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

We get the following factorizations with all factors being cyclotomic, except one ( $A$ and $B$ have been translated so as to begin with 0 as usual):

A(x) = P4P6P3P12P24P36

B(x) = P2P8P9P18P72 .(1- x + x2- x3+ x7- x13+ x14- x15+ x16- x17+ x18)

We get $SA = {4,3}$ and $SB = {2,8,9}$ . Thus:

Condition $(T1)$ for $A$ reads $A(1) = 2.3 = 6$ , and for $B$ it is $B(1) = 2.2.3 = 12$ .
Condition $(T2)$ for $B$ means that factors $P2.9$ and $P8.9$ are in $B(x)$ , which is true. Similarly for $A$ .

As was first noted by (Andreatta, 1997), $A$ is a perfect palindrome, $B$ almost is a perfect palindrome, as seen on the following picture.

Figure 9:

$A$ (left) is an exact palindrome while $B$ (right) almost is so.

(Andreatta, 1997) first asked whether this is generally the case and why. Though of course it is difficult to define rigourously such an elusive quality as »almost palindromic«, I think the theorems in (Coven and Meyerowitz, 1999) help to understand why this is so.

A motif is palindromic when its polynomial is equal to its reciprocal: say

d P(x) = x P (1/x) where d is the degree of P

But from their definition, all cyclotomic polynomials are palindromic: if $q$ is a $nth$ root of unity, so is $1/q$ . Hence any product of cyclotomic polynomials is palindromic, and the above theorems tell us that in a rhythmic canon the factors of the associated polynomials are mostly cyclotomic. So these factors are (almost) palindromic, which is in my opinion as close as one could get to an answer to Andreatta’s question.

The Coven-Meyerowitz theorems are very gratifying, especially if Theorem 12 is eventually proved for any period.

What they leave in the dark are the following points: