  We study natural conditions on essentially discrete spectral triples by which
the quantum differential $da$ belongs to the ideal generated by the unit length
$ds=D^{-1}$. We also study upper and lower bounds on the singular values of the
$da$'s and apply the general framework to natural spectral triples of Dirichlet
spaces and, in particular, those on dual of discrete groups arising from
negative definite functions.
