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Yes, it is a specific eve, New Year's Eve.
I have seen it listed at well over a ton (total weight) of confetti is dropped on New Year;s Eve on Times square.
Honolulu
"New Year's Eve".
It is the eve of the New Year, so New Year is in the genetive (possessive) case, thus: New Year's.
New Year Eve fell on a Thursday in 2009. Previous to this New Year Eve fell on a Thursday in 1998.