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:~~clumma~~|~~clumma~~]] and made on <tt>2012-07-21 16:49:36 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-21 19:15:29 UTC</tt>.<br>

: The original revision id was <tt>~~354228458~~</tt>.<br>

: The original revision id was <tt>354237462</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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=Applying the Hahn-Banach theorem=

Suppose T = Lp(S) is an Lp tuning for the temperament S, and J is the JI tuning. These are both elements of G-tuning space, which are linear functionals on G-interval space, and hence the error map Ɛ = T - J is also. The norm ||Ɛ|| of Ɛ is minimal among all error maps for tunings since T is the Lp tuning. By the [[http://en.wikipedia.org/wiki/Hahn%E2%80%93Banach_theorem|Hahn–Banach theorem]], Ɛ can be extended to an element Ƹ of the full p-limit tuning space with the same norm; that is, so that ||Ɛ|| = ||Ƹ||. This norm must be minimal for the whole tuning space, or the restriction of Ƹ to G would improve on Ɛ. Hence, Ƹ must be the tuning for the full p-limit for the same group of null elements c generated by the commas of S. Thus to find the Lp tuning for the group G, we may first find the tuning for the corresponding higher-rank temperament for the full p-limit group, and then apply it to the normal interval list giving the standard form of generators for G.

Suppose T = Lp(S) is an Lp tuning for the temperament S, and J is the JI tuning. These are both elements of G-tuning space, which are linear functionals on G-interval space, and hence the error map Ɛ = T - J is also. The norm ||Ɛ|| of Ɛ is minimal among all error maps for tunings since T is the Lp tuning. By the [[http://en.wikipedia.org/wiki/Hahn%E2%80%93Banach_theorem|Hahn–Banach theorem]], Ɛ can be extended to an element Ƹ of the full p-limit tuning space with the same norm; that is, so that ||Ɛ|| = ||Ƹ||. This norm must be minimal for the whole tuning space, or the restriction of Ƹ to G would improve on Ɛ. Hence, Ƹ must be the tuning for the full p-limit for the same group of null elements c generated by the commas of S. Thus to find the Lp tuning for the group G, we may first find the tuning for the corresponding higher-rank temperament for the full p-limit group, and then apply it to the normal interval list giving the standard form of generators for G. Note that while the Hahn-Banach theorem is usually proven using the axiom of choice and does not guarantee any kind of uniqueness, in most cases there is only one Lp tuning and the extension of Ɛ to Ƹ is in that case unique.

=L2 tuning=

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<br />

<h1 id="toc2"><a name="Applying the Hahn-Banach theorem"></a>Applying the Hahn-Banach theorem</h1>

Suppose T = Lp(S) is an Lp tuning for the temperament S, and J is the JI tuning. These are both elements of G-tuning space, which are linear functionals on G-interval space, and hence the error map Ɛ = T - J is also. The norm ||Ɛ|| of Ɛ is minimal among all error maps for tunings since T is the Lp tuning. By the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Hahn%E2%80%93Banach_theorem" rel="nofollow">Hahn–Banach theorem</a>, Ɛ can be extended to an element Ƹ of the full p-limit tuning space with the same norm; that is, so that ||Ɛ|| = ||Ƹ||. This norm must be minimal for the whole tuning space, or the restriction of Ƹ to G would improve on Ɛ. Hence, Ƹ must be the tuning for the full p-limit for the same group of null elements c generated by the commas of S. Thus to find the Lp tuning for the group G, we may first find the tuning for the corresponding higher-rank temperament for the full p-limit group, and then apply it to the normal interval list giving the standard form of generators for G.<br />

Suppose T = Lp(S) is an Lp tuning for the temperament S, and J is the JI tuning. These are both elements of G-tuning space, which are linear functionals on G-interval space, and hence the error map Ɛ = T - J is also. The norm ||Ɛ|| of Ɛ is minimal among all error maps for tunings since T is the Lp tuning. By the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Hahn%E2%80%93Banach_theorem" rel="nofollow">Hahn–Banach theorem</a>, Ɛ can be extended to an element Ƹ of the full p-limit tuning space with the same norm; that is, so that ||Ɛ|| = ||Ƹ||. This norm must be minimal for the whole tuning space, or the restriction of Ƹ to G would improve on Ɛ. Hence, Ƹ must be the tuning for the full p-limit for the same group of null elements c generated by the commas of S. Thus to find the Lp tuning for the group G, we may first find the tuning for the corresponding higher-rank temperament for the full p-limit group, and then apply it to the normal interval list giving the standard form of generators for G. Note that while the Hahn-Banach theorem is usually proven using the axiom of choice and does not guarantee any kind of uniqueness, in most cases there is only one Lp tuning and the extension of Ɛ to Ƹ is in that case unique.<br />

<br />

<h1 id="toc3"><a name="L2 tuning"></a>L2 tuning</h1>

@@ 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:clumma|clumma]] and made on <tt>2012-07-21 16:49:36 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-21 19:15:29 UTC</tt>.<br>
-: The original revision id was <tt>354228458</tt>.<br>
+: The original revision id was <tt>354237462</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 25: / Line 25: @@
 =Applying the Hahn-Banach theorem=
-Suppose T = Lp(S) is an Lp tuning for the temperament S, and J is the JI tuning. These are both elements of G-tuning space, which are linear functionals on G-interval space, and hence the error map Ɛ = T - J is also. The norm ||Ɛ|| of Ɛ is minimal among all error maps for tunings since T is the Lp tuning. By the [[http://en.wikipedia.org/wiki/Hahn%E2%80%93Banach_theorem|Hahn–Banach theorem]], Ɛ can be extended to an element Ƹ of the full p-limit tuning space with the same norm; that is, so that ||Ɛ|| = ||Ƹ||. This norm must be minimal for the whole tuning space, or the restriction of Ƹ to G would improve on Ɛ. Hence, Ƹ must be the tuning for the full p-limit for the same group of null elements c generated by the commas of S. Thus to find the Lp tuning for the group G, we may first find the tuning for the corresponding higher-rank temperament for the full p-limit group, and then apply it to the normal interval list giving the standard form of generators for G.
+Suppose T = Lp(S) is an Lp tuning for the temperament S, and J is the JI tuning. These are both elements of G-tuning space, which are linear functionals on G-interval space, and hence the error map Ɛ = T - J is also. The norm ||Ɛ|| of Ɛ is minimal among all error maps for tunings since T is the Lp tuning. By the [[http://en.wikipedia.org/wiki/Hahn%E2%80%93Banach_theorem|Hahn–Banach theorem]], Ɛ can be extended to an element Ƹ of the full p-limit tuning space with the same norm; that is, so that ||Ɛ|| = ||Ƹ||. This norm must be minimal for the whole tuning space, or the restriction of Ƹ to G would improve on Ɛ. Hence, Ƹ must be the tuning for the full p-limit for the same group of null elements c generated by the commas of S. Thus to find the Lp tuning for the group G, we may first find the tuning for the corresponding higher-rank temperament for the full p-limit group, and then apply it to the normal interval list giving the standard form of generators for G. Note that while the Hahn-Banach theorem is usually proven using the axiom of choice and does not guarantee any kind of uniqueness, in most cases there is only one Lp tuning and the extension of Ɛ to Ƹ is in that case unique.
 =L2 tuning=
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 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Applying the Hahn-Banach theorem"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Applying the Hahn-Banach theorem&lt;/h1&gt;
-Suppose T = Lp(S) is an Lp tuning for the temperament S, and J is the JI tuning. These are both elements of G-tuning space, which are linear functionals on G-interval space, and hence the error map Ɛ = T - J is also. The norm ||Ɛ|| of Ɛ is minimal among all error maps for tunings since T is the Lp tuning. By the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Hahn%E2%80%93Banach_theorem" rel="nofollow"&gt;Hahn–Banach theorem&lt;/a&gt;, Ɛ can be extended to an element Ƹ of the full p-limit tuning space with the same norm; that is, so that ||Ɛ|| = ||Ƹ||. This norm must be minimal for the whole tuning space, or the restriction of Ƹ to G would improve on Ɛ. Hence, Ƹ must be the tuning for the full p-limit for the same group of null elements c generated by the commas of S. Thus to find the Lp tuning for the group G, we may first find the tuning for the corresponding higher-rank temperament for the full p-limit group, and then apply it to the normal interval list giving the standard form of generators for G.&lt;br /&gt;
+Suppose T = Lp(S) is an Lp tuning for the temperament S, and J is the JI tuning. These are both elements of G-tuning space, which are linear functionals on G-interval space, and hence the error map Ɛ = T - J is also. The norm ||Ɛ|| of Ɛ is minimal among all error maps for tunings since T is the Lp tuning. By the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Hahn%E2%80%93Banach_theorem" rel="nofollow"&gt;Hahn–Banach theorem&lt;/a&gt;, Ɛ can be extended to an element Ƹ of the full p-limit tuning space with the same norm; that is, so that ||Ɛ|| = ||Ƹ||. This norm must be minimal for the whole tuning space, or the restriction of Ƹ to G would improve on Ɛ. Hence, Ƹ must be the tuning for the full p-limit for the same group of null elements c generated by the commas of S. Thus to find the Lp tuning for the group G, we may first find the tuning for the corresponding higher-rank temperament for the full p-limit group, and then apply it to the normal interval list giving the standard form of generators for G. Note that while the Hahn-Banach theorem is usually proven using the axiom of choice and does not guarantee any kind of uniqueness, in most cases there is only one Lp tuning and the extension of Ɛ to Ƹ is in that case unique.&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;