Val: Difference between revisions

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== Definition ==

A val is a ~~map~~ representing how to view the intervals in ~~a single chain of generators (→[[periods and generators]])~~ as ~~the tempered~~ versions of intervals in just intonation (JI). They form the link between things like EDOs and JI, and by doing so form the basis for all of regular temperament theory. It's very common for vals to refer to EDOs specifically, although they also show us how to relate larger chains of generators to JI as well (such as a stack of meantone fifths).

A val is a set of assignments representing how to view the intervals in a temperament, such as an edo, as approximate versions of intervals in just intonation (JI). They form the link between things like EDOs and JI, and by doing so form the basis for all of regular temperament theory. It's very common for vals to refer to EDOs specifically, although they also show us how to relate larger chains of generators to JI as well (such as a stack of meantone fifths).

A val accomplishes the goal of mapping all intervals in some [[harmonic limit]] by simply notating how many steps in the chain it takes to get to each of the primes within the limit. Since every positive rational number can be described as a product of primes, any mapping for the primes hence implies a mapping for all of the positive rational numbers within the prime limit. By mapping the primes and letting the composite rationals fall where they may, a val tells us which interval in the chain represents the tempered 3/2, which interval represents the tempered 5/4, and so forth.

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Vals are usually written in the notation {{val| ''a b c d e f'' … }}, where each successive column represents a successive prime, such as 2, 3, 5, 7, 11, 13… etc, in that order, up to some [[Harmonic Limit|prime limit]] ''p''.

Vals are important because they provide a way to mathematically formalize how, specifically, the intervals in a random chain of generators are viewed as the tempered versions of more fundamental just intonation intervals. They can also be viewed as a way to map JI "onto" the chain, imbuing it with a harmonic context. Vals will enable you to figure out what commas your temperament eliminates, what [[comma pump]]s are available in the temperament, what the most consonant chords in the temperament are, how to optimize the octave stretch of the temperament to minimize tuning error, what EDOs support your temperament, and other operations as of yet undiscovered.

Vals are important in regular temperament theory because they provide a way to mathematically formalize how, specifically, the intervals in a random chain of generators are viewed as the tempered versions of more fundamental just intonation intervals. They can also be viewed as a way to map JI "onto" the chain, imbuing it with a harmonic context. Vals will enable you to figure out what commas your temperament eliminates, what [[comma pump]]s are available in the temperament, what the most consonant chords in the temperament are, how to optimize the octave stretch of the temperament to minimize tuning error, what EDOs support your temperament, and other operations as of yet undiscovered.

For a more mathematically intensive introduction to vals, see [[vals and tuning space]]. For the characterization of higher rank temperaments, see [[mapping]].

== Example EDO ==

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 == Definition ==
-A val is a map representing how to view the intervals in a single chain of generators (→[[periods and generators]]) as the tempered versions of intervals in just intonation (JI). They form the link between things like EDOs and JI, and by doing so form the basis for all of regular temperament theory. It's very common for vals to refer to EDOs specifically, although they also show us how to relate larger chains of generators to JI as well (such as a stack of meantone fifths).
+A val is a set of assignments representing how to view the intervals in a temperament, such as an edo, as approximate versions of intervals in just intonation (JI). They form the link between things like EDOs and JI, and by doing so form the basis for all of regular temperament theory. It's very common for vals to refer to EDOs specifically, although they also show us how to relate larger chains of generators to JI as well (such as a stack of meantone fifths).
 A val accomplishes the goal of mapping all intervals in some [[harmonic limit]] by simply notating how many steps in the chain it takes to get to each of the primes within the limit. Since every positive rational number can be described as a product of primes, any mapping for the primes hence implies a mapping for all of the positive rational numbers within the prime limit. By mapping the primes and letting the composite rationals fall where they may, a val tells us which interval in the chain represents the tempered 3/2, which interval represents the tempered 5/4, and so forth.
@@ Line 15: / Line 15: @@
 Vals are usually written in the notation {{val| ''a b c d e f'' … }}, where each successive column represents a successive prime, such as 2, 3, 5, 7, 11, 13… etc, in that order, up to some [[Harmonic Limit|prime limit]] ''p''.
-Vals are important because they provide a way to mathematically formalize how, specifically, the intervals in a random chain of generators are viewed as the tempered versions of more fundamental just intonation intervals. They can also be viewed as a way to map JI "onto" the chain, imbuing it with a harmonic context. Vals will enable you to figure out what commas your temperament eliminates, what [[comma pump]]s are available in the temperament, what the most consonant chords in the temperament are, how to optimize the octave stretch of the temperament to minimize tuning error, what EDOs support your temperament, and other operations as of yet undiscovered.
+Vals are important in regular temperament theory because they provide a way to mathematically formalize how, specifically, the intervals in a random chain of generators are viewed as the tempered versions of more fundamental just intonation intervals. They can also be viewed as a way to map JI "onto" the chain, imbuing it with a harmonic context. Vals will enable you to figure out what commas your temperament eliminates, what [[comma pump]]s are available in the temperament, what the most consonant chords in the temperament are, how to optimize the octave stretch of the temperament to minimize tuning error, what EDOs support your temperament, and other operations as of yet undiscovered.
 For a more mathematically intensive introduction to vals, see [[vals and tuning space]]. For the characterization of higher rank temperaments, see [[mapping]].
 == Example EDO ==