  We discuss nonperturbative radiation for a recently introduced class of
infrared safe event shape weights, which describe the narrow-jet limit.
Starting from next-to-leading logarithmic (NLL) resummation, we derive an
approximate scaling rule that relates the nonperturbative shape functions for
these weights to the shape function for the thrust. We argue that the scaling
reflects the boost invariance implicit in NLL resummation, and discuss its
limitations. In the absence of data analysis for the new event shapes, we
compare these predictions to the output of the event generator PYTHIA.
