1650s, "three successive lines of poetry," from triple; perhaps patterned on couplet. Extended to a set of three of anything by 1733, and to three children at the same birth by 1787 (another word for this was trin, 1831, on the model of twin). Musical meaning "three notes played in the time of two" is from 1801.
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Definitions of sextuplet
the cardinal number that is the sum of five and one;
Synonyms: six / " / vi / sixer / sise / Captain Hicks / half a dozen / sextet / sestet / hexad