The results of the measurements of the double-differential production cross-sections of pions, d(2)sigma(pi)/dpd Omega, in p-C and pi(+/-)-C interactions using the forward spectrometer of the HARP experiment are presented. The incident particles are 12 GeV/c protons and charged pions directed onto a carbon target with a thickness of 5% of a nuclear interaction length. For p-C interactions the analysis is performed using 100,035 reconstructed secondary tracks, while the corresponding numbers of tracks for pi(-)-C and pi(+)-C analyses are 106,534 and 10,122, respectively. Cross-section results are presented in the kinematic range 0.5 GeV/c <= p(pi) < 8 GeV/c and 30 mrad <= theta(pi) < 240 mrad in the laboratory frame. The measured cross-sections have a direct impact on the precise calculation of atmospheric neutrino fluxes and on the improved reliability of extensive air shower simulations by reducing the uncertainties of hadronic interaction models in the low energy range.