Minkowski block: Difference between revisions
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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. | |||
A Minkowski block is a particular kind of [[ | |||
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 | 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= | =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. | 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. | ||
[[Category:fokker_block]] | |||
[[Category:math]] | |||
Revision as of 00:00, 17 July 2018
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 OE seminorm on 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 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 transversals. This very often but not always includes the 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.