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 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.

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).

 

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}}.  

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 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.

Note that when we take the closest approximation of each prime, corresponding to rounding rather than e.g. using the second-best approximation possible, we call it a ''[[patent val]]'', therefore, the 5-limit patent val of 26edo is {{val| 26 41 60 }}, where the limit is 5 because that is the highest prime we are considering. If we somehow want to say prime 5 is 61 steps, then that would be represented by the {{val| 26 41 61 }} val. If we want to say that prime 5 is 59 steps, that would be represented by the {{val| 26 41 59 }} val.

 

=== Other applications ===

Vals are important in regular temperament theory because they provide a way to mathematically formalize how, specifically, the intervals in some set of equally spaced pitches 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.

For a more mathematically intensive introduction to vals, see [[Vals and tuning space]]. For the characterization of higher-rank temperaments, see [[Mapping]].

== Relationship with (equal) temperaments ==

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Despite having no contradictions, stacking the tempered intervals of the val will inevitably cause error to accumulate, when compared to the JI counterpart that is supposed to be represented. This is because temperaments temper out an infinite set of commas, which can be derived from a select set of simple/musically relevant commas that are all nullified in the val.

All temperaments compromise JI by reducing the number of primes used, so for instance, 5-limit requires 2,3,5 to represent any pitch. If a 5-limit comma is tempered out, the structure is collapsed, and error is introduced to compensate for something that was not a unison now being one. In mathematical terms, this is equivalent to making one of the basis vectors of JI linearly dependent.

When tempering out enough commas, JI is collapsed onto a quantized line; an equal temperament or rank-1 tuning. This is where vals come into play. Each of the primes is determined by a certain number of quanta, corresponding to octave divisions (edosteps) in [[EDO|edos]], tritave divisions in [[EDT|edts]], et cetera. There are many applications of vals and monzos disjoint from RTT, discussed in [[#Applications]], though all of them still treat vals as providing ''mappings'' from JI to the numbers, with constraints.

== Patent val and generalized patent val ==

As discussed, a patent val is a val derived from rounding prime harmonics to the nearest edosteps. This process for producing a val does not actually require us to use a purely-tuned 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 exactly like ordinary vals, but instead of plugging integer ''N'' into ''N''⋅log₂(''p'') where ''p'' is a prime, we use a non-integer ''N'' or something to that effect. The val produced by a slight alteration is usually the same, so there are actually continuous ranges where the val produced is the same.

 

 

In this notation, 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 appending a letter, called a ''wart'', after the number:  

* etc.  

So we can refer to {{val| 17 27 40 }} by "17c" (not to be confused with 17{{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".

* Wart letters that ''suffix'' the number specify prime approximations being altered from the patent val. The ''n''-th letter of the alphabet refers to the ''n''-th prime: a~2, b~3, c~5, d~7, e~11, etc. A letter which appears ''m'' times refers to the (''m'' + 1)-th most accurate mapping for that prime. So, if a number representing a val is wartless, it is taken to mean the patent val.

* A wart letter that ''prefixes'' the number specifies the corresponding prime as the [[interval of equivalence]] to be divided by the following number. For example, b13 refers to the patent val of [[13edt|13ed3]]. The octave is assumed, so "a" is typically not written out.  

[[Graham Breed]]'s [https://x31eq.com/temper/ Temperament Finder], [[Sintel]]'s [https://sintel.pythonanywhere.com Temperament Calculator], and [[Flora Canou]]'s [https://github.com/FloraCanou/temperament_evaluator Temperament Evaluator] have implementations of wart notation that differ from each other slightly. For example, in the Temperament Finder, the wart letter "p" is used to make explicit that the number refers to the patent val (though the letter originally was intended to stand for "prime"), despite that "p" is logically the letter for prime 53, and the wart letter "q" and after refers to each non-prime basis element of composite/fractional subgroup, respectively and temporarily.  

For a full specification and a deeper dive into the nuances in each implementation, see Flora's [https://github.com/FloraCanou/temperament_evaluator/wiki/Commonwart Commonwart] document.  

In 2022 [[Mike Battaglia]] proposed this notation as a way to be explicit about which primes are being affected and in which direction. In 2024 it was further refined by him and [[Lumi Pakkanen]] to be more analogous to [[Ups and downs notation]].

In this notation, patent vals are notated using the number of divisions followed by square brackets e.g. 17[] for {{val| 17 27 39 }}. To indicate a wider mapping for a prime, it is prefixed with a carret (^) e.g. 17[^5] for {{val| 17 27 40 }}. To indicate a narrower mapping for a prime it is prefixed with a vee (v) e.g. 17[v5] for {{val| 17 27 38 }}. The prefixes stack e.g. 17[^^5] corresponds to {{val| 17 27 41 }}. Multiple modifications are separated by commas (and optionally spaces) e.g. 17[v3, ^5] for {{val| 17 26 40 }}.

The subgroup may be made explicit separated by an "at" sign (@) at the end e.g. 46[]@2.3.7.13/5 for {{val| 46 73 129 63 }} (subgroup 2.3.7.13/5). Formal primes are treated the same way as actual primes e.g. 46[^13/5]@2.3.7.13/5 for {{val| 46 73 129 64 }} (subgroup 2.3.7.13/5)

For patent vals the empty square brackets are optional when using an "at" sign. The subgroup itself is optional if its obvious from context e.g. 12@ for {{val| 12 19 28 }} (subgroup 2.3.5).

The 2022 version used a plus sign (+) in place of the caret and a minus sign (-) in place of the vee. Sintel's Temperament Calculator is a notable implementation of this version of the notation.

== 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 ===

 

 

 

 

==== Subgroup vals ====

It is rather intuitive to generalize the concept of monzos and vals from the ''p''-limit (for some prime ''p'') to other [[JI subgroup]]s. This can be useful when considering different edo tunings of [[subgroup temperaments]]. [[Gene Ward Smith]] called these ''[[sval]]s'', short for ''[[subgroup val]]s'', and correspondingly ''[[smonzo]]s'' as short for ''[[subgroup monzo]]s''.

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 do not have to use only ascending primes). For instance, the patent val for 12et on the 2.3.7 subgroup is often notated 2.3.7 {{val| 12 19 34 }}. If the subgroup indicator is not 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 ordinary vals introduced in this article can be seen as entirely contained within this special case.

Note that we could, for instance, use a different basis for the same subgroup – for instance, we could instead write 2.3.21 {{val| 12 19 53 }}, which is the 12et patent val in the "2.3.21" subgroup. Since the "2.3.21" subgroup is the same as the 2.3.7 subgroup, just written with a different basis, these two apparently different subgroup vals represent the same map from this subgroup to a rank-1 generator chain.  

Subgroup vals can also be written using subgroups that do not involve primes, e.g. 2.3.7.13/5 {{val| 46 73 129 63 }}.

Note that the notion of a patent val for a subgroup val may not agree with the patent val on a prime limit. For instance, the [[patent val]] for [[13edo]] in the 2.9.5 subgroup can be written as 2.9.5 {{val| 13 41 30 }}, because the best approximation to 2 is 13 steps, the best approximation to 9 is 41 steps, and the best approximation to 5 is 30 steps. Note that, however, the patent val on the 2.3.5 subgroup instead maps 3/1 to 21 steps, so that the 9 induced from the 5-limit patent val is not the same as the 9 directly derived from the 2.9.5-subgroup patent val.

This notation is also used for subgroup monzos; e.g. [[81/80]] on the 2.9.5 subgroup is 2.9.5 {{monzo| -4 2 -1 }}, and it is thus easy to see that 2.9.5 {{val| 13 41 30 }} above makes 81/80 [[vanish]]: {{vmprod| 13 41 30 | -4 2 -1 }} = 13 × (-4) + 41 × 2 + 30 × (-1) = 0.

==== Tempered vals ====

 

* [http://tonalsoft.com/enc/v/val.aspx Tonalsoft Encyclopedia | ''Val'']

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