  I consider the current sample of galaxy nuclei producing quasiperiodic
eruptions (QPEs). If the quasiperiod results from the orbital motion of a star
around the central black hole, the dearth of associated black hole masses
$\gtrsim 10^6\msun$ places tight constraints on models. It disfavours those
assuming wide orbits and small eccentricities, because there is ample volume
within pericentre to allow significantly more massive holes in QPE systems than
are currently observed. If instead the orbiting star is assumed to pass close
to the black hole, the same lack of large black hole masses strongly suggests
that the stellar orbits must be significantly eccentric, with $1 - e \lesssim
{\rm few}\times 10^{-2}$. This favours a tidal disruption near-miss picture
where QPEs result from repeated accretion from an orbiting star (in practice a
white dwarf) losing orbital angular momentum to gravitational radiation, even
though this is not assumed in deriving the eccentricity constraint. Given the
tight constraints resulting from the current small observed sample, attempts to
find QPE systems in more massive galaxies are clearly important.
