Even if you don’t know “Für Elise,” you know “Für Elise.”

A bagatelle the length of a pop song, Beethoven’s trifle is recognizable from the start: a wobble between E and D sharp that gives way to a tune you’ve heard virtually everywhere. Ringing from cellphones and children’s toys; sampled in rap and featured on Baby Einstein albums; as likely to appear in a serious drama as in a Peanuts cartoon, “Für Elise” is shorthand for classical music itself. In “Bill & Ted’s Excellent Adventure,” it’s used to identify Beethoven without even saying his name.

But you probably haven’t heard “Für Elise” in a concert hall. More likely to inspire eye rolls than awe among the cognoscenti, it’s rarely programmed — unlike, say, Beethoven’s Fifth Symphony, with its famous dun-dun-dun-DUN fate motif, or his Ninth, which ends with the omnipresent “Ode to Joy.”

I’ve been thinking about the puzzling absence of “Für Elise” from professional recitals since I first met the pianist Igor Levit for a concert and interview we conducted over Facebook Live in 2017. He offered the piece as a surprise at the end of the broadcast, withholding the title but saying, “I will play one of the most beautiful pieces I know.”

Hearing the opening bars, I was caught so off guard I nearly laughed. “Für Elise” occasionally pops up in mainstream recordings; Paul Lewis released an aching account on an album of Beethoven bagatelles last summer. But it’s so rarely heard live — outside student concerts, at least — that for a moment I didn’t know how to respond.

Nearly four years later, and using the 250th anniversary of Beethoven’s birth a few weeks ago as an excuse, I asked Mr. Levit whether he could explain the beauty of “Für Elise” in more detail, and make a case for why it warrants deep attention rather than reflexive exasperation.

“It’s not a piece you actually hear,” he said in a video call from his home in Berlin. “It became in a way unperformable, which I think is a shame.”

Mr. Levit added that when he plays it as an encore, people tend to giggle or look visibly confused. Serious musicians aren’t expected to build their careers on this piece, and audiences don’t rush to concert halls for it.

The ubiquity of “Für Elise” — like Mozart’s “Eine Kleine Nachtmusik” — doesn’t void its masterly craft, nor does it preclude the possibility of performances on the level of Mr. Levit’s. Yet the eye rolls continue. In his biography “Beethoven: A Life,” which was recently translated into English, Jan Caeyers writes that the work “has assumed a significance in Beethoven’s oeuvre that is utterly disproportionate to its musical import.”

That may be true, but it’s a severe judgment nevertheless. For the outsize reputation, we can thank the catchy title, an abbreviation of the dedication: “For Elise on 27 April as a remembrance of L. v. Bthvn.” If the piece had come down in history merely as Bagatelle in A minor (WoO 59, from the “Werke ohne Opuszahl” catalog of Beethoven works without official opus numbers), it likely would have remained a lovely obscurity.

Beethoven drafted and dedicated it in 1810, though it remained unpublished in his lifetime. He is thought to have revisited it in the early 1820s, most likely with an eye toward including it in his Op. 119 Bagatelles, but he ultimately left it out. The scholar Ludwig Nohl eventually discovered and published it in the mid-1860s, igniting a debate over the identity of “Elise” that continues to this day.

Becoming a fixture of music lessons, spreading with the rise of mass media, finding new audiences as the line between high and low culture blurred: All led to the ultra-ubiquity of “Für Elise.” By the time I was a toddler, in the early 1990s, all I had to do was push a piano-shaped button on a toy to hear the opening theme. It was so entrenched in my memory that I could play it, crudely, before I could read a note of music.

Mr. Levit recalled similar experiences; he too learned “Für Elise” by ear. Then he became fascinated by, for example, a fleeting dissonance or a passage of enveloping tenderness. “This piece is an absolute jewel,” he said.

I asked him to expand on that, using his copy of the score from G. Henle Verlag. Mr. Levit has remained busy during the pandemic: He streamed a long series of daily concerts from his apartment, put on a marathon performance of Erik Satie’s “Vexations” and appeared around Europe. But like everyone, he has also been unusually homebound, lately baking challah and playing guitar. So he had time to dive deeply into the three pages of “Für Elise.” (All audio clips are excerpted from Mr. Levit’s Sony recording.)

Opening with uncertainty

“Für Elise” is in A minor, but it doesn’t declare its key right away. The first five notes remind Mr. Levit of a later piece, Schumann’s song cycle “Dichterliebe,” which begins dissonantly with a C sharp quickly followed by a D two octaves lower.

In the Beethoven, the notes are an E and a D sharp, a half-step lower. Toggling between them, with an improvisatory feel and the extreme softness of pianissimo, creates a sense of mystery. For a moment, “Für Elise” could go anywhere.

Once upon a time

A more solid sense of the piece’s direction comes once the left hand enters, trading notes with the right hand in upward arpeggios. It has the lure of a fairy tale, Mr. Levit said — or at least that’s how it sounded to him when he once found himself “fooling around” and doubling the tempo of these measures, rendering them flowing and dreamlike.

“You have this almost nondirectional beginning,” he said, “but then this feeling of ‘A long, long time ago. …’”

A musical hug

After the opening repeats, the piece continues with phrases that gently rise and fall, like breathing. Mr. Levit also sees them as a musical hug: “When it goes up you open the arms, and when it goes down you close them.”

The chord progression here, he added, is practically guaranteed to make you melt. “It’s very beautiful,” Mr. Levit said, “but in the simplest way.” It’s the stuff of the Beatles and Elton John — and reminiscent of Pachelbel, whose Baroque-era Canon in D also echoes through pop music today, one of the few challengers to “Für Elise” among overplayed chestnuts.

A glimpse of late style

The opening theme returns by way of a transition of shocking economy: the note E, played repeatedly but given the illusion of variety by jumping octaves. It’s a flash of late Beethoven, his music at its most elemental. And it’s the kind of moment that appears in subsequent piano repertoire: Mr. Levit pointed to the opening of Liszt’s “La Campanella” and the Marc-André Hamelin étude Liszt inspired.

One of Beethoven’s feats here, Mr. Levit added, is how simplicity is made theatrical by passing those E’s back and forth between the left and right hands. “It’s just emptiness,” he said. “How great must a composer be to allow himself to write about nothing?”

Melody, at last

Mr. Levit argues there is no true melody in “Für Elise” until about a minute into the piece. The opening, he said, is not something that could be easily mimicked by the human voice; it’s more about Beethoven creating space. Then comes a more traditionally constructed passage, with a lyrical right-hand line above left-hand accompaniment.

“I don’t think the beginning is espressivo,” he said. “So when the F major comes in, this allows you to really sing it out. It’s in a way easier to play.”

Easier, that is, until an étude-like dash of notes — perhaps the most difficult four measures of the score — leading abruptly back into the opening theme. The transition, or lack thereof, is characteristic of Beethoven; Mr. Levit described it as “a car crash moment.”

A dramatic interlude

After revisiting the opening theme, Beethoven suddenly changes the temperature of the piece with a tempestuous interlude of right-hand chords over a rumbling floor of repeated low notes. Mr. Levit often uses the word “tender” to describe “Für Elise,” but not here.

“It’s quite dramatic,” he said. “And it’s automatically loud because if you use the pedal, just because of the way the piano is built, it gets louder. It’s intense.”

The wind machine

But the drama comes to a quick end with another “car crash” transition: two measures of barely held chords, then a run of triplet 16th notes rising and falling over a span of more than three octaves. It can be easy to read this as a climax — either to the stormy middle section, or the piece as a whole — but Beethoven marks these notes as pianissimo, exactly as soft as the opening. “It’s ghostlike,” Mr. Levit said, “a pianissimo wind machine.”

Closing the book

The opening theme returns one last time, quietly, with no changes in tempo or dynamics that would have given it the grandeur of an ending. The only addition is a single note — a low A — in the brief final chord. If “Für Elise” is a fairy tale, this is its tidy conclusion.

“It’s very touching,” Mr. Levit said. “This is what happened, that’s how it was. The story was told, and now the end. The book is closed.”