Val: Difference between revisions

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A [[val]] – short for ''valuation'' – is like an algorithm or procedure for finding out how to represent intervals of [[just intonation|just intonation (JI)]] with the pitches of an [[equal tuning]] such as an [[edo]]. They are typically written using the notation {{val| ''a''1 ''a''2 ''a''3 ''a''4 ''a''5 ''a''6 … }}, where ''a''''i'' are numbers that represent how the primes 2, 3, 5, 7, 11, 13, etc., in that order, are represented in edosteps, up to some [[harmonic limit|prime limit]].

The basic principle of using a val is to assign [[prime harmonic]]s to edosteps, and then deduce the number of edosteps of an arbitrary interval based on its [[prime factorization]]. This therefore assumes either that you want to use an equal tuning to approximate specific harmonies or that you have some other more indirect use in mind.

The val is one of the fundamental concepts in [[regular temperament theory]]. The basic principle of using a val is to assign [[prime harmonic]]s to edosteps, and then deduce the number of edosteps of an arbitrary interval based on its [[prime factorization]]. This therefore assumes either that you want to use an equal tuning to approximate specific harmonies or that you have some other more indirect use in mind.

== Motivation ==

One obvious way to find an approximation to a just interval is to use [[direct approximation]], that is, rounding the interval to the nearest edostep. While this may seem simple, it can create contradictions in arithmetic. For example, a [[just major triad]] consists of a [[5/4]] major third and a [[6/5]] minor third combining to a [[3/2]] perfect fifth, but the sum of direct approximations of 5/4 and 6/5 might not be the direct approximation of 3/2. More generally, combining the approximations in an edo does not necessarily give you the same result as multiplying their ratios first and then using the direct approximation of that in the edo. ~~When~~ this ~~happens~~, we ~~say that the arithmetic is~~ ''~~inconsistent~~''. ~~Therefore when this does not happen~~, we ~~say~~ that the ~~result~~ is [[~~consistent~~]].

One obvious way to find an approximation to a just interval is to use [[direct approximation]], that is, rounding the interval to the nearest edostep. While this may seem simple, it can create contradictions in arithmetic. For example, a [[just major triad]] consists of a [[5/4]] major third and a [[6/5]] minor third combining to a [[3/2]] perfect fifth, but the sum of direct approximations of 5/4 and 6/5 might not be the direct approximation of 3/2. More generally, combining the approximations in an edo does not necessarily give you the same result as multiplying their ratios first and then using the direct approximation of that in the edo, so direct approximations of chords are not guaranteed to exist.

Rather than giving up and saying that we cannot use this particular harmony in this particular edo, it turns out we ''can'' if we look at interval approximation in a different way.

In direct approximation, we are treating the approximations as isolated, unrelated free variables, but as we see, ''two'' intervals on top of each other form a triad with ''three'' component intervals that cannot be altered individually. That is why it is important to recognize the fact that intervals like 3/2, 5/4 and 6/5 are related to each other: by stacking 5/4 and 6/5, 3/2 is found; by removing 6/5 from 3/2, 5/4 is found; and by removing 5/4 from 3/2, 6/5 is found. It follows that for the ''three'' intervals in the [[5-odd-limit]], there are ''two'' free variables. If we know any two of them, the third can always be derived (even though it might not be the closest approximation).

~~Unfortunately~~, ~~it is not possible~~ to ~~fix inconsistency, except~~ by ~~avoiding this particular harmony in this particular edo~~, ~~but rather than giving up and saying that~~ we ~~cannot use it~~, ~~it turns out~~ we ''~~can~~'' if we are willing to allow one or more of these ratios to use an alternative approximation, especially considering that we probably do not mind using the second-best approximation in more complex intervals if we can guarantee that the arithmetic never fails us. A val will show us how to do that.

To take this idea further, we notice that the infinitely many intervals of JI can be reduced to a few representatives from which the rest can be derived by some form of combination, so we only need to keep track of the steps of the representatives. Usually, we choose the steps of the [[prime interval|prime harmonics]], by which we mean each interval with frequency ratio ''p''/1 where ''p'' is a {{w|prime number}}.

== Definition ==

A val is a list of numbers that shows the approximation of each prime harmonic used in an edo in terms of steps~~, where by ''prime harmonic'' we mean each frequency ratio ''p''/1 where ''p'' is a {{w|prime number}}~~. This list of integers by convention corresponds to all primes up to some largest prime (the [[limit]]) so that we can tell what number represents the ''mapping'' of what prime by its place in the list. First place is prime 2's mapping (a.k.a. the edo), second place is prime 3's mapping, third place is prime 5's mapping, fourth is prime 7's, etc.

A val is a list of numbers that shows the approximation of each prime harmonic used in an edo in terms of steps. This list of integers by convention corresponds to all primes up to some largest prime (the [[limit]]) so that we can tell what number represents the ''mapping'' of what prime by its place in the list. First place is prime 2's mapping (a.k.a. the edo), second place is prime 3's mapping, third place is prime 5's mapping, fourth is prime 7's, etc.

The val is used to ~~understand~~ the edo's approximations to ratios involving those primes, like 2 × 5 / 3 / 3 = [[10/9]] for primes {2, 3, 5}. This list does not have to be the closest approximation for each prime, but it usually is. Thus a val is essentially just a list of numbers that we are interpreting as having a certain meaning.

The val can be used to compute the edo's approximations to ratios involving those primes, like 2 × 5 / 3 / 3 = [[10/9]] for primes {2, 3, 5}. This list does not have to be the closest approximation for each prime, but it usually is. Thus a val is essentially just a list of numbers that we are interpreting as having a certain meaning.

== Examples ==

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-{{interwiki
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+| en = Val
 | de = Val
-| en = Val
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 | ja = ヴァル
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 A [[val]] – short for ''valuation'' – is like an algorithm or procedure for finding out how to represent intervals of [[just intonation|just intonation (JI)]] with the pitches of an [[equal tuning]] such as an [[edo]]. They are typically written using the notation {{val| ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> ''a''<sub>4</sub> ''a''<sub>5</sub> ''a''<sub>6</sub> … }}, where ''a''<sub>''i''</sub> are numbers that represent how the primes 2, 3, 5, 7, 11, 13, etc., in that order, are represented in edosteps, up to some [[harmonic limit|prime limit]].
-The basic principle of using a val is to assign [[prime harmonic]]s to edosteps, and then deduce the number of edosteps of an arbitrary interval based on its [[prime factorization]]. This therefore assumes either that you want to use an equal tuning to approximate specific harmonies or that you have some other more indirect use in mind.
+The val is one of the fundamental concepts in [[regular temperament theory]]. The basic principle of using a val is to assign [[prime harmonic]]s to edosteps, and then deduce the number of edosteps of an arbitrary interval based on its [[prime factorization]]. This therefore assumes either that you want to use an equal tuning to approximate specific harmonies or that you have some other more indirect use in mind.
 == Motivation ==
-One obvious way to find an approximation to a just interval is to use [[direct approximation]], that is, rounding the interval to the nearest edostep. While this may seem simple, it can create contradictions in arithmetic. For example, a [[just major triad]] consists of a [[5/4]] major third and a [[6/5]] minor third combining to a [[3/2]] perfect fifth, but the sum of direct approximations of 5/4 and 6/5 might not be the direct approximation of 3/2. More generally, combining the approximations in an edo does not necessarily give you the same result as multiplying their ratios first and then using the direct approximation of that in the edo. When this happens, we say that the arithmetic is ''inconsistent''. Therefore when this does not happen, we say that the result is [[consistent]].
+One obvious way to find an approximation to a just interval is to use [[direct approximation]], that is, rounding the interval to the nearest edostep. While this may seem simple, it can create contradictions in arithmetic. For example, a [[just major triad]] consists of a [[5/4]] major third and a [[6/5]] minor third combining to a [[3/2]] perfect fifth, but the sum of direct approximations of 5/4 and 6/5 might not be the direct approximation of 3/2. More generally, combining the approximations in an edo does not necessarily give you the same result as multiplying their ratios first and then using the direct approximation of that in the edo, so direct approximations of chords are not guaranteed to exist.
+Rather than giving up and saying that we cannot use this particular harmony in this particular edo, it turns out we ''can'' if we look at interval approximation in a different way.
+In direct approximation, we are treating the approximations as isolated, unrelated free variables, but as we see, ''two'' intervals on top of each other form a triad with ''three'' component intervals that cannot be altered individually. That is why it is important to recognize the fact that intervals like 3/2, 5/4 and 6/5 are related to each other: by stacking 5/4 and 6/5, 3/2 is found; by removing 6/5 from 3/2, 5/4 is found; and by removing 5/4 from 3/2, 6/5 is found. It follows that for the ''three'' intervals in the [[5-odd-limit]], there are ''two'' free variables. If we know any two of them, the third can always be derived (even though it might not be the closest approximation).
-Unfortunately, it is not possible to fix inconsistency, except by avoiding this particular harmony in this particular edo, but rather than giving up and saying that we cannot use it, it turns out we ''can'' if we are willing to allow one or more of these ratios to use an alternative approximation, especially considering that we probably do not mind using the second-best approximation in more complex intervals if we can guarantee that the arithmetic never fails us. A val will show us how to do that.
+To take this idea further, we notice that the infinitely many intervals of JI can be reduced to a few representatives from which the rest can be derived by some form of combination, so we only need to keep track of the steps of the representatives. Usually, we choose the steps of the [[prime interval|prime harmonics]], by which we mean each interval with frequency ratio ''p''/1 where ''p'' is a {{w|prime number}}.
 == Definition ==
-A val is a list of numbers that shows the approximation of each prime harmonic used in an edo in terms of steps, where by ''prime harmonic'' we mean each frequency ratio ''p''/1 where ''p'' is a {{w|prime number}}. This list of integers by convention corresponds to all primes up to some largest prime (the [[limit]]) so that we can tell what number represents the ''mapping'' of what prime by its place in the list. First place is prime 2's mapping (a.k.a. the edo), second place is prime 3's mapping, third place is prime 5's mapping, fourth is prime 7's, etc.
+A val is a list of numbers that shows the approximation of each prime harmonic used in an edo in terms of steps. This list of integers by convention corresponds to all primes up to some largest prime (the [[limit]]) so that we can tell what number represents the ''mapping'' of what prime by its place in the list. First place is prime 2's mapping (a.k.a. the edo), second place is prime 3's mapping, third place is prime 5's mapping, fourth is prime 7's, etc.
-The val is used to understand the edo's approximations to ratios involving those primes, like 2 × 5 / 3 / 3 = [[10/9]] for primes {2, 3, 5}. This list does not have to be the closest approximation for each prime, but it usually is. Thus a val is essentially just a list of numbers that we are interpreting as having a certain meaning.
+The val can be used to compute the edo's approximations to ratios involving those primes, like 2 × 5 / 3 / 3 = [[10/9]] for primes {2, 3, 5}. This list does not have to be the closest approximation for each prime, but it usually is. Thus a val is essentially just a list of numbers that we are interpreting as having a certain meaning.
 == Examples ==