Fokker block: Difference between revisions

(4 intermediate revisions by 2 users not shown)

Line 1:

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.

All Fokker blocks are weakly [[epimorphic]]~~; if~~ a Fokker block is epimorphic ~~(i.e. it has no negative~~ steps), it is a ''strong'' Fokker block, otherwise it is a ''weak'' Fokker block. ~~An unqualified~~ Fokker block is generally ~~assumed to be~~ strong.

All Fokker blocks are weakly [[epimorphic]], which means that there is a [[val]] that maps each note of the Fokker block onto its own equal temperament scale step and leaves no equal temperament scale step without a mapping towards it. (In other words, this val provides a bijection between the Fokker block and the equal temperament.) If a Fokker block is epimorphic, which means that the val preserves the order of the steps, it is a ''strong'' Fokker block; otherwise it is a ''weak'' Fokker block. The expression "Fokker block" without any qualifier generally denotes a strong block.

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

== Theory ==

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]].

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]]. (Fokker called them ''unison vectors'' in his original text.)

If we place the coordinate vectors of the chromas into a matrix, the number of notes in the Fokker block is equal to the determinant of that matrix. It is also equal to the volume of the parallelepiped formed by the chromas, which is called the '''fundamental domain'''.

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.

Line 14:

Line 16:

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. The constant structure will have no negative steps if and only if the Fokker block is strong.

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).

A Fokker '''arena''' contains all the periodic scales that can be constructed as Fokker blocks from the same list of chromas.

~~== Terminology ==~~

~~{{Todo|add definition |text=Definitions in the "Terminology" section should not utilize mathematical jargon.}}~~

~~=== Arena ===~~

A Fokker arena contains all the periodic scales that can be constructed as Fokker blocks from the same list of ~~commas.~~

~~== Definition ==~~

~~Fokker blocks are defined on octave-equivalent [[lattice]]s, which can be visualized as grids.~~

~~These can be constructed by simply dropping the coordinate corresponding to the octave.~~

~~For example, the 5-limit lattice is three dimensional: each interval with integer coordinates <math>(w, x, y)</math> corresponds to the frequency ratio <math>2^w \cdot 3^x \cdot 5^y</math>.~~

~~If we assume octave equivalence, this reduces to a grid of points with coordinates <math>(x, y)</math>, each representing a class of equivalent intervals.~~

~~In our example, the point <math>(0, 1)</math> now represents the class of just major thirds: [[5/4]], [[5/2]], [[5/1]], etc.~~

~~In general, if the original space is ''n''-dimensional, this reduced space has ''n''-1 dimensions.~~

~~To construct a Fokker block, we take ''n''-1 small intervals in this space, which we will call the ''~~chromas~~'' (called ''unison vectors'' in Fokker's original text).~~

~~These interval vectors form a sublattice inside the original lattice.~~

~~In the two-dimensional case, they form a parallellogram which tiles the plane, and in higher dimensions they form a parallelepiped.~~

~~This tile is called the fundamental domain, and it's volume corresponds to the number of intervals contained inside of it.~~

~~This number is always an integer, and it is the number of notes in the resulting scale.~~

If we assemble the chromas in a matrix, the volume is given by the absolute value of the determinant of this matrix, as per the definition of the determinant as the signed area of the parallelepiped defined by the vectors of the matrix.

== Examples ==

=== Ptolemy's intense diatonic ===

Let's take [[5-limit]] just ~~intation~~, and pick the [[25/24|just chromatic semitone]] (25/24) and the [[syntonic comma]] (81/80) as our chromas.

[[File:Fokker_block_zarlino.png|400px|thumb|Fokker block corresponding to the just diatonic scale. The gray grid is the interval lattice, and the black lines show the sublattice generated by the chromas. The fundamental domain is colored in blue.]]Let's take [[5-limit]] just intonation, and pick the [[25/24|just chromatic semitone]] (25/24) and the [[syntonic comma]] (81/80) as our chromas.

The octave equivalent lattice is generated by fifths and just major thirds.

Since <math>25/24 = 2^{-3} \cdot 3^{-1} \cdot 5^2</math>, it has coordinates <math>(-1, 2)</math> in the octave-equivalent lattice.

The syntonic comma is reached by stacking 4 fifths and going down a major third, so it has coordinates <math>(4, -1)</math>.

[[File:Fokker_block_zarlino.png|400px|thumb|none|Fokker block corresponding to the just diatonic scale. The gray grid is the interval lattice, and the black lines show the sublattice generated by the chromas. The fundamental domain is colored in blue.]]

The corresponding Fokker block is [[Ptolemy's intense diatonic]], also known as Zarlino, specifically the lydian mode.

Line 53:

Line 31:

* Tempering out the chromatic semitone gives the [[mosh]] LsLsLss (a 7-note neutral scale), in [[dicot]].

If we temper out both 25/24 and 81/80, we get [[7edo|7 equal temperament]], which we can interpret as an equalized diatonic scale.

This scale is a Fokker block in multiple ways: it is also possible to arrive at the same set of notes using [[135/128]] together with either 81/80 or 25/24 as the chromas.

=== Duodene and 12 equal temperament ===

Let's now use the [[128/125|diesis]] (128/125) and the [[syntonic comma]] (81/80) as our chromas, in 5-limit JI as above.

[[File:Fokker_block_duodene.png|400px|thumb|Duodene as a Fokker block. Note that the fundamental domain is shifted from the origin to obtain the familiar scale.]]Let's now use the [[128/125|diesis]] (128/125) and the [[syntonic comma]] (81/80) as our chromas, in 5-limit JI as above.

The diesis is the difference between an octave and three major thirds, so it has coordinates <math>(0, -3)</math>.

~~[[File:Fokker_block_duodene.png|400px|thumb|none|Duodene as a Fokker block. Note that the fundamental domain is shifted from the origin to obtain the familiar scale.]]~~

The number of notes in the tile will be 12, since the determinant is:

Line 73:

Line 48:

This should not be surprising, as [[12edo|12 equal temperament]] tempers out exactly these two commas, so each note in the resulting scale will correspond to an interval of 12et.

The resulting just scale is known as [[duodene]].

== See also ==

* [[User:Hkm/Fokker block code|This Python code]] can be used to find Fokker blocks.

* [[Catalog of Fokker blocks]]

* [[List of weak Fokker blocks]]

* [[Fokker chord]]

* [[Minkowski block]]

* [[Minkowski reduced bases for Fokker groups of certain vals]]

== Further reading ==

* [http://www.huygens-fokker.org/docs/fokkerpb.html Unison Vectors and Periodicity Blocks] by [[Adriaan Fokker|A.D. Fokker]]

* [http://www.tonalsoft.com/enc/f/fokker-gentle-1.aspx A gentle introduction to Fokker periodicity blocks], by [[Paul Erlich]]

* [[Fokker chord]]

* [[Catalog of Fokker blocks]]

[[Category:Fokker block| ]]

[[Category:Pitch space]]

{{Todo| add illustration | improve linking | review }}

@@ Line 1: / Line 1: @@
-{{Beginner| Mathematical theory of Fokker blocks }}
+{{Beginner| Mathematical theory of Fokker blocks}}
 {{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.
-All Fokker blocks are weakly [[epimorphic]]; if a Fokker block is epimorphic (i.e. it has no negative steps), it is a ''strong'' Fokker block, otherwise it is a ''weak'' Fokker block. An unqualified Fokker block is generally assumed to be strong.
+All Fokker blocks are weakly [[epimorphic]], which means that there is a [[val]] that maps each note of the Fokker block onto its own equal temperament scale step and leaves no equal temperament scale step without a mapping towards it. (In other words, this val provides a bijection between the Fokker block and the equal temperament.) If a Fokker block is epimorphic, which means that the val preserves the order of the steps, it is a ''strong'' Fokker block; otherwise it is a ''weak'' Fokker block. The expression "Fokker block" without any qualifier generally denotes a strong block.
 The concept of the Fokker block was developed by the physicist and music theorist [[Adriaan Fokker]].
 == Theory ==
-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]].
+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]]. (Fokker called them ''unison vectors'' in his original text.)
+If we place the coordinate vectors of the chromas into a matrix, the number of notes in the Fokker block is equal to the determinant of that matrix. It is also equal to the volume of the parallelepiped formed by the chromas, which is called the '''fundamental domain'''.
 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.
@@ Line 14: / Line 16: @@
 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. The constant structure will have no negative steps if and only if the Fokker block is strong.
-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).
+A Fokker '''arena''' contains all the periodic scales that can be constructed as Fokker blocks from the same list of chromas.
-== Terminology ==
-{{Todo|add definition |text=Definitions in the "Terminology" section should not utilize mathematical jargon.}}
-=== Arena ===
-A Fokker arena contains all the periodic scales that can be constructed as Fokker blocks from the same list of commas.
-== Definition ==
-Fokker blocks are defined on octave-equivalent [[lattice]]s, which can be visualized as grids.
-These can be constructed by simply dropping the coordinate corresponding to the octave.
-For example, the 5-limit lattice is three dimensional: each interval with integer coordinates <math>(w, x, y)</math> corresponds to the frequency ratio <math>2^w \cdot 3^x \cdot 5^y</math>.
-If we assume octave equivalence, this reduces to a grid of points with coordinates <math>(x, y)</math>, each representing a class of equivalent intervals.
-In our example, the point <math>(0, 1)</math> now represents the class of just major thirds: [[5/4]], [[5/2]], [[5/1]], etc.
-In general, if the original space is ''n''-dimensional, this reduced space has ''n''-1 dimensions.
-To construct a Fokker block, we take ''n''-1 small intervals in this space, which we will call the ''chromas'' (called ''unison vectors'' in Fokker's original text).
-These interval vectors form a sublattice inside the original lattice.
-In the two-dimensional case, they form a parallellogram which tiles the plane, and in higher dimensions they form a parallelepiped.
-This tile is called the fundamental domain, and it's volume corresponds to the number of intervals contained inside of it.
-This number is always an integer, and it is the number of notes in the resulting scale.
-If we assemble the chromas in a matrix, the volume is given by the absolute value of the determinant of this matrix, as per the definition of the determinant as the signed area of the parallelepiped defined by the vectors of the matrix.
 == Examples ==
 === Ptolemy's intense diatonic ===
-Let's take [[5-limit]] just intation, and pick the [[25/24|just chromatic semitone]] (25/24) and the [[syntonic comma]] (81/80) as our chromas.
+[[File:Fokker_block_zarlino.png|400px|thumb|Fokker block corresponding to the just diatonic scale. The gray grid is the interval lattice, and the black lines show the sublattice generated by the chromas. The fundamental domain is colored in blue.]]Let's take [[5-limit]] just intonation, and pick the [[25/24|just chromatic semitone]] (25/24) and the [[syntonic comma]] (81/80) as our chromas.
 The octave equivalent lattice is generated by fifths and just major thirds.
 Since <math>25/24 = 2^{-3} \cdot 3^{-1} \cdot 5^2</math>, it has coordinates <math>(-1, 2)</math> in the octave-equivalent lattice.
 The syntonic comma is reached by stacking 4 fifths and going down a major third, so it has coordinates <math>(4, -1)</math>.
-[[File:Fokker_block_zarlino.png|400px|thumb|none|Fokker block corresponding to the just diatonic scale. The gray grid is the interval lattice, and the black lines show the sublattice generated by the chromas. The fundamental domain is colored in blue.]]
 The corresponding Fokker block is [[Ptolemy's intense diatonic]], also known as Zarlino, specifically the lydian mode.
@@ Line 53: / Line 31: @@
 * Tempering out the chromatic semitone gives the [[mosh]] LsLsLss (a 7-note neutral scale), in [[dicot]].
 If we temper out both 25/24 and 81/80, we get [[7edo|7 equal temperament]], which we can interpret as an equalized diatonic scale.
 This scale is a Fokker block in multiple ways: it is also possible to arrive at the same set of notes using [[135/128]] together with either 81/80 or 25/24 as the chromas.
 === Duodene and 12 equal temperament ===
-Let's now use the [[128/125|diesis]] (128/125) and the [[syntonic comma]] (81/80) as our chromas, in 5-limit JI as above.
+[[File:Fokker_block_duodene.png|400px|thumb|Duodene as a Fokker block. Note that the fundamental domain is shifted from the origin to obtain the familiar scale.]]Let's now use the [[128/125|diesis]] (128/125) and the [[syntonic comma]] (81/80) as our chromas, in 5-limit JI as above.
 The diesis is the difference between an octave and three major thirds, so it has coordinates <math>(0, -3)</math>.
-[[File:Fokker_block_duodene.png|400px|thumb|none|Duodene as a Fokker block. Note that the fundamental domain is shifted from the origin to obtain the familiar scale.]]
 The number of notes in the tile will be 12, since the determinant is:
@@ Line 73: / Line 48: @@
 This should not be surprising, as [[12edo|12 equal temperament]] tempers out exactly these two commas, so each note in the resulting scale will correspond to an interval of 12et.
 The resulting just scale is known as [[duodene]].
+== See also ==
+* [[User:Hkm/Fokker block code|This Python code]] can be used to find Fokker blocks.
+* [[Catalog of Fokker blocks]]
+* [[List of weak Fokker blocks]]
+* [[Fokker chord]]
+* [[Minkowski block]]
+* [[Minkowski reduced bases for Fokker groups of certain vals]]
 == Further reading ==
 * [http://www.huygens-fokker.org/docs/fokkerpb.html Unison Vectors and Periodicity Blocks] by [[Adriaan Fokker|A.D. Fokker]]
 * [http://www.tonalsoft.com/enc/f/fokker-gentle-1.aspx A gentle introduction to Fokker periodicity blocks], by [[Paul Erlich]]
-* [[Fokker chord]]
-* [[Catalog of Fokker blocks]]
 [[Category:Fokker block| ]] <!-- main article -->
 [[Category:Pitch space]]
 {{Todo| add illustration | improve linking | review }} <!-- add beginner section that explains how to build Fokker blocks either by hand or using common software, along with visualizations. -->