We argue that the ordinary commutative and associative algebra of spacetime coordinates (familiar from general relativity) should perhaps be replaced, not by a noncommutative algebra (as in noncommutative geometry), but rather by a Jordan algebra (leading to a framework which we term 'Jordan geometry'). We present the Jordan algebra (and representation) that most nearly describes the standard model of particle physics, and we explain that it actually describes a certain (phenomenologically viable) extension of the standard model: by three right-handed (sterile) neutrinos, a complex scalar field φ, and a U(1)B−L gauge boson which is Higgsed by φ. We then note a natural extension of this construction, which describes the SU(4) × SU(2)L × SU(2)R Pati-Salam model. Finally, we discuss a simple and natural Jordan generalization of the exterior algebra of differential forms.