The paper proposes a three-fold classification for pitch-space lattices based on a music-theoretic construct known as the Tonnetz. It argues that applying incremental changes on some of the constructive features of the generic Tonnetz (Cohn 1997) results in a set of coherent and analytically versatile transpositional networks (T-nets), which brings under a focused perspective diverse pitch structures such as Tonnetze, affinity spaces, Alban Berg's ``master array'' of interval-cycles, and other types of networks. The paper also explores the music-modeling potential of progressive and dynamic T-nets by attending to characteristic compositional deployments in the music of Witold Lutosławski.