c. 1400, deschaunt, "a counterpoint added to a given melody," from Anglo-French deschaunt, Old French deschant, from Medieval Latin discantus "refrain, part-song," from Latin dis- "asunder, apart" (see dis-) + cantus "song, a singing; bird-song," from past participle stem of canere "to sing" (from PIE root *kan- "to sing").
The English spelling was partly Latinized in 16c., but it is an exception for its retention of des- in English. It is attested from 1560s in the sense of "the art of composing part-music," also "the upper part or voice." It is attested from 1570s as "a warbled song, a song with various modulations." The transferred sense of "a continued discourse or series of comments on a subject" is recorded from 1590s.
A metaphor taken from musick, where a simple air is made the subject of a composition, and a number of ornamented variations composed upon it. [Hensleigh Wedgwood, "A Dictionary of English Etymology," 1859]
mid-15c., discanten, "to run a variety with the voice in harmony with a musical theme, sing in counterpoint," from descant (n.). Sense of "to comment at length, make copious and varied comments" is attested by 1640s.