  We present a novel spectroscopic technique for second harmonic generation
(SHG) using femtosecond laser pulses at 30~kHz repetition rate, which
nevertheless provides high spectral resolution limited only by the
spectrometer. The potential of this method is demonstrated by applying it to
the yellow exciton series of Cu$_2$O. Besides even parity states with $S-$ and
$D-$ envelope, we also observe odd parity, $P-$ excitons with linewidths down
to 100 $\mu$eV, despite of the broad excitation laser spectrum with a full
width at half maximum of 14~meV. The underlying light-matter interaction
mechanisms of SHG are elaborated by a group theoretical analysis which allows
us to determine the linear and circular polarization dependences, in good
agreement with experiment.
