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In music, syncopation is the stressing of normally unstressed beat in a bar or the failure to sound a tone on an accented beat. For example, in 4/4 time, the first and third beats are normally stressed; if instead the second and fourth beats are stressed and the first and third unstressed, the rhythm is syncopated. Also, if the musician suddenly does not play anything on beat 1, that would also be syncopation.
The stress can also shift by less than a whole beat so it falls on an off-beat, as in the following example where the stress in the first bar is shifted by a quaver (or eighth-note):

Playing a note ever so slightly before or after a beat is another form of syncopation because this produces an unexpected accent.
Syncopation is used on occasion in many music styles, including classical music, but it is a fundamental constant presence in such styles as ragtime and jazz. In the form of a backbeat, syncopation is used in virtually all contemporary popular music.
The term syncopation in dancing is used in two meanings.
Many dance teachers are now abandoning meaning #2, the loose use of the term syncopation and are now using the term "double-time" steps, when that what they mean. They've decided that they don't change the meaning of other musical terms, so they should honor the musical definition of syncopation. In this way, they can enjoy subtle musical syncopations and dance to them as well.
Dance syncopation often matches musical syncopation as when (in West Coast Swing) the leader touches slightly before beat 3 or stomps on beat 6. Kelly Buckwalter (a Two Time US Open WCS Champion) teaches these syncopations. Syncopation in dance
A common incorrect usage of syncopation is to refer to a double time rhythm as syncopation. Incorrect: "In music, spliting the beat into two parts is syncopation." Again, please note this is incorrect, but often taught.