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counterpoint (n.1)

"quilted coverlet," late 15c., early 15c. in Anglo-French, from Old French (cuilte) contrepointe "(quilt) stitched through and through" (15c.), altered (by substitution of contre) from coute pointe, from Medieval Latin culcita puncta "quilted mattress," from Latin culcita "cushion" + puncta, fem. past participle of pungere "to prick, stab" (from suffixed form of PIE root *peuk- "to prick"). As a verb, "quilt by stitching together two pieces of cloth," from 1590s.

counterpoint (n.2)

mid-15c., "art of singing an accompaniment to plain song," from Old French contrepoint, from Medieval Latin cantus contrapunctus, from contrapunctum, from Latin contra "against" (see contra (prep., adv.)) + puncta (see point (n.)). It is a reference to the indication of musical notes by "pricking" with a pointed pen over or under the original melody on a manuscript. Meaning "one or more melodies added, according to fixed rules, to a given melody or theme" is from 1520s.

counterpoint (n.3)

"the opposite point" (in an argument), 1590s, from counter- + point (n.1). As a verb from 1940s.