  (Abridged) We explore, by means of a large ensemble of SPH simulations, how
the level of turbulence affects the collapse and fragmentation of a
star-forming core. All our simulated cores have the same, except that we vary
(a) the initial level of turbulence (as measured by the ratio of turbulent to
gravitational energy, $\alpha_{\rm turb} \equiv U_{\rm turb}/|\Omega| = 0,
0.01, 0.025, 0.05, 0.10 {\rm and} 0.25$) and (b), for fixed $\alpha_{\rm
turb}$, the details of the initial turbulent velocity field (so as to obtain
good statistics).
  A low level of turbulence ($\alpha_{\rm turb} \sim 0.05$) suffices to produce
multiple systems. As $\alpha_{\rm turb}$ is increased, the number of objects
formed and the companion frequency both increase. The mass function is bimodal,
with a flat low-mass segment representing single objects ejected from the core
before they can accrete much, and a Gaussian high-mass segment representing
objects which because they remain in the core grow by accretion and tend to
pair up in multiple systems.
