Much difficulty has so far prevented the emergence of a consistent scenario for the origin of Type Ib and Ic supernovae (SNe). Here, we follow a heuristic approach by examining the fate of helium stars in the mass range 4 to 12Msun, which presumably form in interacting binaries. The helium stars are evolved using stellar wind mass loss rates that agree with observations, and which reproduce the observed luminosity range of galactic WR stars, leading to stellar masses at core collapse in the range 3-5.5Msun. We then explode these models adopting an explosion energy proportional to the ejecta mass, roughly consistent with theoretical predictions. We impose a fixed 56Ni mass and strong mixing. The SN radiation from 3 to 100d is computed self-consistently starting from the input stellar models using the time-dependent non-local thermodynamic equilibrium radiative-transfer code CMFGEN. By design, our fiducial models yield similar light curves, with a rise time of ~20d and a peak luminosity of ~10^42.2erg/s, 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 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 stars stripped by a binary companion can account for the vast majority of ordinary SNe Ib and Ic, and that stellar wind mass loss is the key to remove the helium envelope in SN Ic progenitors. [abridged]
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