Minkowski block
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- This revision was by author genewardsmith and made on 2014-05-19 14:10:57 UTC.
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Original Wikitext content:
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.
Original HTML content:
<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>