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|.