Several indices have been created to measure diversity, and the most frequently used are the Shannon-Wiener (H) and Simpson (D) indices along with the number of species (S) and evenness (E). Controversies about which index should be used are common in literature. However, a generalized entropy (Tsallis entropy) has the potential to solve part of these problems. Here we explore a family of diversity indices (Sq; where q is the Tsallis index) and evenness (Eq), based on Tsallis entropy that incorporates the most used indices. It approaches S when q=0, H when q→1 and gives D when q=2. In general, varying the value of the Tsallis index (q), Sq varies from emphasis on species richness (q<1) to emphasis on dominance (q>1). Similarly, Eq also works as a tool to investigate diversity. In particular, for a given community, its minimum value represents the maximum deviation from homogeneity (Eq=1) for a particular q (herein named q*). It is remarkable that our analysis indicates that q* and its corresponding evenness, Eq*, are negatively affected by S when using simulated data. They may represent an index related to species rarity. Furthermore, Sq* (i.e. the value of Sq for a specific q*) is positively affected by richness that is an important property of any diversity index. In general, our findings indicate that the indices H, D, S, Sq*, E and Eq* are only part of a whole set of possibilities. In addition, the ecological properties of Eq* and Sq*, proposed here for the first time, show promise in ecology.