  We have studied natural purity $^{4}$He single crystals and polycrystals
between 10 and 600 mK using a torsional oscillator with a 2 cm$^{3}$ rigid cell
made of sapphire with a smooth geometry. As the temperature was lowered, we
observed sample dependent but reproducible resonant frequency shifts that could
be attributed to a supersolid fraction of order 0.1%. However, these shifts
were observed with single crystals, not with polycrystals. Our results indicate
that, in our case, the rotational anomaly of solid helium is more likely due to
a change in stiffness than to supersolidity. This interpretation would
presumably require gliding of dislocations in more directions than previously
thought.
