Abstract: A set of pyribole structures can be derived from a model I-beam containing two distinct silicate chains in which the tetrahedra are rotated by different amounts. The model allows some tetrahedral distortion and is not bound by the parity rule (Thompson, 1970). A subset of pyribole structures which includes all the commonly occurring types can be defined using a new rule: that the structure may contain two types of tetrahedral layer, but no tetrahedral layer may contain two types of tetrahedral chain. This rule is more fundamental than the parity rule and has its origin in the optimization of the edge-to-edge packing of the tetrahedral chains into layers. Pnm21 (amphibole) and Pbc21 (pyroxene) emerge as space groups for ‘low protopyriboles’. The approach used here leads naturally to the − and × -chains notation of Thompson used by Veblen and Burnham (1978).