Fokker block: Difference between revisions

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== Theory ==

{{Todo~~|inline=1~~| add definition |text=Either the "strong Fokker block" definition needs to be here, or the second and third paragraphs, which are not true for weak Fokker blocks, need to be removed.}}

{{Todo|add definition |text=Either the "strong Fokker block" definition needs to be here, or the second and third paragraphs, which are not true for weak Fokker blocks, need to be removed.}}

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]].

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== Terminology ==

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

=== Arena ===

A Fokker arena contains all the periodic scales ~~formable~~ as Fokker blocks from the same list of commas.{{~~Todo~~|inline=1| ~~add definition |text=Definitions~~ in the ~~"Terminology" section~~ should not ~~utilize mathematical jargon~~.}}

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.

== Examples ==

=== Duodene and 12 equal temperament ===

Let's take [[5-limit]] just intation, and pick the [[128/125|diesis]] (128/125) and the [[syntonic comma]] (81/80) as our chromas.

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

The diesis can be reached by stacking three major thirds, so it has coordinates <math>(0, 3)</math>.

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

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

:<math>

\begin{vmatrix}

0 & 3 \\

4 & -1

\end{vmatrix}

= 12

</math>

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]].

== Further reading ==

@@ Line 6: / Line 6: @@
 == Theory ==
-{{Todo|inline=1| add definition |text=Either the "strong Fokker block" definition needs to be here, or the second and third paragraphs, which are not true for weak Fokker blocks, need to be removed.}}
+{{Todo|add definition |text=Either the "strong Fokker block" definition needs to be here, or the second and third paragraphs, which are not true for weak Fokker blocks, need to be removed.}}
 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]].
@@ Line 16: / Line 16: @@
 == Terminology ==
+{{Todo|add definition |text=Definitions in the "Terminology" section should not utilize mathematical jargon.}}
 === Arena ===
-A Fokker arena contains all the periodic scales formable as Fokker blocks from the same list of commas.{{Todo|inline=1| add definition |text=Definitions in the "Terminology" section should not utilize mathematical jargon.}}
+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.
+== Examples ==
+=== Duodene and 12 equal temperament ===
+Let's take [[5-limit]] just intation, and pick the [[128/125|diesis]] (128/125) and the [[syntonic comma]] (81/80) as our chromas.
+The octave equivalent lattice is generated by fifths and just major thirds.
+The diesis can be reached by stacking three major thirds, so it has coordinates <math>(0, 3)</math>.
+The syntonic comma is reached by stacking 4 fifths and going down a major third, so it has coordinates <math>(4, -1)</math>
+{{todo| inline=1| insert graphic here}}
+The number of notes in the tile will be 12, since the determinant is:
+:<math>
+	\begin{vmatrix}
+& 3 \\
+& -1
+	\end{vmatrix}
+	= 12
+</math>
+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]].
 == Further reading ==