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 can be made [[constant structure]] (with negative steps allowed) by moving the generator sizes by an arbitrarily small amount. If the logarithmic sizes of the generators are linearly independent (as happens in JI, for example), the generator sizes need not be moved. If the constant structure does not have negative steps, the Fokker block is called '''strong'''; otherwise, it is called '''weak'''.

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

== Theory ==

~~{{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]].

The rank of a Fokker block is the rank of the underlying lattice of pitches including the interval of equivalence. A rank-''n'' Fokker block has {{nowrap| ''n'' - 1 }} chromas: a consequence of this is that a Fokker block of rank ''n'' has [[maximum variety]] at most 2<sup style="white-space: nowrap;">(''n'' − 1)</sup> (since that's the number of combinations of chromas a note can be altered by). For example, a rank-2 Fokker block has max variety at most 2 (hence is a [[MOS scale|mos]]), and a rank-3 Fokker block has max variety at most 4. In this way, Fokker blocks generalize mos scales.

The rank of a Fokker block is the rank of the underlying lattice of pitches including the interval of equivalence. A rank-''n'' Fokker block has {{nowrap| ''n'' - 1 }} chromas: a consequence of this is that a Fokker block of rank ''n'' has [[maximum variety]] at most 2<sup style="white-space: nowrap;">(''n'' − 1)</sup> (since that's the number of combinations of chromas a note can be altered by). For example, a rank-2 Fokker block has max variety at most 2 (hence is a [[MOS scale|mos]]), and a rank-3 Fokker block has max variety at most 4. These results are true for strong and weak Fokker blocks only if we allow negative steps; otherwise they are only true for strong Fokker blocks. In this way, Fokker blocks generalize mos scales.

~~If the ratios of the cent values of two points on a Fokker block's lattice is always irrational, each scale formed from the block is [[constant structure|constant-structure]]~~.

Fokker blocks may be used to describe scales within JI subgroups or regular temperaments, or to describe rank-1 regular temperaments – that is, equal temperaments – themselves (by taking the chromas as commas to be tempered out).

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 {{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 can be made [[constant structure]] (with negative steps allowed) by moving the generator sizes by an arbitrarily small amount. If the logarithmic sizes of the generators are linearly independent (as happens in JI, for example), the generator sizes need not be moved. If the constant structure does not have negative steps, the Fokker block is called '''strong'''; otherwise, it is called '''weak'''.
 The concept of the Fokker block was developed by the physicist and music theorist [[Adriaan Fokker]].
 == Theory ==
-{{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]].
-The rank of a Fokker block is the rank of the underlying lattice of pitches including the interval of equivalence. A rank-''n'' Fokker block has {{nowrap| ''n'' - 1 }} chromas: a consequence of this is that a Fokker block of rank ''n'' has [[maximum variety]] at most 2<sup style="white-space: nowrap;">(''n'' − 1)</sup> (since that's the number of combinations of chromas a note can be altered by). For example, a rank-2 Fokker block has max variety at most 2 (hence is a [[MOS scale|mos]]), and a rank-3 Fokker block has max variety at most 4. In this way, Fokker blocks generalize mos scales.
+The rank of a Fokker block is the rank of the underlying lattice of pitches including the interval of equivalence. A rank-''n'' Fokker block has {{nowrap| ''n'' - 1 }} chromas: a consequence of this is that a Fokker block of rank ''n'' has [[maximum variety]] at most 2<sup style="white-space: nowrap;">(''n'' − 1)</sup> (since that's the number of combinations of chromas a note can be altered by). For example, a rank-2 Fokker block has max variety at most 2 (hence is a [[MOS scale|mos]]), and a rank-3 Fokker block has max variety at most 4. These results are true for strong and weak Fokker blocks only if we allow negative steps; otherwise they are only true for strong Fokker blocks. In this way, Fokker blocks generalize mos scales.
-If the ratios of the cent values of two points on a Fokker block's lattice is always irrational, each scale formed from the block is [[constant structure|constant-structure]].
 Fokker blocks may be used to describe scales within JI subgroups or regular temperaments, or to describe rank-1 regular temperaments – that is, equal temperaments – themselves (by taking the chromas as commas to be tempered out).