Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

remembering that up to a change of the time origin it it always possible to assume that in the canon with inner and outer rhythms $A,B$ , one has

minA = min B = 0.

This will be assumed throughout.

I will begin with some very simple lemmas (not stated in the recent technical papers) without which the greater theorems remain cryptic. Most properties of cyclotomic polynomials are to be found in any good textbook on algebra (Algebra, by S. Lang, for instance).

Definition 6 The $nth$ cyclotomic polynomial is

prod prod Pn(x) = (x - e2ikp/n) = (xn/d - 1)m(d) gcd(k,n)=1 d|n

where $m$ is the Möbius function.

This the monic polynomial whose roots are the primitive units of order $n$ , or equivalently the minimal polynomial of one such root.

Classically they are the irreducible factors of $xn -1$ :

n prod 2 n-1 prod x - 1 = Pd(x) Dn(x) := 1 + x+ x + ...+ x = Pd(x) d|n d|n,d/=1

Usually $Pn$ have coefficients which are $0,1$ or $-1$ . We will need:

Lemma 2 $Pn(1) = 1$ if and only if $n$ is not a power of a prime number. Conversely, $Ppa(1) = p$ .

This follows by induction from the preceding expression and the formula

p- sum 1 kpa-1 p-1 Ppa(x) = x = Pp(x ) k=0

For instance, $P8(x) = 1+ x4$ .

The importance of these particular polynomials lies in

Lemma 3 If $A o+ B o+ nZ = Z$ , then for all $d |n$ ( $d /= 1$ ) $Pd$ is a divisor of either $A(x)$ or $B(x)$ .

The assertion reads

2 n-1 n A(x).B(x) =_ 1+ x+ x + ...+ x (mod x - 1)

Evidently, as $Dn(x)$ is a divisor of $A(x).B(x) = Dn(x)+ Q(x) .(xn- 1)$ ; its cyclotomic factors of $Dn$ , being irreducible, must divide either $A(x)$ or $B(x)$ .