Fokker block: Difference between revisions
Undo revision 203585 by VectorGraphics (talk) Do not edit math you clearly do not understand. Tag: Undo |
Undo revision 203566 by VectorGraphics (talk). Readers aren't obligated to see your poor-tasted operation name |
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{{Wikipedia| Fokker periodicity block }} | {{Wikipedia| Fokker periodicity block }} | ||
A '''Fokker block''' (or periodicity block) is a [[periodic scale|periodic]] [[scale]] that can be thought of as a tile on a lattice of [[pitch class]]es (of a [[JI subgroup]] or a [[regular temperament]]) shaped as a parallelogram, parallelepiped or higher-dimensional analog. It comprises those intervals in the lattice which fall inside the tile after moving the tile on the lattice to a place where no lattice point is on its boundary. (Different positions of the tile can create scales which are not rotations of each other.) The scale repeats at the [[interval of equivalence]], which lies on the [[1/1|unison]] in the lattice of pitch classes. | A '''Fokker block''' (or periodicity block) is a [[periodic scale|periodic]] [[scale]] that can be thought of as a tile on a lattice of [[pitch class]]es (of a [[JI subgroup]] or a [[regular temperament]]) shaped as a parallelogram, parallelepiped or higher-dimensional analog. It comprises those intervals in the lattice which fall inside the tile after moving the tile on the lattice to a place where no lattice point is on its boundary. (Different positions of the tile can create scales which are not rotations of each other.) The scale repeats at the [[interval of equivalence]], which lies on the [[1/1|unison]] in the lattice of pitch classes. | ||
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== Theory == | == Theory == | ||
{{Todo | {{Todo|inline=1| add definition |text=Either the "strong Fokker block" definition needs to be here, or the second and third paragraphs, which are not true for weak Fokker blocks, need to be removed.}} | ||
Fokker blocks have a shape which {{w|tessellation|tiles}} the lattice; an interval between pitches which lie across an edge of two Fokker blocks within the tiling will be altered from its normal value by an interval corresponding to an edge of the parellelepiped. This edge turns out to be the difference between intervals that span the same number of steps in the scale, and so it is called a [[chroma]]. | Fokker blocks have a shape which {{w|tessellation|tiles}} the lattice; an interval between pitches which lie across an edge of two Fokker blocks within the tiling will be altered from its normal value by an interval corresponding to an edge of the parellelepiped. This edge turns out to be the difference between intervals that span the same number of steps in the scale, and so it is called a [[chroma]]. | ||
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=== Arena === | === Arena === | ||
{{Todo | {{Todo|inline=1| add definition |text=Definitions in the "Terminology" section should not utilize mathematical jargon.}} | ||
== Mathematical description == | == Mathematical description == | ||
{{ | {{Inacc}} | ||
{{Todo|inline=1| reduce mathslang |comment=Try explaining without wedgies. }} | |||
=== 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. | 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. | ||
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[[Category:Math]] | [[Category:Math]] | ||
[[Category:Pitch space]] | [[Category:Pitch space]] | ||
{{Todo| add illustration | improve linking | review }} <!-- add beginner section that explains how to build Fokker blocks either by hand or using common software, along with visualizations. --> | |||