Minkowski block
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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 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.
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 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 /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Example"></a><!-- ws:end:WikiTextHeadingRule:0 -->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>