, ..., xn~~, and take the dot product of that with M~~; call that X.</li><li>Maximize X^2 subject to the constraint that the M.M, the dot product of M with itself, is equal to 1. This is maximinzing a quadric subject to a quadratic constraint, and so ~~this~~ is easily done via Lagrange multipliers.</li><li>The result of the maximization process is a positive definite quadratic form Q(x1, x2, ..., xn) in n indeterminates x1, x2, ..., xn.</li><li>Find the point T on the subspace of G-tuning space spanned by the <a class="wiki_link" href="/Gencom">gencom mapping</a> closest to the JIP~~, which as~~ usual is &lt;1 1 1 ... 1|.</li><li>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.</li></ol></body></html></pre></div>

<ol><li>Convert the gencom into monzo form.</li><li>Convert the monzos to weighted coordinates.</li><li>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.</li><li>Take the dot product of M with a vector &lt;x1 x2 ... xn| consisting of n new indeterminates x1, x2, ..., xn; call that X.</li><li>Maximize X^2 subject to the constraint that the M.M, the dot product of M with itself, is equal to 1. This is maximinzing a quadric subject to a quadratic constraint, and so is easily done via Lagrange multipliers.</li><li>The result of the maximization process is a positive definite quadratic form Q(x1, x2, ..., xn) in n indeterminates x1, x2, ..., xn.</li><li>Find the point T on the subspace of G-tuning space spanned by the <a class="wiki_link" href="/Gencom">gencom mapping</a> 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.</li><li>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.</li></ol></body></html></pre></div>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2012-06-26 14:01:37 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-06-26 14:08:05 UTC</tt>.<br>
-: The original revision id was <tt>348181842</tt>.<br>
+: The original revision id was <tt>348184558</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>
@@ Line 30: / Line 30: @@
 # Convert the monzos to weighted coordinates.
 # 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 consisting of n new indeterminates x1, x2, ..., xn, and take the dot product of that with M; call that X.
+# 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 maximinzing a quadric subject to a quadratic constraint, and so this is easily done via Lagrange multipliers.
+# Maximize X^2 subject to the constraint that the M.M, the dot product of M with itself, is equal to 1. This is maximinzing a quadric 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, which as usual is &lt;1 1 1 ... 1|.
+# 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.</pre></div>
 <h4>Original HTML content:</h4>
@@ Line 59: / Line 59: @@
 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. Starting from a &lt;a class="wiki_link" href="/gencom"&gt;gencom&lt;/a&gt; for the temperament, we may find the tuning by the following proceedure:&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 consisting of n new indeterminates x1, x2, ..., xn, and take the dot product of that with M; 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 maximinzing a quadric subject to a quadratic constraint, and so this 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, which as usual is &amp;lt;1 1 1 ... 1|.&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;/ol&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
+&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 maximinzing a quadric 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;/ol&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>