Between Hill evenness and Hill diversity: how fixed-range evenness indices disagree

Ecologists use evenness to describe the distribution of species abundances and compare communities across space and time. Yet the indices in use are not interchangeable. Chao and Ricotta (2019) identified five fixed-range evenness classes built on Hill numbers. At each Hill order α they are values of one normalized Sharma–Mittal family indexed by a scale parameter β. I show that this family is bounded by two rankings: the ranking by Hill evenness D/S (β → −∞) and the ranking by Hill diversity D (β → +∞). For two assemblages at a common order, the indices change their verdict at most once, at a unique β*, and they do so if and only if D and D/S disagree. The linear and reciprocal classes (β = 0, 2) therefore decide whether any of the five classes can disagree. The same parameter governs changes of scale. Replicating an assemblage K times drives indices with β < 1 towards a power of D/S, and indices with β ≥ 1, including Pielou's J, towards one. Along a richness gradient D ∼ cS^γ the limits are 0, γ and 1, so the sign of an evenness–richness trend depends on the index. Published Alpine-succession and Miocene–Pliocene ostracod data show verdicts that reverse with β. The results concern population-level quantities; sampling error is not treated.