User:VectorGraphics/Periodicity - Revision history

VectorGraphics at 22:01, 22 March 2025

2025-03-22T22:01:06Z

← Older revision		Revision as of 22:01, 22 March 2025
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	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.		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).		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).

VectorGraphics at 21:58, 22 March 2025

2025-03-22T21:58:47Z

← Older revision		Revision as of 21:58, 22 March 2025
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	== From temperaments to scales ==		== 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.)]]		[[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'''. So, for example, our 12-well temperament has the chroma basis {81/80, 128/125}.

VectorGraphics at 21:57, 22 March 2025

2025-03-22T21:57:32Z

← Older revision		Revision as of 21:57, 22 March 2025
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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 ==		== From temperaments to scales ==

VectorGraphics at 21:56, 22 March 2025

2025-03-22T21:56:35Z

← Older revision		Revision as of 21:56, 22 March 2025
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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 ==		== In terms of temperaments ==

VectorGraphics at 21:18, 13 March 2025

2025-03-13T21:18:26Z

@@ 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.
-== Method ==
+== In terms of temperaments ==
 As an example, we will be working in the 5-limit. Octave-equivalence is assumed, so our lattice has 2 dimensions:
 Now, the purpose of doing this is now to look for places where stacking intervals gets you close to zero - that is, commas.

VectorGraphics: WIP

2025-03-13T20:26:23Z

WIP

New page

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

== Method ==
As an example, we will be working in the 5-limit. Octave-equivalence is assumed, so our lattice has 2 dimensions:

Now, the purpose of doing this is now to look for places where stacking intervals gets you close to zero - that is, commas.