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 (aka. [[frequency ratio]]s) 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.

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, though practically this is usually true for tunings of interest, so the realistic examples appear for more complex cases like (the direct approximations of) [[~]][[9/8]] and [[~]][[5/4]] combined not being equal to the direct approximation of [[45/32]] (= 9/8 * 5/4), as discussed in the 26edo example below. 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]].  

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.

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.  

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.

== Relationship with equal temperaments ==

The guarantee that there are no contradictions comes with an interesting feature: somehow, you have managed to approximate JI in an internally-consistent way despite the fact that the approximations get worse the more you combine the errors so can get arbitrarily inconsistent. This corresponds to [[tempering out]] an infinite set of [[comma]]s, though there is a finite number of simple/musically relevant commas; that set is simply the set of all intervals that are mapped to 0 steps ([[1/1]]) by the val. This explains where the additional "structure" went – if there are two or more primes, then you need to specify two or more integers in the exponents of the prime factorization (i.e. in the monzo). So we have lost information and structure by simplifying everything to a single integer coordinate; exactly the information that corresponds to ''equating'' any two intervals whose difference is one of the commas tempered, so we have found a precise sense in which we can equate two nearby intervals that are not actually equal – by mapping according to a val that maps the difference to zero. In fact, you do not have to use an [[edo]] tuning as you could use multiple vals ''simultaneously'' to map a single interval if you want to preserve more of the information in JI rather than just increasing the size of the edo; this corresponds to [[regular temperaments]] generally rather than just the 1-dimensional ("rank-1") case that vals correspond to. Therefore, a val specifies a rank-1 temperament a.k.a. an equal temperament.

 

 

Mathematically, a val is a type of function that inputs a {{w|rational number}} (ratio) and outputs a number of steps that represents what interval of the edo we use to approximate that frequency ratio. If ''a''/''b'' is our ratio and ''k'' is the output of the function, then the interval* of ''N''-edo is 2^''k''/''N'' if we assume a pure octave tuning which is often written with the backslash notation.

It is not just any such function though; it is a function with a special property called {{w|linearity}} that allows our arithmetic to be internally consistent (having an internal logic) in the way described above; here ''internally consistent'' is meant in the English sense, so should not be confused with [[consistency]] in the aforediscussed technical sense. The most obvious use of a val (the one discussed in the example) is to algorithmically determine JI interpretations of intervals in edo, which is called using the edo as an equal temperament or rank-1 temperament, where ''rank-1'' means that it corresponds to a 1-dimensional grid of notes related by the same (usually {{w|irrational number|irrational}}) frequency ratios.

Also note that in practice vals are ''very far'' from just any list of positive integers; rather, they are generally equal to or one off from the lists of integers that correspond to a ''patent val''.