  We consider the algebra $A$ of bounded operators on $L^2(\mathbb{R}^n)$
generated by quantizations of isometric affine canonical transformations. The
algebra $A$ includes as subalgebras all noncommutative tori and toric
orbifolds. We define the spectral triple $(A, H, D)$ with $H=L^2(\mathbb R^n,
\Lambda(\mathbb R^n))$ and the Euler operator $D$, a first order differential
operator of index $1$. We show that this spectral triple has simple dimension
spectrum: For every operator $B$ in the algebra $\Psi(A,H,D)$ generated by the
Shubin type pseudodifferential operators and the elements of $A$, the zeta
function ${\zeta}_B(z) = {\rm Tr} (B|D|^{-2z})$ has a meromorphic extension to
$\mathbb C$ with at most simple poles. Our main result then is an explicit
algebraic expression for the Connes-Moscovici cyclic cocycle. As a corollary we
obtain local index formulae for noncommutative tori and toric orbifolds.
