A holographically designed, aperiodic distributed feedback grating is used as a multi-resonance filter and embedded within an existing Fabry-Pérot (FP) terahertz (THz) quantum cascade laser (QCL) cavity. Balancing the feedback strengths of the filter resonances and the FP cavity creates a system capable of a high degree of single-mode selectivity, which is sensitive to changes in driving current. Multi-moded QCLs operating around 2.9 THz are thus modified to achieve purely electronic discrete tuning spanning over 160 GHz with an average tuning resolution of 30 GHz. Applying the same multi-resonance filter to QCLs with gain peaks around 2.65 and 2.9 THz leads to dual-mode lasing with an electrically controlled frequency separation of between 190 and 267 GHz. A phase sensitive mode selection mechanism is experimentally confirmed by the observation of divergent fine-tuning of the lasing modes.