Spectral triple on $\mathcal C(M)$ where $M$ is a compact Riemannian manifold, not necessarily spin

Question

I have been reading Alain Connes' Compact metric spaces, Fredholm modules and hyperfiniteness.

In proposition 1, it is mentioned that an unbounded Fredholm module (nowadays: spectral triple) over $C(M)$, where $M$ is a compact, spin Riemannian manifold is as follows: the representing Hilbert space $L^2(M, S)$ is the space of spinors acting on $M$ and the self-adjoint operator $D$ is the Dirac operator.

If we drop the spin assumption, what would be a "standard" unbounded Fredholm module over $C(M)$?

Answer 1

One standard example is formed by considering the space $L^2(M;\Lambda^\ast T^\ast M)$ of differential forms with operator $D=d+d^\ast$, where $d$ is the exterior derivative and $d^\ast$ is its formal $L^2$-adjoint.