In a marvelous confluence of scheduling, two of the most important works of the last century, one indirectly inspired by the other, were offered this weekend in New York. On Thursday, pianist Pierre-Laurent Aimard played the first work ever written in the 12-tone style, the Five Pieces for Piano from 1923 of Arnold Schoenberg. And on Saturday night, Itzhak Perlman presented a reading of the String Quartet No. 3 of Béla Bartók from 1927, a work that is the godchild of the Schoenberg original.

Performing on the main stage at Carnegie Hall, Mr. Aimard, who grew up in the home of composer Olivier Messiaen, began with Bach, specifically the first 11 chapters of “The Art of the Fugue.” This is music that was neglected for more than 100 years, with sales so poor when it was first published just after the composer’s death that the copper plates on which it was engraved were sold as scrap. Throughout the 19th century, this potentially powerful performance piece was treated strictly as an academic exercise.

Mr. Aimard gave us the simple and the stretto — one subject overlapping onto an- other — fugues, ceasing before the more complex structures commenced. The result was a taste of purity, played in a unique style. Reproducing Bach on a modern piano is always a challenge and Mr. Aimard took his stab at it by invoking the sound of the harpsichord (Bach did not specify any instrumentation for this monumental effort). Relying rather heavily on the pedal, Mr. Aimard created an interwoven acoustic, many comings and goings of overtones keeping the ear on alert. He is not the most accurate of pianists, but he did imbue this mathematical music with the added overlay of poetry.

The Schoenberg was a fine performance, Mr. Aimard taking the trouble to play the pieces as music, not at all like historical relics. The waltz that ends the quintet — strictly speaking the first of the dodecaphonic works to see the light of day — was charming, the others dramatic or playful as appropriate. In these capable hands, Schoenberg sounded very much like Brahms, offering a varied set of mood pictures that just happen to be in an entirely new language.

Ending with Beethoven, Mr. Aimard presented his conception of a kinder, gentler Sonata in A Flat major, Op. 110, one of the final three essays in the genre from this greatest of all composers. Careful to keep his emotions in check, he delivered a somewhat dispassionate but reflective rendition that satisfied if not thrilled. This recital was aimed more toward the brain than the heart, and hit its mark precisely.

For an encore, Mr. Aimard chose one of Elliott Carter’s newest piano pieces. Even if you haven’t heard this particular piece, you have surely heard it all before: a rapid perpetual motion machine, busily going nowhere, often living, as so many contemporary pieces do, at the lowest and highest ends of the keyboard. Mr. Aimard played gymnastically. It is quite remarkable that Mr. Carter has lasted for 100 years; it is doubtful that his music will last 100 more.

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Schoenberg went underground in the Vienna of the 1920s, founding the Society for Private Musical Performances to allow his music and that of his colleagues to breathe “the air of another planet.” A guest at some of these events was Béla Bartók, who, after hearing a rendition of the strict dodecaphonic Lyric Suite of Alban Berg, went home to create the most harmonically revolutionary of his six groundbreaking string quartets. On Saturday at the Metropolitan Museum of Art, a fine gathering of young musicians, under the auspices of their mentor Itzhak Perlman, navigated these extremely treacherous waters.

Considering the age of the musicians, this was a remarkable performance. The LK String Quartet (Sean and Kristen Lee, violins, Laura Seay, viola, and Jordan Han, cello) first got together at the Perlman summer music program on Shelter Island. Classical music has been very good to Mr. Perlman, and part of giving back is this now year-long course of study for highly talented string players between the ages of 12 and 18. New Yorkers have the opportunity to hear many impressive student performances, but this night was exceptional.

In addition to Bartók’s handling of melodic progression, the third quartet is noted for its bag of technical tricks that emphasize the percussive. The group was able to play using sur ponticello (bowing near the bridge), col legno (slapping the strings with the bow), and what has come to be known as the Bartók pizzicato (plucking the string so hard that the sound of the wood vibrating is part of the overall sonority) without allowing these effects to dominate the flow of this progressive music (big-band era fans who remember Gene Krupa’s version of “Big Noise From Winnetka” will have these sounds in their memory’s ear).

Mr. Perlman joined as first violinist for the other two pieces on the program. Rather zippy with his new motorized scooter, he played with that unique buttery sound but, unlike many other veterans who perform with rookies, never dominated or stole the show. With Michelle Ross, second violin, Megan Griffin and Jessica Oudin, violas, and Jia Kim, cello, he intoned the lovely Quintet in B Flat major, Op. 87, of Felix Mendelssohn. Of particular note here was the blending of the group, which, after all, does not play as a unit on a regular basis. This may have simply been a function of where I was sitting, but the dyad sound of the two violas was quite thrilling. Did I mention that the age limit for this band is 18?

The Mendelssohn is a bit of a rarity and the Bartók the stuff of specialists, but it was very easy to rank these performances after hearing a superb Piano Quintet in F minor of Brahms, a piece played often by professionals, but not that often better than this current version. Mr. Perlman and Ms. Oudin were joined by Sharon Roffman, second violin, Yves Dharamraj, cello, and Orion Weiss, piano, who, not being a string player, was accepted for the evening even though he is already at the ridiculously advanced age of 25.

Again, the performance was top-notch, Mr. Weiss leading forcefully but not overwhelmingly. In fact, the separation of the piano from the string line was extremely effective where often in a rendition by more experienced musicians, the net result can be a little soupy, tensions bumping heads in an overwrought manner. Of course, hearing Mr. Perlman play is always a treat, but listening to him with these attentive aspirants was pure joy.

One side note: Last week, I wrote that Mr. Weiss was playing at the Chamber Music Society of Lincoln Center and world-class pianists Menahem Pressler and Wu Han were in the audience. At the museum it was Emanuel Ax. These young people attract sophisticated listeners, and many of them — the stage was also filled with seated patrons — are especially pleased to catch a glimpse of the future of classical music, which, at least on this night, looked hale and hearty.