  We perform a systematic study of the generic Gatto-Sartori-Tonin relation,
$\tan^2\theta_{ij}= m_i/m_j$. This study of fermion mixing phenomena leads us
to the necessary conditions that are needed in order to obtain it without any
approximation. We begin by considering two Dirac fermion families. By means of
the hierarchy in the masses, it is found that a sufficient and necessary
condition is to have a normal matrix with $m_{11}=0$. This matrix can be
decomposed into two different linearly independent contributions. The origin
for such two independent contributions can be naturally explained by what we
shall call the flavor-blind principle. This principle states that Yukawa
couplings shall be either flavor-blind or decomposed into several sets obeying
distinct permutation symmetries. In general, it is shown that the symmetry
properties of the introduced set of Yukawa matrices follow for $n$ fermion
families the unique sequential breaking $S_{nL}\otimes S_{nR} \rightarrow
S_{(n-1)L}\otimes S_{(n-1)R} \rightarrow \cdots \rightarrow S_{2L}\otimes
S_{2R} \rightarrow S_{2A}$. The particular case of three fermion families
explains why the four mass ratios parametrization that we recently proposed can
be used even in the case of no hierarchical masses.
