This paper presents a new switching strategy controlling a new three-level voltage source inverter, resulting in different 25 space voltage-vectors. The new inverter combines some desirable features of two and three-level inverters. The proposed open-leg switching strategy (OLSS) intentionally opens one of the three inverter-legs per-time, in a pre-planned sequence. Thus, the number of allowed switching states becomes 64 compared to 27 originally proposed by other modulation strategies. For a wye-connected load operating at low modulation index (M<0.5 p.u.), the output phase-voltage becomes a seven-level, 12-step waveform, instead of the 6-step one. Thus, the total harmonic distortion is highly reduced. Three new conduction-modes are presented, each verifies a reduced common-mode-voltage time-variation (dv/dt), low third harmonic content, and reduced switching frequency harmonics. For higher modulation indexes, a forth conduction mode is presented, eliminating the low frequency harmonics in the neutral-point current, consequently reducing the dc-link capacitors sizes, and cost. Analytical models, computer simulation results, and experimental verification are presented.