Fokker block: Difference between revisions
m Split links section at the end, add two internal links |
No edit summary |
||
| (2 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
{{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]] | 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]]. | ||
| Line 20: | Line 20: | ||
== 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 | [[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. | ||
| Line 31: | Line 31: | ||
* 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 === | ||
| Line 50: | Line 50: | ||
== 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]] | ||