“The first time I saw it, I was so overwhelmed,” Miranda says of “Satisfied,” the song Angelica Schuyler, who loves Alexander Hamilton from afar, sings at a powerful moment in “Hamilton.” “I mean, it’s still the number that every time I see it, regardless of where I am in my life or which company I’m seeing in the world, I am completely overwhelmed. It is so much bigger than all of us.”

The observation is an especially moving one, coming from the man at the vortex of a team of cyclonic talents that has reminded the world that musical theater is unarguably a great American art form. On a gray day in October, his thoughts on “Satisfied” are being recorded as he sits in a conference room of a Bronx movie studio with those three other artists: Thomas Kail, Alex Lacamoire and Andy Blankenbuehler. Together, they are being recognized by the Kennedy Center as the creators of “Hamilton,” the first work in any performing arts discipline to be singled out as an honoree in the awards’ 40-year history.

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The four men, ranging in age from 38 (Miranda) to 48 (Blankenbuehler), have forged one of the most significant creative alliances in the contemporary world of the stage. With Miranda as composer and star, Kail as director, Blankenbuehler as choreographer and Lacamoire as music director, they’re collectively two-for-two in building Tony Award-winning musicals for Broadway, the first being 2008’s “In the Heights.” And the second: a musical of such international influence that it’s playing simultaneously in New York, London and through three American touring companies; has been showered with awards; and reached so deeply into global culture that it’s sung everywhere from block parties in Brooklyn to dance academies in Beijing.

On this afternoon, an effort to gather the men in one place for a conversation about what this award memorializes — the art of collaboration — has resulted in a sit-down at Silvercup Studios North, where they are participating in another project they all have a hand in: a limited series for FX about director-choreographer Bob Fosse and dancer Gwen Verdon. When it was suggested that the interview focus on one galvanizing interlude of “Hamilton,” and how each played their part in making it happen, the reaction was immediate and electric. It was an opportunity for them to reflect on the profound psychic-income aspect of their group achievement and the ineffable bond that feeds artistic success.

“This speaks to a sense of trust that I think is evident, as you watch all of us kind of lean forward in our seats, getting a chance to talk about this one particular thing,” Kail, 40, says. “The difference between a show that might have an opportunity to be its full expression — when ideas are allowed to flow and be identified — and when they’re squelched, because it’s not your job or you shouldn’t be saying this or you don’t feel the comfort of being able to say it.”

“I’ve worked on other shows where you propose a change because the moment isn’t working, and all of a sudden, you feel the grins tightening, you just see the arms lock. With this group, that has never happened,” adds Lacamoire, 43. “There’s always been a thing about, ‘You know, this isn’t quite landing,’ and then we all think about, how can we make it better?”

Trust and faith in the team

“Piece by piece, only way to make a work of art,” say the lyrics of that Sondheim song, “Putting it Together,” from “Sunday in the Park With George,” that seem so appropriate for the process by which this quartet of creators merged skills and sensibilities to make“Hamilton.” And to achieve the complex assemblage of rhyme, musical style, narrative playfulness, dance and emotional effect that conjoin in a number like “Satisfied.”

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The vignette-filled “Satisfied,” which comes smack-dab in the middle of Act 1, proves to be a wonderful springboard for discussion, because it embodies so many of the musical’s irresistible attributes: its restless, energetic resourcefulness; its ability to paint a historical mural and apply a modern varnish of commentary at the same time; its perspective shifts, its wit, its rigor. It’s no wonder the song drew on and conjured for Miranda and company all manner of cultural references, including “West Side Story,” “A Chorus Line,” “The Matrix” and “Ratatouille.”

It’s a song that was the breakthrough indicator of how much story their deeply researched musical, based largely on Ron Chernow’s best-selling biography, could pack into a conventional two-act structure. Because before “Satisfied,” Angelica Schuyler did not exist in the show. “The question of whether Eliza’s sister would be a character was up for debate,” Miranda recalls. “I mean, she is a confidante of her sister, she had these letters with Hamilton, and it’s, ‘Do I have time to get into that?’ ”

Devising “Satisfied” for Angelica, a role originated in 2015 at off-Broadway’s Public Theater and on Broadway by Renee Elise Goldsberry, who would win a Tony for it, proved crucial to developing the emotional core of “Hamilton.” It took Miranda about a month to write it, and it sets in motion the show’s tragic element, how passion unfulfilled — in this case, Angelica’s for Alexander — eventually tears apart Hamilton and those around him. The song, which includes ingeniously rhymed rap to dramatize the dizzying sophistication of Angelica’s own intellect, begins as Angelica’s toast to the marriage of Hamilton to her sister, Eliza; Angelica has introduced her to Hamilton during the previous song, Eliza’s “Helpless.”

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“I liked the idea of a wedding toast,” Miranda says. “I’ve been to enough wedding toasts where the wrong things tumble out.”

“To your union,” Angelica sings in the five-minute-plus number’s opening segment, “And the hope that you provide/May you always/Be satisfied.” What then follows is what the song identifies as a “rewind”: going back to the events of “Helpless,” but told now from Angelica’s anguished perspective, in a way that crystallizes a pivotal facet of her character. “I remember that night, I might regret that night for the rest of my days,” she sings, in the song’s defining line.

In returning to that moment, Kail says, “we realized that there was an opportunity for Lin to play with the timeline, and the way that we moved through time.” That concept would repeat itself at another climactic moment of “Hamilton,” in the freeze-frame rendering of the bullet that fatally strikes Hamilton in his duel with Aaron Burr. It was not an original idea, actually: The creative team was borrowing a cinematic technique, one of many they use in the show.

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Kail says: “This is something I talk to the actors playing Angelica a lot — about ‘Ratatouille’ ” — the 2007 animated movie about the rat that becomes a Parisian chef.

“Totally ‘Ratatouille’ ” Blankenbuehler interjects. “We use it as a verb and an adjective.”

Kail explains that the freezing of time in “Hamilton” has its parallel in the moment in the film when the food critic, voiced by the late Peter O’Toole, has an epiphany as he savors a piece of food.

“When the critic takes the bite, and you go into the critic’s eye, that’s what we’re doing,” the director says. In other words, the instant in which the cartoon critic samples the food stops time; viewers are given a protracted, imagistic impression of what is happening in the critic’s mind. That same stopping of time occurs in “Satisfied.”

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“I love that, sonically, it takes us someplace we haven’t been to before,” adds Lacamoire, the music director and orchestrator. For Blankenbuehler, the choreographer, “Satisfied” was a feast of new possibility, too: “I think the first time I heard the song was at a reading,” he says. “And I just remember the right hand on the piano, and the tinkles, and I instantly saw women suspended, like on top of a cake, like on pointe, like how things rotate on a wedding cake.”

But perhaps the most complicated choreographic element of “Satisfied” is what happens in the interlude in which prerecorded voices take us into the “rewind” portion of the song. Because the dancing ensemble, assembled for the wedding, physically rewinds, too, to the movement of “Helpless.”

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“All of ‘Helpless’ goes counterclockwise,” Blankenbuehler explains. “So when you rewind in ‘Satisfied,’ and for just a moment you go clockwise, you understand it. When they dance in ‘Helpless’ and ‘Satisfied,’ the same dance movement matches both lyrics.”

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Lacamoire says: “The first time we saw what it looked like, with the lights and the turntable, our jaws literally dropped.”

Asked how they could yield to one another in developing a song with so many working parts, Kail replies: “A tremendous amount of trust. The other research and development that happens over 10 or 12 or 15 years of working with someone, is faith. One of the real benefits of talking to anybody in this group is we can talk about emotion, and it can be translated and distilled into action.”

“We’ve been in situations together,” Blankenbuehler adds, “where we have failed, as well as succeeded, and I think all of us have a good memory for both of those things. So that we get in the next situation, we just fall into the pattern of what it felt like when it went right. Or we remember the roadblocks of when it has gone wrong. Because we’ve done both together.”

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It feels as though these guys could talk about this one song all day. But they all have other places to be. So maybe Miranda captures the essence of collaboration best as he listens to his longtime colleagues talk about their approaches to “Satisfied,” and then says with a laugh: