  A measure on a locally compact Abelian group is said to have a natural
spectrum if its spectrum is equal to the closure of the range of the
Fourier-Stieltjes transform. In this paper we continue the study of spectrally
reasonable measures (measures perturbing any measure with a natural spectrum to
a measure with a natural spectrum) initiated in \cite{ow}. Particularly, we
provide a full characterization of such measures for certain class of locally
compact Abelian groups which includes the circle and the real line. We also
elaborate on the spectral properties of measures with non-natural but real
spectra constructed by F. Parreau.
