Minkowski block: 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>2014-05-19 14:10:57 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-19 23:29:04 UTC</tt>.<br>

: The original revision id was <tt>~~509878950~~</tt>.<br>

: The original revision id was <tt>509980780</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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A Minkowski block is a particular kind of [[Fokker blocks|Fokker block]] which tends to be a good candidate for tempering by a particular regular temperament T. Suppose we have a val v supporting T, and the [[Tenney-Euclidean metrics|OE seminorm]] on [[Monzos and Interval Space|interval space]] defined from the temperament; that is, the seminorm defined by orthogonal projection in interval space orthogonal to the commas of T and 2, the octave.

We can find a subspace of interval space in which every note of T has a unique representative, giving a transversal for the temperament in the form of a sublattice of the lattice of intervals T tempers. In that subspace, the seminorm becomes a norm. The commas of v belonging to T have a unique [[http://www.farcaster.com/papers/sm-thesis/node6.html|Minkowski basis]] in terms of this norm, and we may use these commas~~, and v~~ to define Fokker blocks in the usual way. The tempering of these blocks by T are the Minkowski blocks, for which the correspondong Fokker blocks are therefore [[transversal|transversals]]. This very often but not always includes the [[Hobbits|hobbit]] associated with T and v, in which case we may call them hobbit blocks.</pre></div>

We can find a subspace of interval space in which every note of T has a unique representative, giving a transversal for the temperament in the form of a sublattice of the lattice of intervals T tempers. In that subspace, the seminorm becomes a norm. The commas of v belonging to the transversal sublattice have a unique [[http://www.farcaster.com/papers/sm-thesis/node6.html|Minkowski basis]] in terms of this norm, and we may use these commas to define Fokker blocks in the usual way. The tempering of these blocks by T are the Minkowski blocks, for which the correspondong Fokker blocks are therefore [[transversal|transversals]]. This very often but not always includes the [[Hobbits|hobbit]] associated with T and v, in which case we may call them hobbit blocks.

=Example=

Consider marvel, which is supported by the 11-limit 19et patent val, since it tempers out both 225/224 and 385/384. The transversal sublattice can be taken to be 5-limit JI: every note of marvel has a unique 5-limit JI representative. The 5-limit commas of 19et, in order of the OE seminorm, are 81/80, 4428675/4194304, 273375/262144, 16875/16384, 3125/3072, 15625/15552, 78732/78125... . In terms of the associated 5-limit temperaments, that's meantone, hogzilla, stump, negri, magic, hanson, sensi... . The meantone-hogzilla arena is therefore the arena of the Minkowski blocks for 19-note marvel.</pre></div>

<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"><html><head><title>Minkowski blocks</title></head><body><br />

A Minkowski block is a particular kind of <a class="wiki_link" href="/Fokker%20blocks">Fokker block</a> which tends to be a good candidate for tempering by a particular regular temperament T. Suppose we have a val v supporting T, and the <a class="wiki_link" href="/Tenney-Euclidean%20metrics">OE seminorm</a> on <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">interval space</a> defined from the temperament; that is, the seminorm defined by orthogonal projection in interval space orthogonal to the commas of T and 2, the octave.<br />

<br />

We can find a subspace of interval space in which every note of T has a unique representative, giving a transversal for the temperament in the form of a sublattice of the lattice of intervals T tempers. In that subspace, the seminorm becomes a norm. The commas of v belonging to T have a unique <a class="wiki_link_ext" href="http://www.farcaster.com/papers/sm-thesis/node6.html" rel="nofollow">Minkowski basis</a> in terms of this norm, and we may use these commas~~, and v~~ to define Fokker blocks in the usual way. The tempering of these blocks by T are the Minkowski blocks, for which the correspondong Fokker blocks are therefore <a class="wiki_link" href="/transversal">transversals</a>. This very often but not always includes the <a class="wiki_link" href="/Hobbits">hobbit</a> associated with T and v, in which case we may call them hobbit blocks.</body></html></pre></div>

We can find a subspace of interval space in which every note of T has a unique representative, giving a transversal for the temperament in the form of a sublattice of the lattice of intervals T tempers. In that subspace, the seminorm becomes a norm. The commas of v belonging to the transversal sublattice have a unique <a class="wiki_link_ext" href="http://www.farcaster.com/papers/sm-thesis/node6.html" rel="nofollow">Minkowski basis</a> in terms of this norm, and we may use these commas to define Fokker blocks in the usual way. The tempering of these blocks by T are the Minkowski blocks, for which the correspondong Fokker blocks are therefore <a class="wiki_link" href="/transversal">transversals</a>. This very often but not always includes the <a class="wiki_link" href="/Hobbits">hobbit</a> associated with T and v, in which case we may call them hobbit blocks.<br />

<br />

<h1 id="toc0"><a name="Example"></a>Example</h1>

Consider marvel, which is supported by the 11-limit 19et patent val, since it tempers out both 225/224 and 385/384. The transversal sublattice can be taken to be 5-limit JI: every note of marvel has a unique 5-limit JI representative. The 5-limit commas of 19et, in order of the OE seminorm, are 81/80, 4428675/4194304, 273375/262144, 16875/16384, 3125/3072, 15625/15552, 78732/78125... . In terms of the associated 5-limit temperaments, that's meantone, hogzilla, stump, negri, magic, hanson, sensi... . The meantone-hogzilla arena is therefore the arena of the Minkowski blocks for 19-note marvel.</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>2014-05-19 14:10:57 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-19 23:29:04 UTC</tt>.<br>
-: The original revision id was <tt>509878950</tt>.<br>
+: The original revision id was <tt>509980780</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 9: / Line 9: @@
 A Minkowski block is a particular kind of [[Fokker blocks|Fokker block]] which tends to be a good candidate for tempering by a particular regular temperament T. Suppose we have a val v supporting T, and the [[Tenney-Euclidean metrics|OE seminorm]] on [[Monzos and Interval Space|interval space]] defined from the temperament; that is, the seminorm defined by orthogonal projection in interval space orthogonal to the commas of T and 2, the octave.
-We can find a subspace of interval space in which every note of T has a unique representative, giving a transversal for the temperament in the form of a sublattice of the lattice of intervals T tempers. In that subspace, the seminorm becomes a norm. The commas of v belonging to T have a unique [[http://www.farcaster.com/papers/sm-thesis/node6.html|Minkowski basis]] in terms of this norm, and we may use these commas, and v to define Fokker blocks in the usual way. The tempering of these blocks by T are the Minkowski blocks, for which the correspondong Fokker blocks are therefore [[transversal|transversals]]. This very often but not always includes the [[Hobbits|hobbit]] associated with T and v, in which case we may call them hobbit blocks.</pre></div>
+We can find a subspace of interval space in which every note of T has a unique representative, giving a transversal for the temperament in the form of a sublattice of the lattice of intervals T tempers. In that subspace, the seminorm becomes a norm. The commas of v belonging to the transversal sublattice have a unique [[http://www.farcaster.com/papers/sm-thesis/node6.html|Minkowski basis]] in terms of this norm, and we may use these commas to define Fokker blocks in the usual way. The tempering of these blocks by T are the Minkowski blocks, for which the correspondong Fokker blocks are therefore [[transversal|transversals]]. This very often but not always includes the [[Hobbits|hobbit]] associated with T and v, in which case we may call them hobbit blocks.
+=Example=
+Consider marvel, which is supported by the 11-limit 19et patent val, since it tempers out both 225/224 and 385/384. The transversal sublattice can be taken to be 5-limit JI: every note of marvel has a unique 5-limit JI representative. The 5-limit commas of 19et, in order of the OE seminorm, are 81/80, 4428675/4194304, 273375/262144, 16875/16384, 3125/3072, 15625/15552, 78732/78125... . In terms of the associated 5-limit temperaments, that's meantone, hogzilla, stump, negri, magic, hanson, sensi... . The meantone-hogzilla arena is therefore the arena of the Minkowski blocks for 19-note marvel.</pre></div>
 <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;Minkowski blocks&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;br /&gt;
 A Minkowski block is a particular kind of &lt;a class="wiki_link" href="/Fokker%20blocks"&gt;Fokker block&lt;/a&gt; which tends to be a good candidate for tempering by a particular regular temperament T. Suppose we have a val v supporting T, and the &lt;a class="wiki_link" href="/Tenney-Euclidean%20metrics"&gt;OE seminorm&lt;/a&gt; on &lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;interval space&lt;/a&gt; defined from the temperament; that is, the seminorm defined by orthogonal projection in interval space orthogonal to the commas of T and 2, the octave.&lt;br /&gt;
 &lt;br /&gt;
-We can find a subspace of interval space in which every note of T has a unique representative, giving a transversal for the temperament in the form of a sublattice of the lattice of intervals T tempers. In that subspace, the seminorm becomes a norm. The commas of v belonging to T have a unique &lt;a class="wiki_link_ext" href="http://www.farcaster.com/papers/sm-thesis/node6.html" rel="nofollow"&gt;Minkowski basis&lt;/a&gt; in terms of this norm, and we may use these commas, and v to define Fokker blocks in the usual way. The tempering of these blocks by T are the Minkowski blocks, for which the correspondong Fokker blocks are therefore &lt;a class="wiki_link" href="/transversal"&gt;transversals&lt;/a&gt;. This very often but not always includes the &lt;a class="wiki_link" href="/Hobbits"&gt;hobbit&lt;/a&gt; associated with T and v, in which case we may call them hobbit blocks.&lt;/body&gt;&lt;/html&gt;</pre></div>
+We can find a subspace of interval space in which every note of T has a unique representative, giving a transversal for the temperament in the form of a sublattice of the lattice of intervals T tempers. In that subspace, the seminorm becomes a norm. The commas of v belonging to the transversal sublattice have a unique &lt;a class="wiki_link_ext" href="http://www.farcaster.com/papers/sm-thesis/node6.html" rel="nofollow"&gt;Minkowski basis&lt;/a&gt; in terms of this norm, and we may use these commas to define Fokker blocks in the usual way. The tempering of these blocks by T are the Minkowski blocks, for which the correspondong Fokker blocks are therefore &lt;a class="wiki_link" href="/transversal"&gt;transversals&lt;/a&gt;. This very often but not always includes the &lt;a class="wiki_link" href="/Hobbits"&gt;hobbit&lt;/a&gt; associated with T and v, in which case we may call them hobbit blocks.&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Example"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Example&lt;/h1&gt;
+Consider marvel, which is supported by the 11-limit 19et patent val, since it tempers out both 225/224 and 385/384. The transversal sublattice can be taken to be 5-limit JI: every note of marvel has a unique 5-limit JI representative. The 5-limit commas of 19et, in order of the OE seminorm, are 81/80, 4428675/4194304, 273375/262144, 16875/16384, 3125/3072, 15625/15552, 78732/78125... . In terms of the associated 5-limit temperaments, that's meantone, hogzilla, stump, negri, magic, hanson, sensi... . The meantone-hogzilla arena is therefore the arena of the Minkowski blocks for 19-note marvel.&lt;/body&gt;&lt;/html&gt;</pre></div>