What are vals and what are they for?
A val is a linear map representing how to view the intervals in a temperament, such as an edo, as approximate versions of intervals in just intonation (JI). They form the link between things like EDOs and JI, and by doing so form the basis for all of regular temperament theory. It's very common for vals to refer to EDOs specifically, although they also show us how to relate larger chains of generators to JI as well (such as a stack of meantone fifths).
A val accomplishes the goal of mapping all intervals in some harmonic limit by simply notating how many steps in the chain it takes to get to each of the primes within the limit. Since every positive rational number can be described as a product of primes, any mapping for the primes hence implies a mapping for all of the positive rational numbers within the prime limit. By mapping the primes and letting the composite rationals fall where they may, a val tells us which interval in the chain represents the tempered 3/2, which interval represents the tempered 5/4, and so forth.
Vals are usually written in the notation ⟨a b c d e f …], where each successive column represents a successive prime, such as 2, 3, 5, 7, 11, 13… etc, in that order, up to some prime limit p.
Vals are important in regular temperament theory because they provide a way to mathematically formalize how, specifically, the intervals in a random chain of generators 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 pumps 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.
Consider the 5-limit val ⟨12 19 28]. This val tells us that you should view 12 generator steps as mapping to the octave 2/1. Since the temperament which maps 12 generator steps to the octave is 12-EDO, this means you're describing 12-EDO.
The val ⟨12 19 28], in addition to saying that 12 steps of 12-equal represents 2/1, also states explicitly that 19 steps of 12-equal represents a tempered 3/1, and 28 steps of 12-equal represents a tempered 5/1.
Now assume you'd like to extend 12-EDO into the 7-limit. If you would like to assume the perspective that the 10 step interval in 12-equal (representing 1000 cents) is a very tempered 7/4, then that means that 7/1, which is 7/4 with two octaves stacked on top, is equal to 10 steps + 12 steps + 12 steps = 34 steps. This decision can hence be represented by using the 7-limit ⟨12 19 28 34] val.
If for some strange reason you'd instead like to say that 900 cents is 7/4, then that would be represented by the ⟨12 19 28 33] val, and if you'd like to say that 123000 cents is 7/4, that would be represented by the ⟨12 19 28 1254] val.
This is also known as warts or wart notation.
Given an explicit or assumed limit, any patent val can simply be represented by stating its first coefficient - the digit representing how many generators map to 2/1. For example, the 5-limit patent val for 17-EDO, ⟨17 27 39], can be called simply, "17".
The patent val for any EDO will use the most accurate mapping for each individual prime. However, it may often be the case that one wants to refer to vals other than the patent val. For example, the 5-limit val ⟨17 27 40], which maps the 5/4 to the 424 cent interval rather than the 353 cent interval, is not the patent val for 17-EDO but may be preferred because it is lower in overall error. Non-patent vals are specified by adding a “wart” to the end of their name which specifies their deviation from the patent val. In this case, we want to specify that the 5/1 has been changed to use its second-most accurate mapping. Since 5 is the third prime number, we add the third letter of the alphabet to the end of the EDO number, to form "17c".
If we wanted to use the third-most accurate mapping for 5, ⟨17 27 38], we'd write "17cc". In 17-EDO, the approximation of the prime-5 component is raised for an odd, and lowered for an even, amount of c letters:
c = 40,
cc = 38,
ccc = 41,
cccc = 37.
The general rules:
- Wart letters 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.
Vals in JI subgroups
We can generalize the concept of monzos and vals from the p-limit to other JI subgroups. This can be useful when considering different edo tunings of subgroup temperaments. Gene Ward Smith called these "svals" for short.
To notate a subgroup val, we typically precede the "bra" notation with an indicator regarding the subgroup (and choice of basis). For instance, the patent val for 12 equal on the 2.3.7 subgroup is often notated "2.3.7 ⟨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. ⟨a b c] would represent a 5-limit val.
Note that we could, for instance, use a different basis for the same subgroup - for instance, we could instead write "2.3.21 ⟨12 19 43]", which is the 12 equal 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" svals represent the same map from this subgroup to a rank-1 generator chain. (It is a matter of semantics if these are thought of as "different" svals or "the same sval" written using a different basis.)
Svals can also be written using subgroups that don't involve primes, e.g. 18.104.22.168/5 ⟨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, 13edo's "2.9.5 patent val" can be written as "2.9.5 ⟨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 "induced 9" from the 2.3.5 patent val is not the same as the "direct 9" from the 2.9.5 patent val.
This notation is also used for subgroup monzos; e.g. 81/80 on the 2.9.5 subgroup is "2.9.5 [-4 2 -1⟩", and it is thus easy to see that the 2.9.5 13p val above tempers out 81/80:
⟨13, 41, 30|2^-4, 9^2, 5^-1⟩ = 13*-4 + 41*2 + 30*-1 = 0.
Vals in Regular Temperaments
There is also a notion of a "tempered val" on a group of "tempered monzos," representing intervals in some regular temperament. This name is sometimes abbreviated as "tmonzos" and "tvals". Typically, this is made explicit by writing the generators beforehand. When the tempered intervals have accepted names, such as in meantone temperament, we can use names like "P8" and "P5," so that the tval "P8.P5 ⟨12 7]" represents the 12-edo "patent tval" in meantone temperament (given that particular basis). If the intervals don't have names, a Transversal can be given instead, preceded with the temperament name, so that we have "(meantone) 2.3/2 ⟨12 7], or "(meantone) 2.3/2 ⟨31 18]".