Robot touch can benefit from how humans perceive tactile textural information, from the stimulation mode to which tactile channels respond, then the tactile cues and encoding. Using a soft biomimetic tactile sensor (the TacTip) based on the physiology of the dermal-epidermal boundary, we construct two biomimetic tactile channels based on slowly-adapting SA-I and rapidly-adapting RA-I afferents, and introduce an additional sub-modality for vibrotactile information with an embedded microphone interpreted as an artificial RA-II channel. These artificial tactile channels are stimulated dynamically with a set of 13 artificial rigid textures comprising raised-bump patterns on a rotating drum that vary systematically in roughness. Methods employing spatial, spatio-temporal and temporal codes are assessed for texture classification insensitive to stimulation speed. We find: (i) spatially-encoded frictional cues provide a salient representation of texture; (ii) a simple transformation of spatial tactile features to model natural afferent responses improves the temporal coding; and (iii) the harmonic structure of induced vibrations provides a pertinent code for speed-invariant texture classification. Just as human touch relies on an interplay between slowly-adapting (SA-I), rapidly-adapting (RA-I) and vibrotactile (RA-II) channels, this tripartite structure may be needed for future robot applications with human-like dexterity, from prosthetics to materials testing, handling and manipulation.