Fokker block: Difference between revisions

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A '''Fokker block''' (or periodicity block) is ~~a [[periodic scale|periodic]] [[Constant structure|constant-structure]] [[scale]] that can be thought of as~~ 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 fall upon the lattice with one vertex at the origin~~. A Fokker block ~~comprises~~ those 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 of pitch classes. 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 '''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. 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'''.

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

== Theory ==

Fokker blocks have a shape which 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]].

(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]].

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 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 (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.

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.

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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 <!-- 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 '''Fokker block''' (or periodicity block) is a [[periodic scale|periodic]] [[Constant structure|constant-structure]] [[scale]] that can be thought of as 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 fall upon the lattice with one vertex at the origin. A Fokker block comprises those 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 of pitch classes. 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 '''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. 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'''.
 The concept of the Fokker block was developed by the physicist and music theorist [[Adriaan Fokker]].
 == Theory ==
-Fokker blocks have a shape which 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]].
+(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]].
-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 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 (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.
+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.
 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).