Well the Melodious deck revolves all around the monsters special summoning each other to activate effects and combos so the first place you could look are cards that negate special summoning like Jowgen the Spiritualist. They are all Light attribute too I believe so you could go for anti Light cards like Ally of Justice Catastor before they're able to get a fusion off. The Aria/Elergy lock can be dealt with by cards that don't target or destroy such as Stellarknight Triverr which sends all other cards back to the hand when summoned. Stellarknights are DLC only right now but they should shut down Melodious pretty easily.
As for non-DLC decks, check out the Madolche Archetype as they have a card (Madolche Queen Tiaramisu) which shuffles cards back into the deck without targeting. Or if you want to vent your anger, you can always try one turn kill Exodia!
I got all this off of the Melodious wiki page so look there for more info.
Thank you!