Val: Difference between revisions

A [[val]] — short for ''valuation'' — is like an algorithm or procedure for finding out how to approximate [[frequency ratio]]s ([[interval]]s of [[just intonation|just intonation (JI)]]) with the pitches of an [[edo]]. This therefore assumes either that you want to use an [[edo]] to approximate specific harmonies or that you have some other more indirect use in mind.

Unfortunately, when this happens, it is not possible to fix the inconsistency, except by using a different edo that approximates the intervals in question better so that the multiplication or division of their approximations is consistent, but that is not actually necessary. Plus, even if you did that, there would still be inconsistent ratios because an approximation cannot be perfect, so you cannot truly eliminate the inconsistency completely. Rather than giving up and saying that we cannot guarantee that (~6/5)(~5/4) = ~3/2 or (~9/1)(~5/1) = ~45/1 or (~135/128)(~24/25) = ~81/80, etc. in our chosen edo, it turns out we ''can'' actually guarantee this if we are willing to allow one or more of these ratios to '''not''' use the closest approximation by using a val. This may seem strange in this example, as one likely wants at least (~6/5)(~5/4) = ~3/2, but in principle we probably do not mind if something more complex is inconsistent, like (~11)(~11)(~75) = ~9075, if we can guarantee that the arithmetic never fails us.  

== Using a val ==

Using the 26edo val {{val| 26 41 60 }} as our example, say we want to figure out how [[9/8]], [[5/4]] and (9/8)(5/4) = [[45/32]] are mapped.

First we have to find the {{w|Integer factorization|prime factorization}} of our intervals (then we deduce the mapping from the prime factorization and the val):

* 9/8 = (3 × 3)/(2 × 2 × 2)

* 5/4 = 5/(2 × 2)

* 45/32 = (3 × 3 × 5)/(2 × 2 × 2 × 2 × 2)

Now all we do is substitute each occurrence of each prime with adding (or subtracting if we are dividing) the corresponding number of ''steps'' for that prime given by our val:

* 9/8 is mapped to (41 + 41) - (26 + 26 + 26) = 82 - 78 = 4 steps (so represented by a frequency ratio of 2^4/26 = 1.112…/1 = 4\26)

* 5/4 is mapped to 60 - (26 + 26) = 60 - 52 = 8 steps (so represented by a frequency ratio of 2^8/26 = 1.237…/1 = 8\26)

* 45/32 is mapped to (41 + 41 + 60) - (26 + 26 + 26 + 26 + 26) = 142 - 130 = 12 steps (so represented by a frequency ratio of 2^12/26 = 1.377…/1 = 12\26)

That is a successful use of a val. The arithmetic works out nicely: the approximation of 5/4 times the approximation of 9/8 is the approximation of 45/32. What about that funny [[backslash notation]]? That's just a shorthand: ''k''\''N'' = 2^''k''/''N'' (the ''N''th root of 2, to the ''k''th power). Note that it can also be used ambiguously, as 2^4/26 + 2^8/26 = 2^12/26 is clearly invalid (the correct statement is 2^4/26 × 2^8/26 = 2^12/26) but 4\26 + 8\26 = 12\26 need not be. It is used a lot in the xen community so is provided here for familiarization.

Now let us compare to the closest approximations:

* log₂(9/8) × 26 = 4.418… which rounds to 4 steps

* log₂(5/4) × 26 = 8.370… which rounds to 8 steps

* log₂(45/32) × 26 = 12.788… which rounds to 13 steps

So here we can see that (9/8)(5/4) = 45/32 is inconsistent because 4 + 8 is not equal to 13, in other words, 4 + 8 = 12 fails. So we can see that the val "fixes" the 13 to 12 by obeying an ''internal'' logic provided by the val; in different contexts it may be different intervals that are fixed; not necessarily the more complex one, though usually that is how it works for edos of interest.

So you may have noticed that there was quite a bit of repeated addition we can simplify, so let us note how we can simplify it exactly. Specifically, using exponent notation and not omitting implicit "to the power of 0"'s, the prime factorizations we worked out above are equivalent to:

* 9/8 = 2^-3 × 3² × 5⁰ which we can notate as {{monzo| -3 2 0 }}

* 5/4 = 2^-2 × 3⁰ × 5¹ which we can notate as {{monzo| -2 0 1 }}

* 45/32 = 2^-5 × 3² × 5¹ which we can notate as {{monzo| -5 2 1 }}

These notations are called [[monzo]]s; in other words, all a monzo really is is a shorthand for the prime factorization of an interval. Notice that the angle bracket on a monzo is on the right while on a val it is on the left, to indicate the naturalness of combining vals with monzos and to distinguish them, though often you can just use only square brackets if it is clear from context.

Did you notice that {{monzo| -3 2 0 }} + {{monzo| -2 0 1 }} = {{monzo| -5 2 1 }}? Specifically, try adding corresponding numbers in the lists; -3 with -2 gives -5, 2 with 0 gives 2, 0 with 1 gives 1. That is not a coincidence, but the same thing as multiplication except we are doing it with an additive notation.

Now equipped with our funny notation for a val and a monzo, we can do the exact same thing we did before — calculating the mappings — but using this very dense but efficient notation, where we express "mapping the interval described by the monzo by a val" as simple juxtaposition with angle brackets on the outskirts, where we ''multiply'' corresponding numbers in the list and then add them together, like so:

* mapping of 9/8: {{val| 26 41 60 }}{{monzo| -3 2 0 }} = 26 × -3 + 41 × 2 + 60 × 0 = -78 + 82 + 0 = 4 (steps of 26edo)

* mapping of 5/4: {{val| 26 41 60 }}{{monzo| -2 0 1 }} = 26 × -2 + 41 × 0 + 60 × 1 = -52 + 0 + 60 = 8 (steps of 26edo)

* mapping of 45/32: {{val| 26 41 60 }}{{monzo| -5 2 1 }} = 26 × -5 + 41 × 2 + 60 × 1 = -130 + 82 + 60 = 12 (steps of 26edo)

This is all very tedious, but in practice using a val is much simpler, because you do not need to do this, all you need to know is [[5/4]] is mapped to 8\12 and [[3/2]] is mapped to 15\26, therefore [[9/4]] is mapped to 30\26, therefore [[9/8]] is mapped to (30 - 26)\26 = 4\26, so that since we know (9/8)(5/4) = 45/32, the mapped version of 45/32 will just be 4 + 8 = 12. This method guarantees that you never contradict yourself, even if you are technically using suspicious approximations.

For the mathematically inclined, note that this operation is the same as taking the {{w|dot product}} between the monzo and val interpreted as ordinary vectors.

== Importance ==

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 musically relevant/simple 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 (a.k.a. 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.

Furthermore, there is actually a lot of applications of vals and monzos that are not necessarily about approximating things in edos or even regular temperaments for that matter, discussed in [[#Applications]], though all of them do still use the idea of the ''mapping'' provided by the val, so really, a val is a ''mapping'' from JI to the numbers with certain properties.  

== Mathematical definition ==

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''.  

{{Main| Generalized patent val }}

{{See also| patent val }}

The algorithm/process for producing a val does not actually require us to use a purely-tuned 2/1 (octave); instead we can stretch or compress the octave, resulting in potentially different mappings for primes, which is more common the more off the prime is and the more we alter the octave. This can give us a sense in which certain vals which are not patent vals are patent in a more broad sense, hence ''generalized''.

This works by instead of doing log₂(''p'') (where ''p'' is prime) we use log_2.01…(''p'') or something to that effect, where 2.01/1 is our altered version of 2/1. The ''val'' produced by a slight alteration is usually the same, so there are actually continuous ranges where the val produced is the same.

 

We can specify the patent val of ''N''-edo as just ''N'', then we can specify each prime we want to map to the second best approximation by a cryptic letter shorthand:

So we can refer to "104c" (not to be confused with 104{{cent}} (cents)), where we mnemonically think "a, b, '''c'''; 3rd letter; 3rd prime is 2, 3, '''5'''; there is one c so we make the mapping of prime 5 worse (further from just) ''once'' compared to patent".

Perhaps a much clearer notation is to specify explicitly which primes are altered how many times in which direction from the patent val; the notation used by [https://sintel.pythonanywhere.com sintel's temperament finder] would be: 104[+5] and if we also "''warted''" prime 23 (which is similarly flat to prime 5), it'd be 104[+5, +23], which corresponds in the other notation to 104ci where i is the 9th letter of the alphabet and 23 is the 9th prime. They differ on how to notate more than one wart though; with [+5, +23] the direction is ''always'' sharpwards, and [-5, -23] would be flatwards, while with "ci" it's based on "second-best, third-best, etc."; so repeated warts with letters have alternating error while repeated +'s and -'s always have the same kind of error. Slight variations of the notation that [[User:sintel|sintel]] uses have been reinvented and suggested by multiple people, so the one in use by a temperament finder is prioritized here.