Behind the final two doors on the Faust Omega floor in the Orochi Tower, you get to choose knowledge or music. If you choose music, you have to repeat the notes played to open the final gate. If you play all the tunes correctly, you will get to keep the flute. Here is the sequence.
Tune 1 - 345666 234555
Tune 2 - 345666 234555 43255
Tune 3 - 6533 432
Tune 4 - 6533432 2132
Tune 5 - 345666 234555 43255 6533 432 2132 (all of them at once)
This still needs to be checked for accuracy but I'm 95% sure it's correct. I will confirm it tomorrow.