Val
What are vals and what are they for?
Definition
A val is a set of assignments 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.
For a more mathematically intensive introduction to vals, see vals and tuning space. For the characterization of higher rank temperaments, see mapping.
Example EDO
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.
Shorthand notation
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: = 39, 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.
Proposed notation: To write a JI ratio as a monzo in a JI subgroup, we choose a basis for the subgroup and factor an interval into the basis elements as we factor an interval in the p-limit into primes at most p. Then we write the monzo so as to explicitly state what basis elements we factor the intervals into and how many of each basis element the interval has in the factorization. For example, we can write 81/80 = 92/(24 51) in the 2.9.5 subgroup as [2^-4, 9^2, 5^-1⟩. (We reserve the notation [a b c ...⟩ and ⟨a b c ...] for the p-limit.)
Vals can be defined the same way in other subgroups as well; they represent how a subgroup is (viewed as being) tuned in terms of that edo's steps. For example, 13edo's "2.9.5 patent val" can be written as ⟨2~13, 9~41, 5~30]. To see that this val "tempers out 81/80", we do the same operation (of matching up and multiplying the components and summing the products) as described in the previous section:
⟨2~13, 9~41, 5~30][2^-4, 9^2, 5^-1⟩ = 13*-4 + 41*2 + 30*-1 = 0.
Vals in regular temperaments
Proposed notation: We write a tempered interval (an interval in a regular temperament) as a (generalized) monzo by taking a set of generators (for rank-2 temperaments, this will be the period and the generator), then writing what JI ratio each generator approximates (distinguished from pure-JI subgroups by putting a tilde before it), followed by the number of that specified generator that the interval has. For example, the major third in meantone temperament can be written as [~2: ‐2, ~3/2: 4⟩, meaning "4 perfect fifths minus 2 octaves".
Similarly, edo tunings of a temperament can be given in terms of (a generalized version of) vals, by specifying how many edo steps are used for each generator of the temperament. For example, 31edo's tuning of meantone temperament can be written as ⟨~2: 31, ~3/2: 18].