Val: Difference between revisions

A [[val]] — short for ''valuation'' — is like an algorithm or procedure for finding out how to represent [[frequency ratio]]s ([[interval]]s of [[just intonation|just intonation (JI)]]) with the pitches of an [[equal tuning]] such as an [[edo]]. The basic principle of using a ''patent'' val (the most common usecase) is that you round prime harmonics to edosteps, and then deduce the "mapping" of an arbitrary interval based on its prime factorization. (The "patent" refers to the method of choosing the approximations to the prime harmonics.)

 

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.

If you want to find an approximation to a just interval, the immediate question is: why would you need an algorithm instead of just looking at the [[direct approximation]] possible in the edo? The answer is to avoid contradictions. For example, it might not be true that [[~]][[6/5]] × [[~]][[5/4]] = [[~]][[3/2]] or that [[~]]9/1 × [[~]]5/1 = [[~]]45/1 if you are just always using the direct approximation of each of these frequency ratios (6:5, 5:4, 3:2, 9:1, etc.) in the edo, because of something called ''inconsistency'', which means if you know what the intervals that you want to combine are, then combining their approximations in the edo does not 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, when this happens, it is not possible to fix the inconsistency, except by using a different edo that approximates the intervals 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 other 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, 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 allow us to do that, which brings us to…

A [[val]] is a list of numbers telling you 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. 1st place is prime 2's mapping (a.k.a. the edo), 2nd place is prime 3's mapping, 3rd place is prime 5's mapping, 4th 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 [[#Warts and generalized patent vals|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.

=== Example: 26edo ===

* 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)

* 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)

* 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)

 

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. Using [[backslash notation]] to denote the number of steps in an edo, these are 4\26, 8\26, and 12\26, respectively, so that the underlying "logic" of the approximations being followed is 4\26 + 8\26 = 12\26.

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\26 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.

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 (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.

== Vals vs. mappings ==

# A val can be thought of as a mapping with one row. Put another way, the rows of mappings are vals. To be mathematically precise, a val is a specific type of [[Wikipedia:Linear_map|(linear) mapping]] called a [[Wikipedia:Linear_form|"linear form", or "linear functional"]], which means that its output is a [[Wikipedia:Scalar_(mathematics)|scalar]], or in other words, a single number. This corresponds to the fact that a val must be a 1xM array of numbers, or in other words a [[Wikipedia:Vector_(mathematics_and_physics)|vector]] (specifically a [[Wikipedia:Row_and_column_vectors|row vector]], AKA covector).

# In standard usage, vals must have only integer entries (when expressed in the standard, non-weighted coordinate basis), although t[[Tuning map|uning maps]] are sometimes considered a kind of val.

# Being short for "[[Wikipedia:Valuation_(algebra)|valuation]]", a val is a formal linear sum of [[Wikipedia:P-adic_order|p-adic valuations]].

 

 

 

''Main article: [[Subgroup monzos and vals]]''

 

To notate a subgroup val, we typically precede the "bra" (angle bracket) notation with an indicator regarding the subgroup (and choice of basis, as we don't have to use only ascending primes). For instance, the patent val for 12 equal on the 2.3.7 subgroup is often notated "2.3.7 {{val|12 19 34}}". If the subgroup indicator isn't present, the subgroup can be inferred from context. It is very typical for a val with no explicit subgroup indicator to be interpreted as representing some prime limit, e.g. {{val| a b c }} would represent a 5-limit val (in fact, the normal vals introduced in this page can be seen as entirely contained within this special case).

=== In regular temperaments ===

''Main article: [[Tempered monzos and vals]]''

 

== Generalizations ==

The entries of a val measure equal-tempered steps, which can be thought of either as a generator for a rank-1 temperament (and thus the structure can be generalized to account for multiple generators, resulting in a mapping matrix) or as a logarithmic interval size measure (and thus the entries can be generalized to non-integer values to create a tuning map).  

 

=== Mapping matrix ===

 

''Main article: [[Tuning map]]''

A tuning map generalizes a val in a different way. Instead of treating the entries of a val as equal temperament steps, it treats them as a logarithmic interval size measure (usually cents). Thus, the entries of a tuning map may be any real number. ⟨1200 1901.955] is the tuning map for the justly-tuned 3-limit, and ⟨1200 1896.8 2787.1] is the tuning map for the 5-limit tuned to meantone (specifically, 31edo).