Still, when the Quatuor Mosaïques presented its second program at the Y on Thursday evening, the group’s ingratiating sound and impeccable interpretive unity gave the sense of a unit that lives and breathes together constantly. You could hear it in the way the players deftly negotiated the asymmetrical theme, shifts between major and minor, and rash asides in the Allegro con spirito of Haydn’s Quartet in G minor (Op. 20, No. 3), which opened the concert. They avidly embraced the jerky oddness of the Menuet; Mr. Coin sounded especially soulful in the tender Poco adagio.

With works like this, Haydn transformed the string quartet from a medium of genteel diversion to one of innovation and rigorous discourse. In Mozart’s Quartet in B flat (K. 458, “Hunt”), you heard Haydn’s advances allied to a fervid young imagination, as well as a penchant for singing lines to which Mr. Höbarth’s lithe touch and lilting tone proved ideally suited.

From a start in Haydn’s fundamentals Beethoven proceeded to erect mountains, wrestle demons and gaze on eternity within the sublime sprawl of his late quartets. In his final Quartet No. 16 in F (Op. 135), he returned to Classical form with flinty concision. The Quatuor Mosaïques made the most of this mercurial work, with a ravishing Lento assai that was tantamount to secular hymnody.

Recalled for an encore, the group complied with a similarly rapt account of the Cavatina from Beethoven’s Quartet No. 13 in B flat (Op. 130).