Intervals and Vals
For p an odd prime, the intervals of the p-limit Np may be taken as the set of all frequency ratios which are positive rational numbers whose factorization involves only primes less than or equal to p. If q is such a ratio, it may be written in factored form as
q = 2^e2 3^e3 ... p^ep
where e2, e3, ... ep are integer exponents. We may write this in factored form as a ket vector of the exponents, or monzo:
|e2 e3 ... ep>
The p-limit rational numbers Np form an abelian group, or Z-module, under multiplication, so that it acts on itself as a transformation group of a musical space; this becomes an additive group using vector addition when written additively as a monzo.
Np is a free abelian group of rank pi(p), where pi(p) is the number of primes less than or equal to p. The rank is the dimension of the vector space in which Np written additively can be embedded as a lattice; saying it is free means this embedding can be done, since there are no torsion elements, meaning there are no positive rational numbers q (called roots of unity) other than 1 itself, with the property that for some positive power n, q^n = 1.
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 v2(5/4) = -2, v3(5/4) = 0, v5(5/4) = 1. If intervals are written as ket vectors, or monzos, vals are denoted by the corresponding bra vector. The 5-limit 12-et val, for instance, would be written <12 19 28|.