It was a rainy afternoon in Copenhagen, and four exhausted ballet dancers were hard at work in a large rehearsal room on the fourth floor of the Royal Danish Theatre. The choreographer Alexei Ratmansky was pacing the studio, intensely focused. An assistant stood beside him holding a score of TheArt of the Fugue, Johann Sebastian Bach’s great unfinished work.

The dancers had been at it for hours. They looked sallow, too tired even to sweat, but Ratmansky was not yet satisfied with what he saw. He asked the rehearsal pianist to play a passage from the score, took a moment to think, then blurted out two phrases of choreography for the dancers. He showed the steps, and they repeated them back. Then he showed them again, illuminating this or that nuance, and they did it again. Everyone in the room was trying things out, fixing, sculpting, connecting the movements until a phrase emerged, clear and new. It was a week before the premiere, and the ballet was still very much a work in progress.

Ratmansky, one of the world’s most admired ballet choreographers, was born in 1968 in St. Petersburg and raised in Kyiv. His mother is Russian, and his father is Ukrainian. He studied ballet in Moscow and served as director of the city’s Bolshoi Ballet for five years in the early 2000s, after dancing in Ukraine, Canada, and Denmark. Today he is an artist in residence at both New York City Ballet (he has lived primarily in New York since 2009) and the Dutch National Ballet in Amsterdam, and he continues to create new ballets, and remount older works, all over the world. Two days before this rehearsal, he had been in Vienna for the Austrian premiere of his ballet Callirhoe, made in 2020 for New York’s American Ballet Theatre. The day after the premiere at the Royal Danish Theatre, he flew to Miami to create yet another.

But the ballet he was making in Copenhagen stood apart. For one, Ratmansky was taking on a daunting, cerebral piece of music, left unfinished when Bach died of a stroke in July 1750. In The Art of the Fugue,Bach was attempting to explore every permutation, variation, multiplication, and expansion of the fugue, a composition in which a musical “subject,” or theme, is presented, then imitated and elaborated on by different “voices” at varying intervals. The final fugue, into which Bach planned to weave four different melodic combinations, including one made up of the notes that spell out his name in German musical notation, cuts off after eleven minutes, mid-page. This abrupt end is the subject of much conjecture. Either Bach died while composing—a romantic notion put forward by his sons but more or less debunked by scholars—or the final page went missing. Whatever the true story, the piece has an aura of mystery and sadness.

Second, and perhaps more emotionally significant, was the fact that in Copenhagen Ratmansky was returning to a project that had been painfully interrupted when Russia invaded Ukraine. In early 2022 he had begun to develop his interpretation of The Art of the Fugue at the Bolshoi. Then, on February 24, Russian missiles rained down on Ukrainian cities. Ratmansky had known that Russia was amassing troops on Ukraine’s border—his wife, Tatiana, who is Ukrainian, had begged him not to go to Moscow. But Russian president Vladimir Putin had denied that he was planning to invade, and even Ratmansky’s parents and sister, who live in Kyiv, thought it only a remote possibility. When the tanks entered Ukraine, Ratmansky gathered his artistic team and left for New York, severing ties with the Bolshoi and with Russia. He has since been a vigorous defender of Ukraine and a vocal critic of Russia, comparing it in an interview with Nazi Germany. And he has said he now feels remorse for having occasionally continued to work there after the annexation of Crimea in 2014. “I was thinking art is outside of politics, or could be,” he told The Guardian. “But the invasion opened my eyes.”

The Ratmansky ballets performed in Russia after the invasion were shown without their choreographer credited. The companies he had worked for removed his name from their publicity materials. He put his unfinished ballet away in a remote corner of his mind. “This door was closed,” he told me. But the abandoned project nagged at him, and in 2024, when Nikolaj Hübbe, then director of the Royal Danish Ballet, asked him whether he would like to create a full evening-length work, he leaped at the chance.

There was a certain rightness, he felt, to finally making the ballet in Denmark, a country that has been particularly supportive of Ukraine’s cause. The president of his adopted country, Donald Trump, has singled out Denmark, one of NATO’s staunchest and most loyal members, for derision as he sets his sights on Greenland, an autonomous territory within the Danish kingdom. Ratmansky also has a strong personal connection with the country: from 1997 to 2003, before his stint at the Bolshoi, he and Tatiana were dancers at the Royal Danish Ballet. In that period he made some of his first large works and immersed himself in the choreography of August Bournonville, a nineteenth-century Danish master whose affable technique found its way into Ratmansky’s own. The ballet that made Ratmansky famous, The Bright Stream (2003), is full of ideas from Bournonville. And the choreography of The Art of the Fugue, too, includes homages, like a sideways step performed as if the dancer were gliding over a rainbow, with a lilt and roundedness typical of nineteenth-century Danish style.

It was in Denmark, too, that Ratmansky’s son, Vasily, now twenty-seven, was born. Ratmansky was surprised at the emphasis that Vasily’s Danish schools placed on personal autonomy for children—the opposite of his Soviet schooling. He sees this focus on mutual respect and dignity reflected in Denmark’s support for Ukraine. “It takes me two minutes to walk from the hotel to the theater,” he told the dancers. “And in those two minutes I can see two Ukrainian flags.” The theater, located around the corner from the colorful façades of Copenhagen’s seventeenth-century Nyhavn harbor, expresses these ideals. Inscribed above the stage in gold letters is the Danish Golden Age dictum “EI BLOT TIL LYST”—“Not just for pleasure.” “In the hard times,” Ratmansky told me, “art means quite a lot.”

This is not to say that Ratmansky’s Art of the Fugue is explicitly about the war or about Ukraine. In fact there was never meant to be a story behind it. “This is the most abstract ballet I’ve ever made,” he said before the premiere. He divided the ballet into five sections, following Bach’s composition: seven simple fugues (fugues that develop a single subject), four double and four triple fugues (fugues that contain two or three subjects), two mirror fugues (in which one musical line goes up when the other goes down, and vice versa), four canons (purer constructions in which a melody is introduced and then imitated exactly by another voice), and the final, grand, incomplete fugue. Each component of Bach’s hour-and-a-half-long suite has a different character, and Ratmansky’s choreography reflected this. Just as Bach used all the notes of the D minor key to create his fugues, Ratmansky worked with basic steps from the ballet canon but combined them in hundreds of ways to create seemingly infinite juxtapositions. Also as in Bach’s fugues, Ratmansky created resonances by echoing earlier sections of his own dance.

Because there was no obvious overriding idea, it was sometimes hard for the dancers to find their bearings during rehearsals. Danish ballet has a strong tradition of storytelling, and the dancers there are accustomed to depicting characters. Ratmansky kept insisting that the secret instead lay in listening closely to the music and finding a way to embody it, rather than simply following its rhythmic structure. “It’s important that the music come from your movement,” he exhorted them after one particularly difficult session. “The body, mind, and spine express the music.” The dancers nodded wearily. Over time, though, his tiny adjustments, many of which were about timing and reflecting the highs and lows of a phrase, had a striking combined effect. What at first looked flat and blurry became complicated, with an accent here, an unexpected pause there. “Guys,” he told a group of male dancers rehearsing an energetic section marked by a strong rhythm, “in the running there is strength and athleticism, but it’s as if it were underwater, like currents moving.” They tried again, and the motions became heavier. “The movement initiates in the lower back.” They did it again, and now the motions had more drive. “Every movement has to start with an impulse, and we need to see this impulse.” Again, and now the motions had intention.

The week before the premiere, as he was still coming up with steps to fill remaining gaps in the ballet, the choreography met the stage design and lighting. The set, by the German designer Moritz Junge—this is his fifth collaboration with Ratmansky—was completely different from what had been imagined for the Moscow production. Those designs, by George Tsypin, were architectural and ornate, like an alternate Baroque world, but that kind of grandeur, so typical of the Bolshoi, no longer made sense. Instead Ratmansky asked Junge to design a sober, abstract set hinting at both cosmology and spiritual reflection. Junge produced a world of triangles, circles, squares, dark passageways, and reflected surfaces. Monumental black walls slid across the stage to form different spaces: a square room with many doors; a nave-like space with an elevated platform that served as both an altar and a harpsichord nook; a room with a large ovoid mirror in the back, as in a ballet studio; and an open space with two interlocking spheres that dangled above the stage, one of which revolved slowly, giving it the appearance of a waxing and waning moon. The lighting designer Brandon Stirling Baker, who often works with the choreographer Justin Peck in New York, created a series of contrasting atmospheres: a golden glow at the beginning of the ballet, shadowy recesses in the church-like harpsichord section, and a cool lunar landscape for the ending. The dancers often had an almost angelic aura.

Bach did not specify instrumentation for The Art of the Fugue, though many believe it was meant for either harpsichord or organ. Ratmansky assigned a different instrument or combination of instruments for each section. For the first he chose a pensive piano, positioned at the back of the stage. For the second he used a joyful brass quintet, playing in the pit. The two mirror fugues were assigned to a grave, courtly harpsichord, and the four canons were given to a melancholy string trio, with only two instruments playing together at any time. (Canons have just two voices.) The final fugue was performed by eight members of the Danish early music ensemble Musica Ficta, who stood on pyramidal platforms placed just behind the dancers.

Ratmansky’s vocal arrangement included a portion from a late Bach chorale: “Grant me a blessèd end, and wake me, Lord, at the Day of Judgment, that I might behold Thee forever more. Amen.” The otherworldly interweaving of eight voices imbued the steps of the dancers below them with a very human longing for transcendence. It felt for a moment as if the fugue could go on, as if the dancers could keep moving across the stage on mysterious pathways forever. Instead, about eleven minutes into the fugue, the chorus suddenly fell silent. The dancers kept moving as the curtain descended. The first time I saw this, during a run-through, tears filled my eyes.

The Art of the Fugue begins quietly, with a line of four women, each with her hand placed amicably on the shoulder of the woman next to her. They stretch one foot forward in a tendu, the most basic step in ballet, to the first note of a D minor triad: D, A, F, D. (It is from this simple triad that Bach’s entire compositionis constructed.) They begin to dance in various combinations: three and one, two and two, each dancer alone. Connections arise and dissolve. One woman supports another, holding her waist as she turns. Another crouches on the ground, as if in thought; another helps her up. Two hover on pointe, side by side, like heavenly bodies passing each other in space. As the fugue comes to an end, the four women walk away from the audience, arm in arm, and open their chests to the sky.

The next section is danced by four men. Their energy is different, more grounded, but they too lift and assist one another. A theme of mutual support begins to emerge. From then on, the human combinations multiply little by little, alternating between solitude and connection, playfulness and meditation, presence and absence. Each section ends with a striking image, a momentary freeze-frame. In one passage, three women dance with one man, pulling and pushing one another until in the final moment they lean over him and lift their legs into a sunburst formation, an immediately recognizable image from George Balanchine’s Apollo. There are nods to Ratmansky’s own work: a movement that looks like the dancer is shaking out linens, lifted from his Callirhoe; slow processionals on pointe, from Songs of Bukovina (2017); a succession of men assisting a woman in an extended balance, as in the famous “Rose Adagio” from Marius Petipa’s Sleeping Beauty, which Ratmansky staged in 2015; torsos that slowly curve to one side, from his Solitude (2024); a pose like a runner preparing for a sprint, from Voices (2020).

The brass quintet, which follows the quiet, introspective opening dances set to piano, ushers in an outburst of youthful exuberance and brashness. The music here is playfully syncopated, the costumes bright. Two men, backed by an ensemble of nine more, fly across the stage, their arms carving through the air. One of them performs a series of turns on one leg, traveling forward on a diagonal path and then back again, a spinning comet zigzagging through space. A lone woman, powerful and high flying, is tossed through the air before joining one of the men in a jagged, syncopated duet. They flick their arms at each other as if tossing a ball.

After the intermission, three dancers, dressed in simple gray and bathed in Baker’s light and shadows, return, accompanied by the sharp, dry sound of the harpsichord. After a solo danced by a woman (in Copenhagen, Holly Dorger) who paws at the floor and executes sprightly jumps in a gargouillade (in which a dancer takes off, draws a little circle with one foot in the air, and then draws another with the other foot), Ratmansky shows us the physical equivalent of a fugue: a man jumps straight into the air and flicks his foot three times, then another man jumps up and flicks his foot. Ratmansky avoided including too many of these literal fugues in the ballet, finding the idea too obvious and not particularly congenial to dance. “It would have been boring,” he said. The dancing isn’t meant to be a visual replica of the music at all but a kind of counterpoint to it.

From this point on the ballet is far more somber. A mournful string trio begins to play, and the black walls move again. The male dancer who led the joyful brass section returns, climbs atop a pyramid that has just appeared at the back of the stage, and watches the goings-on below, as if remembering a scene from his own life. He sees a man and a woman dancing together, surrounded by an ensemble of women in long black chiffon dresses who stream across the stage, gathering in pools, like dark sylphs. The choreography now hints more directly at something darker—loss, perhaps death. The man lifts the woman onto his shoulder, and she peers into the darkness, bearing witness to something only she can see. A three-note phrase played by the viola sounds like weeping. Later two dancers execute a lift in which the woman is held upside down across the man’s body as he turns. She looks like one of Saturn’s rings.

When the eight singers enter for the final fugue, taking their positions on the two pyramids, the man who joyfully flew across the stage early in the ballet emerges again from the wings and crosses to center stage. There he stands perfectly still. The others arrive, in ones and twos, with that lovely rocking sideways step Ratmansky borrowed from Bournonville. They move around him, oblivious to his presence. Is he the man from Bach’s chorale, facing his own mortality? (“He told us to think of a memory from childhood,” Tobias Praetorius, one of the dancers who performed the role in Copenhagen, told me of that moment.) The surrounding dancers repeat gestures and steps we’ve seen before. These flashes are like fragments of a life, or of many lives, passing before our eyes, out of order, all at once. Finally, after several minutes, the man begins to move, reaching and falling, allowing himself to be raised up into the air and dragged across the stage. He seems caught between anguish and acceptance. The dancers stray from one another, becoming disconnected, alone, lost in their own orbits. First one voice falls silent, and then the others. The dancers continue in the vacuum until the curtain falls.

For all of Ratmansky’s protestations during rehearsals, the ballet did turn out to have a kind of story, a quietly devastating one. The dance follows an arc: from a cosmos of steps danced by an almost impersonal cast to a single person, a single trajectory, a single life, surrounded by many other individual lives. This ending seemed to surprise even Ratmansky. He had resisted creating a protagonist or, as he put it, “a lonely hero figure.” But the ballet needed an anchor, and now it had one.

After the invasion of Ukraine, Ratmansky is a changed artist. One of his greatest attributes as a choreographer had once been lightness; many of his ballets are leavened with humor, even absurdity. (His most recent, which opened at New York City Ballet on February 5, is inspired by Hans Christian Andersen’s “The Emperor’s New Clothes.”) He hasn’t lost that quality, thankfully, but the war has deepened and clarified his morality. He speaks often of the double consciousness he feels now, with the war always on his mind even as he goes on working. In 2024 he made a ballet, Solitude, inspired by a photograph of a Ukrainian father holding the hand of his dead son, killed in a Russian attack. For ten minutes the father kneels on the stage as the dance continues around him. I wonder whether there is a thread that runs from that man to the man in The Art of the Fugue. Or perhaps that man is Bach, or a stand-in for Ratmansky, a human being trying to make sense of the world.

Bach’s music, which Ratmansky first heard as a child, has taken him to a place he has not been to before. Having stumbled through the meandering corridors of these fugues, he has found his way to a peaceable kingdom in which each person, each action, each note has its place. At a moment when the world is in chaos, he has built a parallel universe in Denmark, ruled by freedom and sanity and sustained by the beauty and order of Bach.