  We define a new statistic on any Weyl group which we call the odd length and
which reduces, for Weyl groups of types $A$, $B$, and $D$, the the statistics
by the same name that have already been defined and studied in [10], [13],
[14], and [3]. We show that the signed (by length) generating function of the
odd length always factors nicely except possibly in type $E_8$, and we obtain
multivariate analogues of these factorizations in types $B$ and $D$.
