This paper studies the population dynamics of multiple preference traits in a model of intergenerational cultural transmission. Parents socialize and transmit their preferences to their children with endogenous intensities. Populations concentrated on a single cultural group are in general not stable. There is a unique stable stationary distribution, and it supports two or more cultural groups, in particular those with greater intolerance with respect to others' traits. The larger the heterogeneity of intolerance levels across cultural groups, the smaller the number of traits that are supported in the stable stationary distribution.