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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | A '''Minkowski block''' is a particular kind of [[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. |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-10-30 05:12:29 UTC</tt>.<br>
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| : The original revision id was <tt>174934257</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| 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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| <h4>Original Wikitext content:</h4>
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| <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;white-space: pre-wrap ! important" class="old-revision-html">
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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. | |
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| We can find a subgroup of just intonation in which every member of the notes of the temperament, for a particular just tuning, has a unique representative. In that case, the seminorm becomes a norm. The commas of the val v belonging to the subgroup 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 the reduction of v to the subgroup, to define Fokker blocks in the usual way. The tempering of these blocks by the temperament are the Minkowski blocks, for which the correspondong Fokker blocks arre therefore [[transverals]]. 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]]s. This very often but not always includes the [[hobbit]] associated with T and v, in which case we may call them hobbit blocks. |
| <h4>Original HTML content:</h4>
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| <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 />
| | == Example == |
| 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="/%2ATenney-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 />
| | 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. |
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| We can find a subgroup of just intonation in which every member of the notes of the temperament, for a particular just tuning, has a unique representative. In that case, the seminorm becomes a norm. The commas of the val v belonging to the subgroup 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 the reduction of v to the subgroup, to define Fokker blocks in the usual way. The tempering of these blocks by the temperament are the Minkowski blocks, for which the correspondong Fokker blocks arre therefore <a class="wiki_link" href="/transverals">transverals</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>
| | [[Category:Math]] |
| | [[Category:Fokker block]] |