In english when one hears a beautiful piece of music, the feeling can often be described that it "makes my heart sing".
Is there an equivalent description for this feeling in german?
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There are a lot of possible german translations; how they fit is very much based on your specific context.
I can offer some more:
Das bringt meine Seele zum Klingen
Das lässt mein Herz schneller schlagen
I don't know an equivalent description for that. It would be I love this song - "Ich liebe diesen Song".
But if the song won't go out of your head. So if you have to sing it all the time in your head, because you like it, you can say "einen ohrwurm haben". In english that means "to have an ear worm" (sounds weird in english) Not exactly the same but you can use it in the same context.