  Inspired by recent work arXiv:2301.11227 on massive ambitwistor strings this
paper examines the spectrum of such models using oscillator expansions. The
spectrum depends heavily on the constant related to the normal ordering of the
zero mode operator ${L_0}$ of the Virasoro algebra. The supergravity model is
investigated in more detail, and two anomaly-free variations are presented,
both with a rich spectrum and with tree scattering amplitudes that include a
kinematic Parke-Taylor factor for particles other than gravitons without a need
for an external current algebra. The spectrum of some of the models can also be
interpreted as containing three generations of the Pati-Salam model.
