  For periodic Toda chains with a large number $N$ of particles we consider
states which are $N^{-2}-$close to the equilibrium and constructed by
discretizing any given $C^2-$functions with mesh size $N^{-1}$. For such states
we derive asymptotic expansions of the Toda frequencies $(\omega^N_n)_{0 < n <
N}$ and the actions $(I^N_n)_{0 < n < N},$ both listed in the standard way, in
powers of $N^{-1}$ as $N \to \infty$. %listed in accordance with the ordering
of the frequencies at the equilibrium, %$(2 \sin \frac{n\pi} {N})_{0 < n < N}$.
At the two edges $n \sim 1$ and $N -n \sim 1$, the expansions of the
frequencies are computed up to order $N^{-3}$ with an error term of higher
order. Specifically, the coefficients of the expansions of $\omega^N_n$ and
$\omega^N_{N-n}$ at order $N^{-3}$ are given by a constant multiple of the n'th
KdV frequencies $\omega^-_n$ and $\omega^+_n$ of two periodic potentials,
$q_{-}$ respectively $q_+$, constructed in terms of the states considered. The
frequencies $\omega^N_n$ for $n$ away from the edges are shown to be
asymptotically close to the frequencies of the equilibrium. For the actions
$(I^N_n)_{0 < n < N},$ asymptotics of a similar nature are derived.
