In this paper, we propose harmony generation according to user input parameters based on fundamental harmonic properties as a new approach to the problem of automatic music completion (the automatic generation of music pieces from any incomplete fragments of music), which we have proposed as a generalization of conventional music information problems such as automatic melody generation and harmonization. The goal is enabling possibly inexperienced users to turn partial musical ideas into complete pieces for quick exploration of musical possibilities. Therefore, the focus lies on response to intuitive modes of input, allowing the user to intentionally shape the generated music. To that end, parameterized harmony generation utilizes fundamental musical principles which are understandable by both user and computer, instead of conventional probabilistic models (which imply imitation of a style or data corpus) or restrictive rule-based models. We apply this approach to the automatic completion of four-part chorales, using the harmonic concepts of active tones, cadences and key modulation. We implemented a system that jointly optimizes harmony and voicing considering both user input and music theory. Our system was evaluated by a professional composer and in a subjective evaluation experiment. We also invite the reader to use our system at http://160.16.202.131/automatic_music_completion.