Fokker block: Difference between revisions

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A '''~~Weak~~ Fokker block''' (or ~~weak~~ periodicity block) is a ~~region~~ on a lattice of [[pitch class]]es (of a [[JI subgroup]] or a [[regular temperament]]) shaped as a ~~parellelogram, parellelepiped~~, or higher-dimensional analog ~~whose vertices lie in the lattice~~. ~~Scales can be formed from a weak Fokker block comprising~~ those intervals in the lattice which fall inside the ~~parellelepiped~~ (after moving the ~~parellepiped~~ on the lattice to a place where no lattice point is on its boundary). ~~These scales repeat~~ at the [[interval of equivalence]], which lies on the [[1/1|unison]] in the lattice of pitch classes. ~~If the edges of the parellelepiped correspond to intervals so small that when a unit vector on the lattice is stacked it forms a [[mos]], the parellelepiped forms a (strong) '''Fokker 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). The scale repeats at the [[interval of equivalence]], which lies on the [[1/1|unison]] in the lattice of pitch classes.

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

== Theory ==

~~(Weak)~~ Fokker blocks have a shape which tiles the lattice; an interval between pitches which lie across an edge of two ~~(weak)~~ 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]].

The rank of a ~~(weak)~~ Fokker block is the rank of the underlying lattice of pitches including the interval of equivalence. A rank-n Fokker block has 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.

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.

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~~. No scale formed from a weak Fokker block is~~ constant-structure.

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).

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).

== Mathematical description ==

=== 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''.

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 <!-- add beginner section that explains how to build Fokker blocks either by hand or using common software, along with visualizations? -->
 {{Wikipedia| Fokker periodicity block }}
-A '''Weak Fokker block''' (or weak periodicity block) is a region on a lattice of [[pitch class]]es (of a [[JI subgroup]] or a [[regular temperament]]) shaped as a parellelogram, parellelepiped, or higher-dimensional analog whose vertices lie in the lattice. Scales can be formed from a weak Fokker block comprising those intervals in the lattice which fall inside the parellelepiped (after moving the parellepiped on the lattice to a place where no lattice point is on its boundary). These scales repeat at the [[interval of equivalence]], which lies on the [[1/1|unison]] in the lattice of pitch classes. If the edges of the parellelepiped correspond to intervals so small that when a unit vector on the lattice is stacked it forms a [[mos]], the parellelepiped forms a (strong) '''Fokker 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). The scale repeats at the [[interval of equivalence]], which lies on the [[1/1|unison]] in the lattice of pitch classes.
 The concept of the Fokker block was developed by the physicist and music theorist [[Adriaan Fokker]].
 == Theory ==
-(Weak) Fokker blocks have a shape which tiles the lattice; an interval between pitches which lie across an edge of two (weak) 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]].
-The rank of a (weak) Fokker block is the rank of the underlying lattice of pitches including the interval of equivalence. A rank-n Fokker block has 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.
+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.
-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. No scale formed from a weak Fokker block is constant-structure.
+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).
+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).
 == Mathematical description ==
 {{Expert|inline=1}}
 === 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''.