User:VectorGraphics/Periodicity: Difference between revisions

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This page covers a framework for thinking of temperaments in terms of JI lattices, independently derived by [[User:VectorGraphics]], Adriaan Fokker, and likely others.

This page covers a framework for thinking of temperaments and scales in terms of JI lattices, independently derived by [[User:VectorGraphics]], Adriaan Fokker, and likely others.

== In terms of temperaments ==

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~~The~~ repeating block ~~representing an equal temperament~~ is called a "periodicity block", or a "Fokker block", after the mathematician who originally described them, Adriaan Fokker.

Such a repeating block is called a "periodicity block", or a "Fokker block", after the mathematician who originally described them, Adriaan Fokker.

== From temperaments to scales ==

[[File:12 well 5limit.png|left|thumb|337x337px|A well temperament of 12-edo defined from a Fokker periodicity block, in ratios and in cents. (Both possible mappings of 600c have been provided.)]]

Such a block can then be placed back into a less restrictive temperament or into just intonation, to define a scale, sort of like a MOS scale but generalized to a higher dimension. For example, one might describe a 12-tone well temperament in 5-limit just intonation by placing a periodicity block representing 12edo into a JI lattice, and seeing what intervals lie within it.

Such a block can then be placed back into a less restrictive temperament or into just intonation, to define a scale, sort of like a MOS scale but generalized to a higher dimension. For example, one might describe a 12-tone well temperament in 5-limit just intonation by placing a periodicity block representing 12edo into a JI lattice, and seeing what intervals lie within it. This can be treated as a scale on its own (a just chromatic scale), outside of the context of being a 12edo detemper.

In the context of constructing a scale (as opposed to defining a temperament), the intervals chosen at which to repeat the table are called '''chromas''' instead of commas, and they define a '''chroma basis'''. So, for example, our 12-well temperament has the chroma basis {81/80, 128/125}.

In the context of constructing a scale (as opposed to defining a temperament), the intervals chosen at which to repeat the table are called '''chromas''' instead of commas, and they define a '''chroma basis''', which tells you which pairs of accidentals you'll need to notate music in that scale. So, for example, our 12-well temperament has the chroma basis {81/80, 128/125}.

A lot of other JI scales (and scales in general, using a tempered lattice) can be constructed as periodicity blocks. For example, the just zarlino scale can be constructed with the chroma basis {81/80, 25/24}. Porcupine temperament equates these two chromas by tempering out the porcupine comma, so the zarlino periodicity block can be placed into a lattice representing porcupine temperament to obtain the porcupine zarlino scale (a MODMOS of porcupine's "Onyx" MOS itself).

@@ Line 1: / Line 1: @@
-This page covers a framework for thinking of temperaments in terms of JI lattices, independently derived by [[User:VectorGraphics]], Adriaan Fokker, and likely others.
+This page covers a framework for thinking of temperaments and scales in terms of JI lattices, independently derived by [[User:VectorGraphics]], Adriaan Fokker, and likely others.
 == In terms of temperaments ==
@@ Line 20: / Line 20: @@
-The repeating block representing an equal temperament is called a "periodicity block", or a "Fokker block", after the mathematician who originally described them, Adriaan Fokker.
+Such a repeating block is called a "periodicity block", or a "Fokker block", after the mathematician who originally described them, Adriaan Fokker.
 == From temperaments to scales ==
 [[File:12 well 5limit.png|left|thumb|337x337px|A well temperament of 12-edo defined from a Fokker periodicity block, in ratios and in cents. (Both possible mappings of 600c have been provided.)]]
-Such a block can then be placed back into a less restrictive temperament or into just intonation, to define a scale, sort of like a MOS scale but generalized to a higher dimension. For example, one might describe a 12-tone well temperament in 5-limit just intonation by placing a periodicity block representing 12edo into a JI lattice, and seeing what intervals lie within it.
+Such a block can then be placed back into a less restrictive temperament or into just intonation, to define a scale, sort of like a MOS scale but generalized to a higher dimension. For example, one might describe a 12-tone well temperament in 5-limit just intonation by placing a periodicity block representing 12edo into a JI lattice, and seeing what intervals lie within it. This can be treated as a scale on its own (a just chromatic scale), outside of the context of being a 12edo detemper.
-In the context of constructing a scale (as opposed to defining a temperament), the intervals chosen at which to repeat the table are called '''chromas''' instead of commas, and they define a '''chroma basis'''. So, for example, our 12-well temperament has the chroma basis {81/80, 128/125}.
+In the context of constructing a scale (as opposed to defining a temperament), the intervals chosen at which to repeat the table are called '''chromas''' instead of commas, and they define a '''chroma basis''', which tells you which pairs of accidentals you'll need to notate music in that scale. So, for example, our 12-well temperament has the chroma basis {81/80, 128/125}.
 A lot of other JI scales (and scales in general, using a tempered lattice) can be constructed as periodicity blocks. For example, the just zarlino scale can be constructed with the chroma basis {81/80, 25/24}. Porcupine temperament equates these two chromas by tempering out the porcupine comma, so the zarlino periodicity block can be placed into a lattice representing porcupine temperament to obtain the porcupine zarlino scale (a MODMOS of porcupine's "Onyx" MOS itself).