Fokker block: Difference between revisions
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{{Wikipedia| Fokker periodicity block }} | {{Wikipedia| Fokker periodicity block }} | ||
A '''Fokker block''' (or periodicity block) or is a [[periodic scale|periodic]] constant-structure [[scale]] that can be thought of as region of [[pitch class]]es on a lattice (of a [[JI subgroup]] or a [[regular temperament]]) shaped as a parellelogram, parellelepiped, or higher-dimensional analog whose vertices fall upon the lattice with one vertex at the origin. It comprises intervals in the lattice which fall inside the parellelepiped or on the faces of the parellelepiped which intersect the origin and no others (or equivalently, those intervals which fall inside the parellelepiped after it is moved a very small amount while keeping the origin inside it). The scale repeats at the [[interval of equivalence]] (which lies on the [[1/1|unison]] in the lattice due to [[equivalence]]). If the edges of the parellelepiped correspond to intervals which are too large, the Fokker block will not be constant structure and hence a '''weak Fokker block'''. | A '''Fokker block''' (or periodicity block) or is a [[periodic scale|periodic]] constant-structure [[scale]] that can be thought of as region of [[pitch class]]es on a lattice (of a [[JI subgroup]] or a [[regular temperament]]) shaped as a parellelogram, parellelepiped, or higher-dimensional analog whose vertices fall upon the lattice with one vertex at the origin. It comprises intervals in the lattice which fall inside the parellelepiped or on the faces of the parellelepiped which intersect the origin and no others (or equivalently, those intervals which fall inside the parellelepiped after it is moved a very small amount while keeping the origin inside it). The scale repeats at the [[interval of equivalence]] (which lies on the [[1/1|unison]] in the lattice due to [[equivalence]]). If the edges of the parellelepiped correspond to intervals which are too large, the Fokker block will not be constant structure and hence a '''weak Fokker block'''. | ||
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== Mathematical description == | == Mathematical description == | ||
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=== Preliminaries === | === Preliminaries === | ||
While the idea generalizes easily to [[just intonation subgroups]] and tempered groups, for ease of exposition we will suppose that we are in a [[Harmonic limit|''p''-limit]] situation with {{nowrap|''n'' {{=}} π(''p'')}} primes up to and including ''p''. | While the idea generalizes easily to [[just intonation subgroups]] and tempered groups, for ease of exposition we will suppose that we are in a [[Harmonic limit|''p''-limit]] situation with {{nowrap|''n'' {{=}} π(''p'')}} primes up to and including ''p''. | ||