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 1dimensional case of the more general term multival.
One important function of certain vals is that they give the primemapping, in number of scale degrees, of a tuning system, whether a justintonation periodicityblock 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.
For example: The 2,3,5val for 12edo (which is also a breed) is < 12 19 28 ] . This says that 12edo:
Octaveequivalence means that all notes separated by a 2:1 ratio are considered equivalent  thus, all mappings in 12edo can be reduced mod12 because 12 is the number of 12edo scale degrees spanning the equivalenceinterval, 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 equaltemperaments is a breed.
[The val] <1 * gives the exponent of 2; this is the "2adic valuation" of number theory; <0 1 * is the 3adic 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 primefactor 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 plimit group Np of intervals, there is a noncanonically 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 Zlinear combination of basis elements representing the prime numbers, a val may be regarded as a Zlinear combination of a dual basis, consisting of the padic valuations.
For a given prime p, the corresponding padic valuation vp gives the pexponent 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.
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