Minkowski block
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- This revision was by author genewardsmith and made on 2010-10-26 01:17:49 UTC.
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Original Wikitext content:
A Minkowski block is a particular kind of [[Fokker block]] which tends to be a good candidate form tempering by a particular regular temperament T. Suppose we have a val v supporting T, and the [[http://mathworld.wolfram.com/Seminorm.html|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 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. This often but not always includes the [[Hobbits|hobbit]] associated with T and v.
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%20block">Fokker block</a> which tends to be a good candidate form tempering by a particular regular temperament T. Suppose we have a val v supporting T, and the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/Seminorm.html" rel="nofollow">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 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. This often but not always includes the <a class="wiki_link" href="/Hobbits">hobbit</a> associated with T and v.</body></html>