  For most materials, the symmetry group is known a priori and deduced from the
realization process. This allows many simplifications for the measurements of
the stiffness tensor. We deal here with the case where the symmetry is a priori
unknown, as for biological or geological materials, or when the process makes
the material symmetry axis uncertain (some composites, monocrystals). The
measurements are then more complicated and the raw stiffness tensor obtained
does not exhibit any symmetry in the Voigt's matricial form, as it is expressed
in the arbitrarily chosen specimen's base. A complete ultrasonic measurement of
the stiffness tensor from redundant measurements is proposed. In a second time,
we show how to make a plane symmetry pole figure able to give visual
information about the quasi-symmetries of a raw stiffness tensor determined by
any measurement method. Finally we introduce the concept of distance from a raw
stiffness tensor to one of the eight symmetry classes available for a stiffness
tensor. The method provides the nearest tensor (to the raw stiffness tensor)
possessing a chosen symmetry class, with its associated natural symmetry base.
