A solid slab of crystal naturally oscillates at frequencies at which one of
its three dimensions, length, breadth and thickness, is a mechanical
1/2-wavelength. It can easily be induced to oscillate at harmonics of the
fundamental.

It can also oscillate in one of several mechanical modes, eg., longitudinal,
breadth-wise or in torsion. And in shunt or series-resonant electrical
modes.

The circuit it is embedded in can encourage a preferred frequency. It is
easy to select harmonics. Self-preference is also given to the frequency
which has the highest Q, ie., the least mechanical loss. This is usually the
fundamental.

It does not oscillate EXACTLY at multiples simply because it has three
dimensions and Length, Breadth and Thickness slightly 'interfere' with each
other.

A poorly cut crystal, eg., lack of parallelism, at which there may be no
strong preference may jump erratically between two non-harmonically related
frequencies.

Frequency versus temperature curves depend on oscillation mode and on the
angle at which the slab is cut relative to the direction of the individual
crystals in the bulk material lattice as found by optical means. Cubic
curves are best because they contain a flat horizontal portion.