  The pretty good transmission scenario implies that the probability of sending
one excitation from one extreme of a spin chain to the other can reach values
arbitrarily close to the unity just by waiting a time long enough. The
conditions that ensure the appearance of this scenario are known for chains
with different interactions and lengths. Sufficient conditions for the presence
of pretty good transmission depend on the spectrum of the Hamiltonian of the
spin chain. Some works suggest that the time $t_{\varepsilon}$ at which the
pretty good transmission takes place scales as $1/(|\varepsilon|)^{f(N)}$,
where $\varepsilon$ is the difference between the probability that a single
excitation propagates from one extreme of the chain to the other and the unity,
while $f(N)$ is an unknown function of the chain length. In this paper, we show
that the exponent is not a simple function of the chain length but a power law
of the number of linearly independent irrational eigenvalues of the
one-excitation block of the Hamiltonian that enter into the expression of the
probability of transmission of one excitation. We explicitly provide examples
of a chain showing that the exponent changes when the couplings between the
spins change while the length remains fixed. For centrosymmetric spin chains
the exponent is at most $N/2$.
