Fokker block: Difference between revisions

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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, or at a fraction thereof.

All Fokker blocks are weakly [[epimorphic]], which means that there is a [[val]] that maps each note of the Fokker block onto its own equal temperament scale step and leaves no equal temperament scale step without a mapping towards it. (In other words, this val provides a bijection between the Fokker block and the equal temperament.) If a Fokker block is epimorphic, which means that the val preserves the order of the steps, it is a ''strong'' Fokker block; otherwise it is a ''weak'' Fokker block. The expression "Fokker block" without any qualifier generally denotes a strong block.

All Fokker blocks are weakly [[epimorphic]], which means that there is a [[val]] that maps each note of the Fokker block onto its own [[equal temperament]] scale step and leaves no equal temperament scale step without a mapping towards it. In other words, this val provides a bijection between the Fokker block and the equal temperament. If a Fokker block is epimorphic, which means that the val preserves the order of the steps, it is a ''strong'' Fokker block; otherwise it is a ''weak'' Fokker block. The expression "Fokker block" without any qualifier generally denotes a strong block.

The concept of the Fokker block was developed by the physicist and music theorist [[Adriaan Fokker]].

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 {{Beginner| Mathematical theory of Fokker blocks}}
 {{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, or at a fraction thereof.
-All Fokker blocks are weakly [[epimorphic]], which means that there is a [[val]] that maps each note of the Fokker block onto its own equal temperament scale step and leaves no equal temperament scale step without a mapping towards it. (In other words, this val provides a bijection between the Fokker block and the equal temperament.) If a Fokker block is epimorphic, which means that the val preserves the order of the steps, it is a ''strong'' Fokker block; otherwise it is a ''weak'' Fokker block. The expression "Fokker block" without any qualifier generally denotes a strong block.
+All Fokker blocks are weakly [[epimorphic]], which means that there is a [[val]] that maps each note of the Fokker block onto its own [[equal temperament]] scale step and leaves no equal temperament scale step without a mapping towards it. In other words, this val provides a bijection between the Fokker block and the equal temperament. If a Fokker block is epimorphic, which means that the val preserves the order of the steps, it is a ''strong'' Fokker block; otherwise it is a ''weak'' Fokker block. The expression "Fokker block" without any qualifier generally denotes a strong block.
 The concept of the Fokker block was developed by the physicist and music theorist [[Adriaan Fokker]].

Fokker block: Difference between revisions

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