An Etymological Dictionary of the German Language/Flöte
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Flöte, f., from the equiv. MidHG. floite, vloite, f., ‘flute’; corresponds to Du. fluit, from OFr. flaüte, ModFr. flûte (whence also E. flute, Du. fluit); comp. Ital. flauto, ‘flute.’ In the idiom flötengehen, ‘to come to nothing,’ a LG. fleuten, ‘to flow’ (OLG. fliotan), appears; it meant orig. (in the 18th cent.) ‘to go through, run away.’