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:

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-~~24 17~~:09:17 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-07-10 13:09:56 UTC</tt>.

: The original revision id was <tt>~~511016642~~</tt>.

: The original revision id was <tt>515966118</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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Suppose T = Tp(S) is an Tp 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 of S since T is the Tp tuning. By the [[http://en.wikipedia.org/wiki/Hahn%E2%80%93Banach_theorem|Hahn–Banach theorem]], Ɛ can be extended to an element Ƹ in the space of full p-limit tuning maps with the same norm; that is, so that ||Ɛ|| = ||Ƹ||. Additionally, due to a [[http://www.math.unl.edu/%7Es-bbockel1/928/node25.html|corollary of Hahn-Banach]], the set of such error maps valid for S can be extended to a larger set which is valid for an extended temperament S*; this temperament S* will be of rank greater than or equal to S, and will share the same kernel.

||Ƹ||, the norm of the full p-limit error map, must also be minimal among all valid error maps for S*, or the restriction of Ƹ to G would improve on Ɛ. Hence, as ||Ƹ|| is minimal, J* + Ƹ, where J* is the full p-limit JIP, must equal the Tp tuning for S*. Thus to find the Tp tuning of S for the group G, we may first find the Tp tuning T* for S*, and then apply it to the normal interval list giving the standard form of generators for G.

||Ƹ||, the norm of the full p-limit error map, must also be minimal among all valid error maps for S*, or the restriction of Ƹ to G would improve on Ɛ. Hence, as ||Ƹ|| is minimal, J* + Ƹ, where J* is the full p-limit [[JIP]], must equal the Tp tuning for S*. Thus to find the Tp tuning of S for the group G, we may first find the Tp tuning T* for S*, 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 Zorn's lemma 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.

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Suppose T = Tp(S) is an Tp 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 of S since T is the Tp 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 in the space of full p-limit tuning maps with the same norm; that is, so that ||Ɛ|| = ||Ƹ||. Additionally, due to a <a class="wiki_link_ext" href="http://www.math.unl.edu/%7Es-bbockel1/928/node25.html" rel="nofollow">corollary of Hahn-Banach</a>, the set of such error maps valid for S can be extended to a larger set which is valid for an extended temperament S*; this temperament S* will be of rank greater than or equal to S, and will share the same kernel.

||Ƹ||, the norm of the full p-limit error map, must also be minimal among all valid error maps for S*, or the restriction of Ƹ to G would improve on Ɛ. Hence, as ||Ƹ|| is minimal, J* + Ƹ, where J* is the full p-limit JIP, must equal the Tp tuning for S*. Thus to find the Tp tuning of S for the group G, we may first find the Tp tuning T* for S*, and then apply it to the normal interval list giving the standard form of generators for G.

||Ƹ||, the norm of the full p-limit error map, must also be minimal among all valid error maps for S*, or the restriction of Ƹ to G would improve on Ɛ. Hence, as ||Ƹ|| is minimal, J* + Ƹ, where J* is the full p-limit <a class="wiki_link" href="/JIP">JIP</a>, must equal the Tp tuning for S*. Thus to find the Tp tuning of S for the group G, we may first find the Tp tuning T* for S*, 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 Zorn's lemma 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.

@@ 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>2014-05-24 17:09:17 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-07-10 13:09:56 UTC</tt>.<br>
-: The original revision id was <tt>511016642</tt>.<br>
+: The original revision id was <tt>515966118</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 28: / Line 28: @@
 Suppose T = Tp(S) is an Tp 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 of S since T is the Tp tuning. By the [[http://en.wikipedia.org/wiki/Hahn%E2%80%93Banach_theorem|Hahn–Banach theorem]], Ɛ can be extended to an element Ƹ in the space of full p-limit tuning maps with the same norm; that is, so that ||Ɛ|| = ||Ƹ||. Additionally, due to a [[http://www.math.unl.edu/%7Es-bbockel1/928/node25.html|corollary of Hahn-Banach]], the set of such error maps valid for S can be extended to a larger set which is valid for an extended temperament S*; this temperament S* will be of rank greater than or equal to S, and will share the same kernel.
-||Ƹ||, the norm of the full p-limit error map, must also be minimal among all valid error maps for S*, or the restriction of Ƹ to G would improve on Ɛ. Hence, as ||Ƹ|| is minimal, J* + Ƹ, where J* is the full p-limit JIP, must equal the Tp tuning for S*. Thus to find the Tp tuning of S for the group G, we may first find the Tp tuning T* for S*, and then apply it to the normal interval list giving the standard form of generators for G.
+||Ƹ||, the norm of the full p-limit error map, must also be minimal among all valid error maps for S*, or the restriction of Ƹ to G would improve on Ɛ. Hence, as ||Ƹ|| is minimal, J* + Ƹ, where J* is the full p-limit [[JIP]], must equal the Tp tuning for S*. Thus to find the Tp tuning of S for the group G, we may first find the Tp tuning T* for S*, 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 Zorn's lemma 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.
@@ Line 61: / Line 61: @@
   Suppose T = Tp(S) is an Tp 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 of S since T is the Tp 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 in the space of full p-limit tuning maps with the same norm; that is, so that ||Ɛ|| = ||Ƹ||. Additionally, due to a &lt;a class="wiki_link_ext" href="http://www.math.unl.edu/%7Es-bbockel1/928/node25.html" rel="nofollow"&gt;corollary of Hahn-Banach&lt;/a&gt;, the set of such error maps valid for S can be extended to a larger set which is valid for an extended temperament S*; this temperament S* will be of rank greater than or equal to S, and will share the same kernel.&lt;br /&gt;
 &lt;br /&gt;
-||Ƹ||, the norm of the full p-limit error map, must also be minimal among all valid error maps for S*, or the restriction of Ƹ to G would improve on Ɛ. Hence, as ||Ƹ|| is minimal, J* + Ƹ, where J* is the full p-limit JIP, must equal the Tp tuning for S*. Thus to find the Tp tuning of S for the group G, we may first find the Tp tuning T* for S*, and then apply it to the normal interval list giving the standard form of generators for G.&lt;br /&gt;
+||Ƹ||, the norm of the full p-limit error map, must also be minimal among all valid error maps for S*, or the restriction of Ƹ to G would improve on Ɛ. Hence, as ||Ƹ|| is minimal, J* + Ƹ, where J* is the full p-limit &lt;a class="wiki_link" href="/JIP"&gt;JIP&lt;/a&gt;, must equal the Tp tuning for S*. Thus to find the Tp tuning of S for the group G, we may first find the Tp tuning T* for S*, and then apply it to the normal interval list giving the standard form of generators for G.&lt;br /&gt;
 &lt;br /&gt;
 Note that while the Hahn-Banach theorem is usually proven using Zorn's lemma 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;