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I want to use Omnispell Adapt's ability

2U, Tap: You may cast an instant or sorcery card from your hand without paying its mana cost.

to cast Entrancing Melody. Its effect reads

Gain control of target creature with converted mana cost X.

What is X for the effect on that card?

Unfortunately for you, X=0. The Comprehensive Rules state:

202.3e When calculating the converted mana cost of an object with an {X} in its mana cost, X is treated as 0 while the object is not on the stack, and X is treated as the number chosen for it while the object is on the stack.

Since Entrancing Melody is in your hand and not on the stack, X is treated as 0.

This is also mentioned in the Rulings under the card in Gatherer:

If a spell has {X} in its mana cost, you must choose 0 as the value of X when casting it without paying its mana cost.

The value of X is chosen when the spell is cast.
But the only legal choice for X is zero in this case.

107.3b If a player is casting a spell that has an {X} in its mana cost, the value of X isn’t defined by the text of that spell, and an effect lets that player cast that spell while paying neither its mana cost nor an alternative cost that includes X, then the only legal choice for X is 0. This doesn’t apply to effects that only reduce a cost, even if they reduce it to zero. See rule 601, “Casting Spells.”

This is confirmed by a ruling on Omnispell Adapt.

If a spell has {X} in its mana cost, you must choose 0 as the value of X when casting it without paying its mana cost.

I just found the rules clarification on Omnispell Adapts:

If a spell has {X} in its mana cost, you must choose 0 as the value of X when casting it without paying its mana cost.

In that case, Omnispell Adapt can only cast Entrancing Melody for X=0, which makes the effect on both cards quite useless.

Except on targets like Endless One, because its cost is X=0, when targeted by Entrancing Melody...