Fokker block: Difference between revisions

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== 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.

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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 intation, 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.

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=== 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 8: / Line 8: @@
 == 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 intation, 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 56: / Line 34: @@
 === 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: