Fokker block: Difference between revisions

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A '''Fokker ~~periodicity~~ block''', or ~~for short "Fokker~~ block" or ~~"periodicity block",~~ is a [[periodic scale|periodic]] [[scale]] that can be thought of as ~~a {{w|parallelepiped #Parallelotope|parallelogram-shaped}} tile~~ of [[pitch class]]es (in a [[JI subgroup]] or a [[regular temperament]]) ~~that can tessellate~~ the ~~entire [[~~lattice]]. It comprises intervals in the lattice which fall inside or on the ~~boundary~~ of the ~~tile. The edges of~~ the ~~periodicity block correspond to~~ intervals ~~that separate~~ the ~~same scale degree but different qualities, i.e. [[chroma]]s~~. The scale repeats at the [[interval of equivalence]] (which lies on the [[1/1|unison]] in the lattice due to [[equivalence]]). ~~The rank of a Fokker block is~~ the ~~rank~~ of the ~~underlying lattice of pitches including~~ the ~~interval of equivalence,~~ and ~~can be thought of as its "dimensionality"~~.

A '''Fokker block''' (or periodicity block) or is a [[periodic scale|periodic]] constant-structure [[scale]] that can be thought of as region of [[pitch class]]es on a lattice (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. It comprises 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 due to [[equivalence]]). 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'''.

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

A Fokker ~~block is usually~~ a ~~[[constant structure]]~~. ~~However,~~ in ~~certain tunings~~, ~~it may not be, in which case~~ it is called a ~~''weak Fokker block''~~.

== 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]].

The ~~concept~~ of ~~the~~ Fokker block ~~was developed heavily by the physicist and music theorist [[Adriaan Fokker]], and as such is named in his honor.~~

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.

~~== Theory ==~~

~~A periodicity block can be described by a "chroma basis", which~~ is the ~~set~~ of ~~intervals that define~~ the ~~boundaries~~ of the ~~block~~. 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.

~~Periodicity~~ 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 ==

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 {{Inaccessible}} <!-- 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 periodicity block''', or for short "Fokker block" or "periodicity block", is a [[periodic scale|periodic]] [[scale]] that can be thought of as a {{w|parallelepiped #Parallelotope|parallelogram-shaped}} tile of [[pitch class]]es (in a [[JI subgroup]] or a [[regular temperament]]) that can tessellate the entire [[lattice]]. It comprises intervals in the lattice which fall inside or on the boundary of the tile. The edges of the periodicity block correspond to intervals that separate the same scale degree but different qualities, i.e. [[chroma]]s. The scale repeats at the [[interval of equivalence]] (which lies on the [[1/1|unison]] in the lattice due to [[equivalence]]). The rank of a Fokker block is the rank of the underlying lattice of pitches including the interval of equivalence, and can be thought of as its "dimensionality".
+A '''Fokker block''' (or periodicity block) or is a [[periodic scale|periodic]] constant-structure [[scale]] that can be thought of as region of [[pitch class]]es on a lattice (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. It comprises 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 due to [[equivalence]]). 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'''.
+The concept of the Fokker block was developed by the physicist and music theorist [[Adriaan Fokker]].
-A Fokker block is usually a [[constant structure]]. However, in certain tunings, it may not be, in which case it is called a ''weak Fokker block''.
+== 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]].
-The concept of the Fokker block was developed heavily by the physicist and music theorist [[Adriaan Fokker]], and as such is named in his honor.
+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.
-== Theory ==
-A periodicity block can be described by a "chroma basis", which is the set of intervals that define the boundaries of the block. 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.
-Periodicity 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 ==