Juvenile songbirds acquire songs by imitation, as humans do speech. Although imitation should drive convergence within a group and divergence through drift between groups, zebra finch songs sustain high diversity within a colony, but only mild variation across colonies. We investigated this phenomenon by analyzing vocal learning statistics in 160 tutor-pupil pairs from a large breeding colony. Song imitation was persistently accurate in some families, but poor in other families. This could not be attributed to genetic differences, as fostered pupils copied their tutors’ vocal sounds as accurately or as poorly as the tutor’s biological pupils. We discovered two effects that explained the finding: first, even in cases of accurate imitation, pupils often recombined imitated syllables to form new units, and, therefore, distributions of syllable types in pupils’ songs were not well correlated with their tutors’; second, rare vocalizations in tutors’ songs became more abundant in their pupils’ songs, and vice versa. Consequently, cultural transmission of tutor songs that were high in acoustic diversity were stronger than those that were low in diversity. We suggest that a frequency dependent balanced imitation of vocal repertoires prevents the extinction of rare song elements and the overabundance of common ones. Within a group it promotes repertoire diversity, while across groups it constrains drift. Together with syllable recombination, balanced imitation sustains cross-generational homeostasis that prevents the collapse of vocal culture into either complete uniformity or chaos.