“It was the sound of hope to Harry . . . the most beautiful and welcome thing he had ever heard in his life. . . . He felt as though the song were inside him instead of just around him. . . . It was the sound he connected with Dumbledore, and it was almost as though a friend were speaking in his ear. . . .” -- GF34
The song of the phoenix gives strength and hope to those it sings for, ‘increasing the courage of the pure of heart and striking fear into the hearts of the impure’ according to Scamander (FB). When wands sharing a phoenix-feather cores from the same bird are forced to do battle, the rare Priori Incantatem effect that results will be accompanied by the sound of phoenix song (GF34).
In GF34, the connection between the twin cores of Harry Potter and Lord Voldemort's wands cause the sound of a phoenix song during the connection of the wands. This happens in the Little Hangleton Graveyard while Harry Potter and Lord Voldemort duel.