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>2012-07-21 14:15:39 UTC</tt>.<br>

: This revision was by author [[User:clumma|clumma]] and made on <tt>2012-07-21 16:47:21 UTC</tt>.<br>

: The original revision id was <tt>~~354219176~~</tt>.<br>

: The original revision id was <tt>354228342</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 ~~the (or at least a)~~ 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 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 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.

=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 ~~the (or at least a)~~ 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 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 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 />

<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:genewardsmith|genewardsmith]] and made on <tt>2012-07-21 14:15:39 UTC</tt>.<br>
+: This revision was by author [[User:clumma|clumma]] and made on <tt>2012-07-21 16:47:21 UTC</tt>.<br>
-: The original revision id was <tt>354219176</tt>.<br>
+: The original revision id was <tt>354228342</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 the (or at least a) 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 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 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.
 =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 the (or at least a) 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 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 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;
 &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;