"related notes in music," 1590s, ultimately a shortening of accord (or borrowed from a similar development in French) and influenced by corde "string of a musical instrument" (c. 1300), which is Latin chorda "catgut, a string" of a musical instrument (see cord (n.)).
English cord as a shortening of accord is attested from mid-14c.; cord meaning "music" is attested in English from late 14c. The spelling with an -h- is first recorded c. 1600, from further confusion with chord (n.2) and perhaps also classical correction. Originally two notes sounded simultaneously; of three or more from 18c.
Meaning "string of a musical instrument" is from 1660s (earlier this was cord). The geometry sense "straight line intersecting a curve" is from 1550s; figurative meaning "feeling, emotion" first attested 1784, from the notion of the heart or mind as a stringed instrument.