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Douglas Harper's Etymology Dictionary
off-beat

also offbeat, "unusual," 1938, from off (adv.) + beat (n.). From earlier sense in reference to from music rhythm (1927).

Wikipedia
Off-beat (music)
  1. redirect Beat (music)#On-beat_and_off-beat
Off-Beat

Off-Beat is a 1960 album by American jazz vocalist June Christy, arranged and conducted by Pete Rugolo.

The tracks on this album were included in a 1997 double-CD re-issue under the title The Song Is June!.

Usage examples of "off-beat".

Not just the usual European ones, but such off-beats as Tagalog, Chinese and Arabic, and those books were well used.

Workshifts were staggered and it was easy for a Reverend Mother to adjust her circadian rhythms to off-beat time.