Mozart completed this work in 1783 for the hornist Joseph Leutgeb; the accompaniment is scored for pairs of oboes and horns, plus strings. Leutgeb, 24 years older than Mozart and a horn player in the court orchestra at Salzburg, had known the composer since childhood. He moved to Vienna in 1777, four years before Wolfgang, to become a cheesemonger (with money from Leopold Mozart, who was normally tightfisted). Leutgeb's virtuosity, acclaimed by Parisians as well as the Viennese, was obviously still intact when Mozart wrote a concert Rondo for him (K. 371), then a quintet with strings (K. 407/386c) and the first two of four misnumbered concertos, all in E flat for the valveless hand horn. If, by 1787, aging had begun to impair his technique, Leutgeb remained a superb legato player.

K. 417, despite its published number, was the first concerto and bore this autograph: "Wolfgang Amadè Mozart takes pity on Leutgeb, ass, ox, and simpleton, at Vienna on May 27, 1783." He would write even more insultingly later on, yet plainly cared very much for this lifelong friend who survived him by a decade.

The music is a treasurehouse of melody, with an expansive opening movement in common time but no tempo marking (Allegro maestoso was a publisher's, or editor's, educated guess). Following an abbreviated first exposition, the soloist enters with a new theme; later on, the development shifts to a minor key. The succeeding Andante, in the dominant key of B flat, has basic song structure with a middle-section melody of rare beauty for the soloist. A concluding, hunt-inspired Rondo in 6/8 time set the pattern for the rest of Mozart's Leutgeb concertos: irrepressibly jolly as well as comic—by intention, of course—this one with an accelerated coda that leaves the hounds to catch up.

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