Using an octonionic formalism, we introduce a new mechanism for reducing ten space–time dimensions to four without compactification. Applying this mechanism to the free, ten-dimensional, massless (momentum space) Dirac equation results in a particle spectrum consisting of exactly three generations. Each generation contains one massive spin-1/2 particle with two spin states, one massless spin-1/2 particle with only one helicity state, and their antiparticles — precisely one generation of leptons. There is also a single massless spin-1/2 particle/antiparticle pair with the opposite helicity and no generation structure. We conclude with a discussion of some further consequences of this approach, including those which could arise when using the formalism on a curved space–time background, as well as the implications for the nature of space–time itself.