A major challenge in evolutionary biology is explaining the origins of complex phenotypic diversity. In animal communication, complex signals may evolve from simpler signals because novel signal elements exploit preexisting biases in receivers’ sensory systems. Investigating the shape of female preference functions for novel signal characteristics is a powerful, but underutilized, method to describe the adaptive landscape potentially guiding complex signal evolution. We measured female preference functions for characteristics of acoustic appendages added to male calling songs in the grasshopper Chorthippus biguttulus, which naturally produces only simple songs. We discovered both hidden preferences for and biases against novel complex songs, and identified rules governing song attractiveness based on multiple characteristics of both the base song and appendage. The appendage's temporal position and duration were especially important: long appendages preceding the song often made songs less attractive, while following appendages were neutral or weakly attractive. Appendages had stronger effects on songs of shorter duration, but did not restore the attractiveness of very unattractive songs. We conclude that sensory biases favor, within predictable limits, the evolution of complex songs in grasshoppers. The function-valued approach is an important tool in determining the generality of these limits in other taxa and signaling modalities.