Classical music doesn’t usually tend to draw comparisons to endurance sports, but pianist András Schiff seems eager to change that. Having previously performed the complete series of Mozart’s piano concertos over a seven-year period ending in 2006, Mr. Schiff then decided to tackle all 32 of Beethoven’s sonatas and is playing them in a two-season, four-city American tour. On Wednesday and Friday, the cycle progresses with the third and fourth concerts at Carnegie Hall.

For Mr. Schiff, a crucial element is the chronological order in which the sonatas are being performed. Normally, in accordance with traditional programming, individual concerts mix the famous with the lesser known, the early with the late. But Mr. Schiff is attracted to the chronological approach. “And the more I play, the more convinced I am,” he said. “The sonatas are all wonderful in quality. Playing them chronologically shows their evolution and development. In the ‘Ring,’ you wouldn’t do ‘Götterdämmerung’ before ‘Die Walküre.’ That is an extreme case, of course, but the sonatas, too, tell a narrative story.”

Mr. Schiff, who was born in Budapest and has lived in the West since 1979, is on to something illuminating with his chronological approach. In a concert of three Mozart piano concertos — No. 15 in E flat, K. 449; No. 16 in B flat, K. 450, and No. 17 in G, K. 453 — Mozart’s writing for winds, an extraordinary feature of his mature works, seemed to leap forward in the third concerto on the program. “It’s true. Something happened to Mozart with these concertos. They were all written in 1784 as was,” Mr. Schiff said, clinching the argument, “the quintet for piano and winds,” whose Köchel number happens to be 452.

As for Beethoven, music history students learn that his output falls into three stylistic periods — early, middle, and late. “The periods are certainly there, but there are logical steps. They don’t come out of nowhere,” Mr. Schiff said. He also insists that sonatas belonging to the same opus number be kept together. “Sets of three tend to include a lyrical sonata, a dramatic one, and one that is humorous.” Humor in Beethoven? “People expect Beethoven to be a heroic conqueror, even a wrestler. He can be, but he can also be very tender or humorous. The sonatas have an enormous range.”

Wednesday’s concert includes sonatas No. 19 in G minor, Op. 49, No. 1; No. 20 in G, Op. 49, No. 2; No. 9 in E, Op. 14, No. 1; No. 10 in G, Op. 14, No. 2, and No. 11 in B flat, Op. 22. The designation Opus 49 is misleading. “They were early, written between 1795 and 1798. But I like playing these ‘easy’ sonatas, and this program is the logical place for them,” Mr. Schiff said. As a byproduct of the chronological approach, a program like Wednesday’s could spell trouble at the box office. But Mr. Schiff is steadfast in asserting that the “lesser-known works are not a bit lesser in value than the famous ones. [The movements of] Opus 14 are very special pieces and completely different in character. The first is lyrical. Beethoven reworked it as a string quartet, and it has a string quartet texture. The second has a melodious first movement, but it turns into one of the funniest sonatas.” Mr. Schiff considers Opus 22 to be “the most perfect classical sonata.”

The famous sonatas pose problems of their own, sometimes requiring, as Mr. Schiff puts it, “restorative work.” Take the so-called “Moonlight Sonata,” No. 14 in C-sharp minor, Op. 27, No. 2, which will be heard on Friday along with its companion, No. 13 in E flat, Op. 27, No. 1, and sonatas No. 12 in A flat, Op. 26, and No. 15 in D, Op. 28 (“Pastoral”). The title “Moonlight,” which was not Beethoven’s, “sentimentalizes this wonderful piece.” Mr. Schiff scrupulously follows the instruction “senza sordino,” meaning “with pedal,” depressing it a quarter to halfway and holding it for the first movement. “It creates a cloud of mist that contrasts with the dry second movement. The first movement is often played twice too slow. The instruction is ‘alla breve’ — two to the bar. We don’t have Beethoven’s metronome indication — he specified metronome markings for only one sonata, the ‘Hammerklavier’ [Op. 106],” and they are disputed because the first movement marking is shockingly fast. Does Mr. Schiff follow it? “Of course,” he said. “I have examined Beethoven’s metronome, and there is nothing wrong with that metronome!” The works on Friday’s concert introduce Beethoven’s middle period. “Opus 26 has four movements, but none is in sonata form. Opus 27, Number 1, is more like a fantasy than a sonata. Beethoven was experimenting.” Mr. Schiff, who has also played cycles of Bach and Schubert, calls the Beethoven undertaking “a bigger challenge than anything — and I’ve had some challenges.”

What is next? “I want to play the Diabelli Variations and “The Art of Fugue” and more Schumann, Debussy, and Bartók. But not as complete cycles,” he said. “I think I am done with them.”