  The most basic characteristic of x-quasiperiodic solutions u(x,t) of the
sine-Gordon equation u_{tt}-u_{xx}+\sin u=0 is the topological charge density
denoted $\bar n$. The real finite-gap solutions u(x,t) are expressed in terms
of the Riemann theta-functions of a non-singular hyperelliptic curve $\Gamma$
and a positive generic divisor D of degree g on $\Gamma$, where the spectral
data $(\Gamma, D)$ must satisfy some reality conditions. The problem addressed
in note is: to calculate $\bar n$ directly from the theta-functional
expressions for the solution u(x,t). The problem is solved here by introducing
what we call the multiscale or elliptic limit of real finite-gap sine-Gordon
solutions. We deform the spectral curve to a singular curve, for which the
calculation of topological charges reduces to two special easier cases.
