The exposition of a sonata-form movement is the first of the movement's three big parts. Especially in eighteenth- and earlier-nineteenth-century music, this part of the three is usually the easiest to recognize, because it is often surrounded by repeat signs. In modern-day practice, sometimes performers take these repeats and sometimes they do not. Occasionally, the composer will provide first and second endings at the end of the exposition.
A sonata-form exposition typically modulates to a secondary key and articulates a perfect authentic cadence in this key toward the end of the exposition. In their 2006 book Elements of Sonata Theory, James Hepokoski and Warren Darcy define a "failed exposition" as an exposition in which the secondary material does not end with a perfect authentic cadence in the secondary key, what the authors call the "essential expositional closure," or EEC. The last movement of Beethoven's Fifth Symphony is a good case in point. After the music modulates from the primary key of C major into the secondary key of G major, the music never articulates a perfect authentic cadence in this key before the end of the exposition, which is marked clearly by first and second endings.
Below are the string parts, which are sufficiently representative of the music's content to recognize formal aspects (and their absence), with some annotations. An earlier opportunity to cadence is avoided; instead, another theme begins at the point of cadential effacement. This theme is set up in a antecedent-consequent periodic design. Toward the end of the antecedent, a G sharp steers the music toward an A-minor triad, which functions as a predominant chord for the half cadence appropriately ending this antecedent phrase. As the louder consequent phrase begins, Classical practice suggests that this music will head toward a perfect authentic cadence. However, when the melody arrives again at the G sharp, Beethoven respells it as A flat and locks the bass first on C, then F, turning this harmony into a minor-mode predominant in C major. Both this theme and this exposition fail to achieve the customary cadential closure.
The first movement of Luise Adolpha Le Beau's Violin Sonata, op. 10, from the second half of the nineteenth century, begins with an exposition that, like that of the first movement of Beethoven's Fifth Symphony, begins in C minor and modulates to E-flat major for the second theme. The entire exposition is shown below, with a couple of annotations. Le Beau does write one perfect authentic cadence in E-flat major at the end of the second theme, but it is within the first ending. So if a performance takes the first ending and the repeat, like this one, the exposition succeeds, or at least, the first pass through it does. If a performance does not take the repeat and opts for the second ending after a first time through, like this one, the exposition fails. I do not know of another sonata-form exposition that puts the only possible EEC in the first ending and not in the second ending, but I would not be surprised to find out that there are others: please let me know with a comment below. (If Schopenhauer wrote a sonata...)
