  The heavy jet mass distribution in e+e- collisions is computed to
next-to-next-to-next-to leading logarithmic (NNNLL) and next-to-next-to leading
fixed order accuracy (NNLO). The singular terms predicted from the resummed
distribution are confirmed by the fixed order distributions allowing a precise
extraction of the unknown soft function coefficients. A number of quantitative
and qualitative comparisons of heavy jet mass and the related thrust
distribution are made. From fitting to ALEPH data, a value of alpha_s is
extracted, alpha_s(m_Z)=0.1220 +/- 0.0031, which is larger than, but not in
conflict with, the corresponding value for thrust. A weighted average of the
two produces alpha_s(m_Z) = 0.1193 +/- 0.0027, consistent with the world
average. A study of the non-perturbative corrections shows that the flat
direction observed for thrust between alpha_s and a simple non-perturbative
shape parameter is not lifted in combining with heavy jet mass. The Monte Carlo
treatment of hadronization gives qualitatively different results for thrust and
heavy jet mass, and we conclude that it cannot be trusted to add power
corrections to the event shape distributions at this accuracy. Whether a more
sophisticated effective field theory approach to power corrections can
reconcile the thrust and heavy jet mass distributions remains an open question.
