  As a first step towards the understanding of the thermodynamical properties
of quasiperiodic structures, we have performed both analytical and numerical
calculations associated with succesive hierarchical approximations to
multiscale fractal energy spectra. We show that, in a certain range of
temperatures, the specific heat displays log-periodic oscillations as a
function of the temperature. We exhibit scaling arguments that allow for
relating the mean value as well as the amplitude and period of the oscillations
to the characteristic scales of the spectrum.
