Much difficulty has so far prevented the emergence of a consistent scenario for the origin of Type Ib and Ic supernovae (SNe). Either the SN rates or the ejecta masses and composition were in tension with inferred properties from observations. Here, we follow a heuristic approach by examining the fate of helium stars in the mass range from 4 to 12 M⊙, which presumably form in interacting binaries. The helium stars were evolved using stellar wind mass loss rates that agree with observations and which reproduce the observed luminosity range of galactic Wolf-Rayet stars, leading to stellar masses at core collapse in the range from 3 to 5.5 M⊙. We then exploded these models adopting an explosion energy proportional to the ejecta mass, which is roughly consistent with theoretical predictions. We imposed a fixed 56Ni mass and strong mixing. The SN radiation from 3 to 100 d was computed self-consistently, starting from the input stellar models using the time-dependent nonlocal thermodynamic equilibrium radiative-transfer code CMFGEN. By design, our fiducial models yield very similar light curves, with a rise time of about 20 d and a peak luminosity of ~1042.2 erg s−1, which is in line with representative SNe Ibc. The less massive progenitors retain a He-rich envelope and reproduce the color, line widths, and line strengths of a representative sample of SNe Ib, while stellar winds remove most of the helium in the more massive progenitors, whose spectra match typical SNe Ic in detail. The transition between the predicted Ib-like and Ic-like spectra is continuous, but it is sharp, such that the resulting models essentially form a dichotomy. Further models computed with varying explosion energy, 56Ni mass, and long-term power injection from the remnant show that a moderate variation of these parameters can reproduce much of the diversity of SNe Ibc. We conclude that massive stars stripped by a binary companion can account for the vast majority of ordinary Type Ib and Ic SNe and that stellar wind mass loss is the key to removing the helium envelope in the progenitors of SNe Ic.