That's called "Opening Theme" in FF1 OST. Also known as "the Bridge-Crossing Theme", "the Prologue" and "Final Fantasy" to the fans. By the way, shouldn't you add the pitch to help with the melody flow? For example, with your last two notes, A and G, it's a bit hard to figure out that G is pointing to a note almost an octave higher than A, not the G just below A.

If mine was correct, I'll post the next one. It should be quite easy too. I'll add numbers after each note to state their pitch (for example, F1-A1-C2 is an F major chord).

Game: FFVII
Melody
C1 A1 G1 A1 C1