The word "discordant" comes from Latin discordans, meaning "disagreeing", evolving from discordare ("to disagree") and discors ("discordant"), which combines dis- ("apart" or "in two") and cor ("heart").  It entered English in the late 14th century via Old French descorder: "to fail to agree, clash"., describing things that conflict, clash, or are harsh to the ear, stemming from a core sense of hearts or minds being apart and in disagreement.  It entered Middle English as discordaunt in the late 1300s, meaning "conflicting in nature or kind, contradictory".  It was originally used for "opinions" or "people in disagreement".  The meaning expanded to describe "sounds that are harsh or jarring" (dissonant).  It is used in geology for "unconformable rock layers" and it's used in medicine, it describes "a lack of agreement" (e.g., in transplantation). 

The Oxford English Dictionary indicates that the adjective "discordant" was first used in English before 1393.  The earliest OED evidence comes from the writing of poet John Gower. Other sources also place its first known use in the late 14th century, deriving from the Old French descordant.  The OED provides historical coverage of the word, showing its use in various contexts over time, including its musical sense of "inharmonious"  "not in accord or harmony", "at variance", "contradictory", or "dissonant" (for sounds).from around 1400.  It is closely related to the noun "discord" ("disagreement, strife"). 

Examples of "discordant" used in a sentence: 
"The only discordant note was a dead cow stinking up the trail." 
"He shut his eyes and prepared himself for the discordant sounds." 
"The sounds of bombs, guns, and engines all mingled in discordant harmony." 
"The musical voice was now a harsh discordant tone that echoed around him."
"She has the difficult task of bringing together a number of discordant elements." 
"Its voice grows harsh, and discordant, sounding more like two people talking at once."
"But even among the top ranks, there were discordant views about how we should move forward." 
"At first, the sound was discordant, a mixture of yells and whistles and the general chaos of the crowd." 

Just the facts:

DISCORDANT Definition & Meaning - Merriam-Webster

DISCORDANT Synonyms: 219 Similar and Opposite Words | Merriam-Webster Thesaurus