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UPBEAT and SYNCOPATION

UPBEAT or ANACRUSIS
Upbeat (Anacrusis) is a term to describe an unestressed note or notes at the beginning of a piece of music or phrase.
When there is an anacrusis at the beginning of a song, often the score ends before the last beat of the last bar, in order to keep the number of bars.
upbeat

SYNCOPATION

It is a rhythmic effect created by playing on the off-beat.
In a bar of 4/4 the off-beat falls on the part of the beat without accent or weak beat
1&2 3&4 5&6 7&8.

Listen to an Off-beat example:

offbeat

A syncopation may involve putting an "important" note on a weak beat, or off the beat altogether.
syncopation