Fokker block - Revision history

Hkm at 17:33, 21 August 2025

2025-08-21T17:33:28Z

← Older revision		Revision as of 17:33, 21 August 2025
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	{{Beginner\| Mathematical theory of Fokker blocks }}		{{Beginner\| Mathematical theory of Fokker blocks}}
	{{Wikipedia\| Fokker periodicity block }}		{{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.		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]].		The concept of the Fokker block was developed by the physicist and music theorist [[Adriaan Fokker]].

Hkm at 00:27, 31 July 2025

2025-07-31T00:27:39Z

← Older revision		Revision as of 00:27, 31 July 2025
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	== See also ==		== See also ==
			* [[User:Hkm/Fokker block code\|This Python code]] can be used to find Fokker blocks.
	* [[Catalog of Fokker blocks]]		* [[Catalog of Fokker blocks]]
	* [[List of weak Fokker blocks]]		* [[List of weak Fokker blocks]]

Hkm: /* Ptolemy's intense diatonic */ fix typo

2025-07-28T20:15:25Z

Ptolemy's intense diatonic: fix typo

← Older revision		Revision as of 20:15, 28 July 2025
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	== Examples ==		== Examples ==
	=== Ptolemy's intense diatonic ===		=== Ptolemy's intense diatonic ===
	[[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.		[[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.		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.		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.
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	* Tempering out the chromatic semitone gives the [[mosh]] LsLsLss (a 7-note neutral scale), in [[dicot]].		* 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.		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.		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 ===		=== Duodene and 12 equal temperament ===

Fredg999: Split links section at the end, add two internal links

2025-07-28T19:08:06Z

Split links section at the end, add two internal links

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

			== See also ==
			* [[Catalog of Fokker blocks]]
			* [[List of weak Fokker blocks]]
			* [[Fokker chord]]
			* [[Minkowski block]]
			* [[Minkowski reduced bases for Fokker groups of certain vals]]

	== Further reading ==		== Further reading ==
	* [http://www.huygens-fokker.org/docs/fokkerpb.html Unison Vectors and Periodicity Blocks] by [[Adriaan Fokker\|A.D. Fokker]]		* [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]]		* [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:Fokker block\| ]] <!-- main article -->
	[[Category:Pitch space]]		[[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. -->		{{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. -->

Hkm: /* Terminology */ consolidated "theory", "terminology", and "definition" sections

2025-07-28T15:43:35Z

Terminology: consolidated "theory", "terminology", and "definition" sections

← Older revision		Revision as of 15:43, 28 July 2025
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	== Theory ==		== 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.		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.		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 ==		== Examples ==
	=== Ptolemy's intense diatonic ===		=== 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.		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.		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>.		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.		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 ===		=== 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>.		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:		The number of notes in the tile will be 12, since the determinant is:

Fredg999: Rework strong/weak paragraph, misc. edits

2025-07-28T04:49:03Z

Rework strong/weak paragraph, misc. edits

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	{{Beginner\| Mathematical theory of Fokker blocks }}		{{Beginner\| Mathematical theory of Fokker blocks }}
	{{Wikipedia\| Fokker periodicity block }}		{{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.		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 [[~~Periodic scale#Epimorphism\|weakly~~ epimorphic]]; if a Fokker block is (~~strongly~~) ~~epimorphic~~, it is ~~called~~ '''~~strong~~'''.		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.

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

	== Theory ==		== Theory ==

Hkm: readded fact about constant structure

2025-07-28T04:36:47Z

readded fact about constant structure

← Older revision		Revision as of 04:36, 28 July 2025
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	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.		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.

			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).		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).

Hkm: framed definition in terms of epimorphicity, not constant structure

2025-07-28T04:27:37Z

framed definition in terms of epimorphicity, not constant structure

← Older revision		Revision as of 04:27, 28 July 2025
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	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.		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.

	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. If the constant structure does not have negative steps, the Fokker block is ~~called '''strong'''; otherwise~~, it is called '''~~weak~~'''.		All Fokker blocks are [[Periodic scale#Epimorphism\|weakly epimorphic]]; if a Fokker block is (strongly) epimorphic, it is called '''strong'''.

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

Hkm: clarify what properties hold for strong vs. non-strong fokker blocks

2025-07-28T01:04:10Z

clarify what properties hold for strong vs. non-strong fokker blocks

← Older revision		Revision as of 01:04, 28 July 2025
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	{{Wikipedia\| Fokker periodicity block }}		{{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.		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.

			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. If the constant structure does not have negative steps, the Fokker block is called '''strong'''; otherwise, it is called '''weak'''.

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

	== Theory ==		== 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.}}~~
	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]].

	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. In this way, Fokker blocks generalize mos scales.		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.

	~~If the ratios of the cent values of two points on a Fokker block's lattice is always irrational, each scale formed from the block is [[constant structure\|constant-structure]]~~.

	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).		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).

VectorGraphics at 08:12, 1 July 2025

2025-07-01T08:12:13Z

← Older revision		Revision as of 08:12, 1 July 2025
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	This tile is called the fundamental domain, and it's volume corresponds to the number of intervals contained inside of it.		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.		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.		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 ==		== Examples ==