  The transition-metal dicyanides M(CN)$_2$ (M = Zn, Cd) are amongst the most
important negative thermal expansion (NTE) materials known, favoured for the
magnitude, isotropy, and thermal persistence of the NTE behaviour they show.
The conventional picture of the NTE mechanism in this family is one of
correlated rotations and translations of M(C/N)$_4$ polyhedra acting to draw
the diamondoid network of M--CN--M linkages in on itself. An implication of
this mechanism is increased transverse vibrational motion of C and N atoms
relative to the isotropic displacements of M atoms, which act as anchors. Here,
we use a combination of neutron total scattering measurements and \emph{ab
initio} calculations to reassess the vibrational behaviour of the M(CN)$_2$
family. We find that M, C, and N atoms all exhibit similar degrees of local
thermal motion, such that the cyanide linkages behave as pseudo-springs
connecting M$\ldots$M pairs. This interpretation leads us to uncover a `hidden'
dispersion in the M(CN)$_2$ phonon dispersions, closely related to that of
diamond and silicon themselves. By virtue of this mapping, a simple geometric
model based on the competing energy scales of network stretching and flexing --
long applied to interpret NTE modes in C and Si -- turns out to capture the key
NTE physics of M(CN)$_2$, especially at low temperatures. Our study highlights
the potential insight gained by coarse-graining the complex lattice dynamics of
framework materials in terms of what we call `framework modes' -- the
correlated distortions of the underlying network structure itself.
