A val is a list of numbers of the form < a b c ... ] , where a,b,c... are integers. It is a covector on the tonespace lattice (i.e., a linear functional on the vectors in the lattice). It is a specific dimensionality referring to the 1-dimensional case of the more general term multival.
One important function of certain vals is that they give the prime-mapping, in number of scale degrees, of a tuning system, whether a just-intonation periodicity-block or a temperament. The numbers are listed according to the prime series:
A mapping to any interval in the scale can be found by multiplying the numbers a,b,c... in the val by their respective exponents x,y,z... in the interval's monzo [ x y, z ... > and adding them -- the sum is the mapping in scale degrees of that interval.
The 2,3,5-val for 12-edo (which is also a breed) is < 12 19 28 ] . This says that 12-edo:
Octave-equivalence means that all notes separated by a 2:1 ratio are considered equivalent -- thus, all mappings in 12-edo can be reduced mod-12 because 12 is the number of 12edo scale degrees spanning the equivalence-interval, therefore:
This can be seen clearly in calculating the mapping by multiplying the val with the monzo and adding the products, thus:
A specific type of val which applies to equal-temperaments is a breed.
[The val] <1 *| gives the exponent of 2; this is the "2-adic valuation" of number theory; <0 1 *| is the 3-adic valuation and so forth. Example:
Equal temperaments are associated to a single val, linear temperaments to a pair of vals, and so forth.
A map from a rational tone group to the integers, which respects multiplication.
If h is a val, then:
h(a*b) = h(a) + h(b);
h(1) = 0; and
h(1/a) = -h(a).
If we write the rational number "a" as a = 2^e2 * 3^e3 * ... * p^ep [that is, if we prime-factor it], we may denote it by a row vector [e2, e3, ...., ep]. In that case we denote a val by a column vector of integers of the same dimension.
In the language of abstract algebra, a val is a homomorphism from the tone group to the integers.
Given the p-limit group Np of intervals, there is a non-canonically isomorphic dual group Vp of vals.
A val is a homomorphism of Np to the integers Z. Just as an interval may be regarded as a Z-linear combination of basis elements representing the prime numbers, a val may be regarded as a Z-linear combination of a dual basis, consisting of the p-adic valuations.
For a given prime p, the corresponding p-adic valuation vp gives the p-exponent of an interval q
So, for instance:
If an interval is written as a row vector of integers, a val is simply a column vector of integers.