  We consider the BPS states of the $E_8$ non-critical string wound around one
of the circles of a toroidal compactification to four dimensions. These states
are indexed by their momenta and winding numbers. We find explicit expressions,
$G_n$, for the momentum partition functions for the states with winding number
$n$. The $G_n$ are given in terms of modular forms. We give a simple algorithm
for generating the $G_n$, and we show that they satisfy a recurrence relation
that is reminiscent of the holomorphic anomaly equations of Kodaira-Spencer
theory.
