Classic theory predicts species richness scales as the quarter-power of area, yet species–area relationships (SAR) vary widely depending on habitat, taxa, and scale range. Because power-law SAR are used to predict species loss under habitat loss, and to scale species richness from plots to biomes, insight into the wide variety of observed SAR and the conditions under which power-law behavior should be observed is needed. Here we derive from the maximum entropy principle, a new procedure for upscaling species richness data from small census plots to larger areas, and test empirically, using multiple data sets, the prediction that up to an overall scale displacement, nested SAR lie along a universal curve, with average abundance per species at each scale determining the local slope of the curve. Power-law behaviour only arises in the limit of increasing average abundance, and in that limit, the slope approaches zero, not ¼. An extrapolation of tree species richness in the Western Ghats to biome scale (60 000 km2) using only census data at plot scale (¼ ha) is presented to illustrate the potential for applications of our theory.