"kind of woolen fabric," mid-15c., perhaps from Middle French tiretaine "strong, coarse fabric" (mid-13c.), from Old French tiret "kind of cloth," from tire "silk cloth," from Medieval Latin tyrius "cloth from Tyre" (see Tyrian).
If this is the source, spelling likely influenced in Middle English by tartaryn "rich silk cloth" (mid-14c.), from Old French tartarin "Tartar cloth," from Tartare "Tartar," the Central Asian people (see Tartar). Specific meaning "woolen or worsted cloth woven with crossing stripes of colors" is from c. 1500, formerly a part of the distinctive dress of Scottish Highlanders, each clan having its particular pattern.