It is quite difficult to trace the origin of this idiom; however, the musical references are clear. It’s easy enough to change a tune if you really, really don’t like a song, but when it comes to changing your mind on something, and you are stubborn as a mule, then changing you tune becomes quite difficult, unless it brings you some sort of advantage or it suits you.

At first Bill was against selling his old jazz records, but he changed his tune when he realized how much money he could make.

Oliver really hated the idea of studying maths with a private maths tutor, but he changed his tune once he saw how beautiful she was.

Kelly was quite rude to me at first, but she changed her tune when someone mentioned how wealthy I was.

The Prime Minister wasn't polite to journalists at first, but he changed his tune when he realised

a. they had a lot of influence

b. they didn't get paid much

c. they were only doing their job

*answer is A

Something to think about:
Do you know someone who changed their tune when they learned some new information about a situation?