Likely answer

Anne McCaffrey's Series Featuring Fire Lizards and Music

In "Dragonsinger" by Anne McCaffrey, the protagonist Menolly finally reaches the Harper Hall as a music apprentice. This direct sequel to "Dragonsong" follows her journey on the planet Pern, where she continues to care for her group of nine telepathic fire lizards while proving her talent as a musician in a male-dominated field.

Confirmed match: Dragonsinger by Anne McCaffrey