We show that it is possibile to produce nearly bandwidth-limited few-cycle attosecond pulses based on periodic resonance interaction of a quasi-monochromatic radiation with the bound states of hydrogenlike atoms. A periodic resonance is provided by a far-off-resonant laser field with intensity much below the atomic ionization threshold via periodic tunnel ionization from the excited states and adiabatic Stark splitting of the excited energy levels. Without external synchronization of the spectral components, it is possible to produce $135\mathrm{as}$ pulses at $13.5\mathrm{nm}$ in ${\mathrm{Li}}^{2+}$ -plasma controlled by radiation of a mode-locked Nd:YAG laser, as well as $1.25\mathrm{fs}$ pulses at $122\mathrm{nm}$ in atomic hydrogen controlled by radiation of a ${\mathrm{CO}}_2$ laser.