Im not sure if this a bug or Im forgetting something.

I think once it’s done being written it becomes an option to publish it using the mailbox, you can also play the song on the violin itself.

Once you finish the song, the book disappears and the song appears on the list when you click the violin. It is also true for any musical instrument. Additionally, songs can be sold for regular income and fame at the mail box. The best part is that the composition does not disappear. It can be player and re-sold. This makes music making a stable source of income, but you can't sell more than one song for each instrument.