Val: Difference between revisions

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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.

Vals are usually written in the notation {{val| a b c d e f … p }}, 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 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.

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 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.
-Vals are usually written in the notation {{val| a b c d e f … p }}, 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 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.