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Nice cover! I actually haven't played Cave Story so I don't have much of an idea about the accuracy, but that's not the point it sounds nice.
There were however clashing notes with the pulse which is caused by two channels playing the same note for example: Pulse 1 playing D#3 and Pulse 2 playing D#4, thankfully this is very easy to fix by just placing a P81 or P82 command at the beginning of the song which puts the channels on different frequencies (this can also give you a cool flanger effect if used correctly) Anyways, welcome to the forums!
(example of Pxx command attached here)
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Pretty ambitious track for a beginner (the drums are really tricky). Not bad! Mostly accurate except for the timing.
The harmony in the intro seems to have been replaced by an octave, which seems a shame because this is the one part of the track that an NES can reproduce quite faithfully. The volume could stand to be softened, and the bass lowered an octave, but maybe that's just personal opinion.
Like Dwedit mentioned, the [url=http://www.famitracker.com/wiki/index.php?title=Gxx]Gxx effect can help you achieve triplets, which will be necessary for covering this particular track.
I've attached an example of what I'm/we're talking about (first two frames only).
Good luck!
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The only things certain in life are death and uncertainty.