Fokker block: Difference between revisions

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{{Inaccessible}}

A '''Fokker 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 ~~elements of~~ the lattice which fall inside or on the boundary of the tile~~, the~~ edges of ~~which~~ 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.

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 of rank ''r'' has [[maximum variety]] at most 2(''r'' − 1). 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.

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''.

The Fokker block ~~is one of the most notable inventions of~~ the physicist and music theorist [[Adriaan Fokker]].

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.

== 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(''n'' − 1) (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).

== 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 an including ''p''.

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''.

Suppose we have {{nowrap|''n'' − 1}} commas, which we will assume are greater than 1, and we form an {{nowrap| ''n'' × ''n'' }} matrix, the top row of which are ''n'' indeterminate elements {{monzo| ''e''2 ''e''3 ''e''5 … ''e''''p'' }}, and the other rows of which are the monzos corresponding to our chosen commas. If we take the determinant of this matrix, we get {{nowrap|''w''2''e''2 + ''w''3''e''3 + … + ''w''''p''''e''''p''}} where the ''w''2, ''w''3 … ''w''''p'' are integers. We interpret this as the [[vals and tuning space|val]] {{nowrap|'''v''' {{=}} {{val| w2 w3 … w''p'' }}}}. If this is a zero vector the commas are not independent, and if the there exists a common divisor we have what is known as a torsion problem, and we discard the comma set. Otherwise, if {{nowrap|''w''2 < 0}} we reverse sign, and we have a val ''V'' which tells us what equal temperament our Fokker block will be approximating. For example, starting with the commas 225/224, 100/99, 176/175, and 385/384, the above procedure gives us {{nowrap|''V'' {{=}} {{val| 22 35 51 62 76 }}}}, and we will be looking at a 22-note scale in the 11-limit. We may call ''V'' the ''epimorph val'', and the {{nowrap|''n'' − 1}} commas, which form a basis for the kernel of ''V'', the ''chroma basis''.

Suppose we have {{nowrap|''n'' − 1}} commas, which we will assume are greater than 1, and we form an {{nowrap| ''n'' × ''n'' }} matrix, the top row of which are ''n'' indeterminate elements {{monzo| ''e''2 ''e''3 ''e''5 … ''e''''p'' }}, and the other rows of which are the monzos corresponding to our chosen commas. If we take the determinant of this matrix, we get {{nowrap|''w''2''e''2 + ''w''3''e''3 + … + ''w''''p''''e''''p''}} where the ''w''2, ''w''3 … ''w''''p'' are integers. We interpret this as the [[vals and tuning space|val]] {{nowrap|'''v''' {{=}} {{val| w2 w3 … w''p'' }}}}. If this is a zero vector the commas are not independent, and if the there exists a common divisor we have what is known as a [[Saturation, torsion, and contorsion|torsion]] problem, and we discard the comma set. Otherwise, if {{nowrap|''w''2 < 0}} we reverse sign, and we have a val ''V'' which tells us what equal temperament our Fokker block will be approximating. For example, starting with the commas 225/224, 100/99, 176/175, and 385/384, the above procedure gives us {{nowrap|''V'' {{=}} {{val| 22 35 51 62 76 }}}}, and we will be looking at a 22-note scale in the 11-limit. We may call ''V'' the ''epimorph val'', and the {{nowrap|''n'' − 1}} commas, which form a basis for the kernel of ''V'', the ''chroma basis''.

Now choose a uniformizing step for the Fokker block, by which is meant a ''p''-limit interval ''c'' such that {{nowrap|''V''(''c'') {{=}} 1}}; that is, if '''m''' is the monzo for ''c'', then {{nowrap|{{vmp|''V''|'''m'''}} {{=}} 1}}. Precisely which interval with this property we choose does not actually matter, so if our chromas are 225/224, 100/99, 176/175, and 385/384, we could for instance choose 22/21, 25/24, 28/27, 33/32, 36/35, 45/44, or 49/48. Having selected a step, form the {{nowrap| ''n'' × ''n'' }} matrix whose last row is the monzo for the step ''c'', and whose other rows are the monzos of the {{nowrap|''n'' − 1}} chromas. Because we have chosen ''c'' so that {{nowrap|''V''(''c'') {{=}} 1}}, the determinant of this matrix will be ±1. It is therefore a {{w|unimodular matrix}}, that is, a square matrix with coefficients which are integers and with determinant ±1. Such a matrix is invertible, and the inverse matrix is also unimodular. If we call ''c'' "''c''''n''", and label the chromas ''c''1, ''c''2, … , ''c''(''n'' − 1); and if we consider the columns of the inverse matrix to be vals and call them '''v'''1, '''v'''2, … , '''v'''''n'', then by the definition of the inverse of a matrix, {{nowrap|'''v'''''i''(''c''''j'') {{=}} δ(''i'', ''j'')}}, where δ(''i'', ''j'') is the {{w|Kronecker delta}}. Stated another way, '''v'''''i''(''c''''j'') is 0 unless {{nowrap|''i'' {{=}} ''j''}}, in which case {{nowrap|'''v'''''i''(''c''''i'') {{=}} 1}}.

@@ Line 1: / Line 1: @@
 {{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 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 elements of the lattice which fall inside or on the boundary of the tile, the edges of which 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.
+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 of rank ''r'' has [[maximum variety]] at most 2<sup style="white-space: nowrap;">(''r'' − 1)</sup>. 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.
 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''.
-The Fokker block is one of the most notable inventions of the physicist and music theorist [[Adriaan Fokker]].
+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.
+== 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).
 == 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 an including ''p''.
+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''.
-Suppose we have {{nowrap|''n'' − 1}} commas, which we will assume are greater than 1, and we form an {{nowrap| ''n'' × ''n'' }} matrix, the top row of which are ''n'' indeterminate elements {{monzo| ''e''<sub>2</sub> ''e''<sub>3</sub> ''e''<sub>5</sub> … ''e''<sub>''p''</sub> }}, and the other rows of which are the monzos corresponding to our chosen commas. If we take the determinant of this matrix, we get {{nowrap|''w''<sub>2</sub>''e''<sub>2</sub> + ''w''<sub>3</sub>''e''<sub>3</sub> + … + ''w''<sub>''p''</sub>''e''<sub>''p''</sub>}} where the ''w''<sub>2</sub>, ''w''<sub>3</sub> … ''w''<sub>''p''</sub> are integers. We interpret this as the [[vals and tuning space|val]] {{nowrap|'''v''' {{=}} {{val| w<sub>2</sub> w<sub>3</sub> … w<sub>''p''</sub> }}}}. If this is a zero vector the commas are not independent, and if the there exists a common divisor we have what is known as a torsion problem, and we discard the comma set. Otherwise, if {{nowrap|''w''<sub>2</sub> &lt; 0}} we reverse sign, and we have a val ''V'' which tells us what equal temperament our Fokker block will be approximating. For example, starting with the commas 225/224, 100/99, 176/175, and 385/384, the above procedure gives us {{nowrap|''V'' {{=}} {{val| 22 35 51 62 76 }}}}, and we will be looking at a 22-note scale in the 11-limit. We may call ''V'' the ''epimorph val'', and the {{nowrap|''n'' − 1}} commas, which form a basis for the kernel of ''V'', the ''chroma basis''.
+Suppose we have {{nowrap|''n'' − 1}} commas, which we will assume are greater than 1, and we form an {{nowrap| ''n'' × ''n'' }} matrix, the top row of which are ''n'' indeterminate elements {{monzo| ''e''<sub>2</sub> ''e''<sub>3</sub> ''e''<sub>5</sub> … ''e''<sub>''p''</sub> }}, and the other rows of which are the monzos corresponding to our chosen commas. If we take the determinant of this matrix, we get {{nowrap|''w''<sub>2</sub>''e''<sub>2</sub> + ''w''<sub>3</sub>''e''<sub>3</sub> + … + ''w''<sub>''p''</sub>''e''<sub>''p''</sub>}} where the ''w''<sub>2</sub>, ''w''<sub>3</sub> … ''w''<sub>''p''</sub> are integers. We interpret this as the [[vals and tuning space|val]] {{nowrap|'''v''' {{=}} {{val| w<sub>2</sub> w<sub>3</sub> … w<sub>''p''</sub> }}}}. If this is a zero vector the commas are not independent, and if the there exists a common divisor we have what is known as a [[Saturation, torsion, and contorsion|torsion]] problem, and we discard the comma set. Otherwise, if {{nowrap|''w''<sub>2</sub> &lt; 0}} we reverse sign, and we have a val ''V'' which tells us what equal temperament our Fokker block will be approximating. For example, starting with the commas 225/224, 100/99, 176/175, and 385/384, the above procedure gives us {{nowrap|''V'' {{=}} {{val| 22 35 51 62 76 }}}}, and we will be looking at a 22-note scale in the 11-limit. We may call ''V'' the ''epimorph val'', and the {{nowrap|''n'' − 1}} commas, which form a basis for the kernel of ''V'', the ''chroma basis''.
 Now choose a uniformizing step for the Fokker block, by which is meant a ''p''-limit interval ''c'' such that {{nowrap|''V''(''c'') {{=}} 1}}; that is, if '''m''' is the monzo for ''c'', then {{nowrap|{{vmp|''V''|'''m'''}} {{=}} 1}}. Precisely which interval with this property we choose does not actually matter, so if our chromas are 225/224, 100/99, 176/175, and 385/384, we could for instance choose 22/21, 25/24, 28/27, 33/32, 36/35, 45/44, or 49/48. Having selected a step, form the {{nowrap| ''n'' × ''n'' }} matrix whose last row is the monzo for the step ''c'', and whose other rows are the monzos of the {{nowrap|''n'' − 1}} chromas. Because we have chosen ''c'' so that {{nowrap|''V''(''c'') {{=}} 1}}, the determinant of this matrix will be ±1. It is therefore a {{w|unimodular matrix}}, that is, a square matrix with coefficients which are integers and with determinant ±1. Such a matrix is invertible, and the inverse matrix is also unimodular. If we call ''c'' "''c''<sub>''n''</sub>", and label the chromas ''c''<sub>1</sub>, ''c''<sub>2</sub>, … , ''c''<sub style="white-space: nowrap;">(''n'' − 1)</sub>; and if we consider the columns of the inverse matrix to be vals and call them '''v'''<sub>1</sub>, '''v'''<sub>2</sub>, … , '''v'''<sub>''n''</sub>, then by the definition of the inverse of a matrix, {{nowrap|'''v'''<sub>''i''</sub>(''c''<sub>''j''</sub>) {{=}} δ(''i'', ''j'')}}, where δ(''i'',&nbsp;''j'') is the {{w|Kronecker delta}}. Stated another way, '''v'''<sub>''i''</sub>(''c''<sub>''j''</sub>) is 0 unless {{nowrap|''i'' {{=}} ''j''}}, in which case {{nowrap|'''v'''<sub>''i''</sub>(''c''<sub>''i''</sub>) {{=}} 1}}.