General:
* Cultural participation
* Cultural involvement
* Cultural interaction
* Cultural immersion
* Cultural exchange
* Cultural dialogue
* Cross-cultural understanding
* Cultural appreciation
* Cultural awareness
Focus on active participation:
* Cultural activism
* Cultural advocacy
* Cultural leadership
* Community engagement
Focus on learning and understanding:
* Cultural exploration
* Cultural learning
* Cultural education
* Cultural sensitivity
* Cultural competence
Focus on artistic expression:
* Cultural production
* Creative expression
* Artistic engagement
Focus on bridging differences:
* Intercultural understanding
* Diversity appreciation
* Inclusion
The best word will depend on the specific context and what you want to emphasize. For example, "cultural participation" might be appropriate for a general discussion of how people interact with different cultures, while "cultural activism" would be a better choice for a discussion of how people are actively working to promote cultural change.
It's also worth considering using a more specific phrase, like "engagement with indigenous art" or "engagement with local music scenes," which provides more context for the reader.