Gone are the days when immigrants such as Izzy Baline (Irving Berlin) and sons of immigrants such as George Gershwin felt the need to yank up their roots and create musical images of an America completely divorced from their own backgrounds. Lin-Manuel Miranda, a 27-year-old who grew up in a Puerto Rican household in northern Manhattan and clearly kept his eyes and ears open, has turned his own experiences into the overstuffed, extremely likable “In the Heights.”

Mr. Miranda’s story may be as familiar as his setting is specific, and he and book writer Quiara Alegría Hughes overreach at a few key moments. But they and director Thomas Kail have woven the polyglot sounds of Washington Heights (which also has a substantial Dominican population) into a vibrant, tuneful celebration of community. It may not be the tightest or most complicated musical in town, but its boisterous tug is all but irresistible.

Our narrator, a hapless bodega owner named Usnave de la Vega (Mr. Miranda, who also wrote the music and lyrics), is “stuck to the corner like a streetlight,” a status that he both relishes and bemoans. His childhood pal Nina Rosario (Mandy Gonzalez) made it all the way to Stanford University from West 181st Street, but the tuition bills are proving burdensome to her parents, Kevin and Camila (John Herrera and Priscilla Lopez), who run a struggling neighborhood livery car stand. The ambitious but poor Benny (Christopher Jackson) carries a torch for Nina, while Usnave does the same for the sweet but sultry Vanessa (Karen Olivo). Meanwhile, Usnave’s saintly abuela casts her gauzy memories back to life in Havana, and a $96,000 lottery ticket promises to make several lives a bit easier.

The show begins with the sound of a radio flipping past numerous stations, and Mr. Miranda’s score is equally mercurial, a virtual primer of modern-day Latin pop. Salsa, merengue, mambo, and even reggaeton coexist comfortably alongside hip-hop and, in songs such as the winning duet “When You’re Home,” more traditional Broadway pop balladry. His lyrics land more confidently in the context of Usnave’s helter-skelter freestyle raps than they do in the more traditional songs, but even when the occasional rhyme wobbles, he brings a welcome strain of unpredictability to his craft. And Andy Blankenbuehler’s elaborate, loose-hipped choreography lends a bit of oomph to several of these tunes, particularly a delectable nightclub sequence in which a sizzling Vanessa and a stumbling Usnave each try to make the other jealous.

Unpredictability is in short supply in Ms. Hughes’s book: Nina’s future at Stanford is never really in question, and audience members at a recent show yelled out who won the lottery ticket well before the show got around to disclosing the winner. And once, just once, I’d like to have a character cough in the first act and still be alive for the curtain call. Still, Ms. Hughes and Mr. Miranda — to say nothing of set designer Anna Louizos, with her spot-on street scene overshadowed by the George Washington bridge — do show a knack for specificity when it comes to the titular neighborhood. Last year’s collapse of a retaining wall onto the West Side Highway warrants a mention, as does the sudden absence of the 9 train. (Nina, who’s been away for a year, still calls the Seventh Avenue local the 1/9.)

While the most topical material typically spills from the excitable Usnave, Mr. Miranda has created for himself a curiously extraneous role. Usnave guides the audience through a community so tight-knit that few introductions are necessary. Neither the romantic lead nor the comic relief — Mr. Jackson and Robin De Jesús (as a goofy younger cousin) fill those roles, and quite capably — he lands somewhere in the middle instead.

And yet it’s impossible to imagine “In the Heights” without Usnave’s goofy/cute demeanor and delightful mostly-spoken-word patter (as when he tries to charm a woman at the bar with “an amaretto sour for this ghetto flower”). In fact, Mr. Manuel’s graciousness is the show’s biggest hindrance. He continually deflects attention away from Usnave and toward a half dozen other characters, none of whom has his lyric bite or offkilter charm. A fairly maudlin plot development near the end nudges him into a central role, but a sharper Act I structure might have made this shift feel less obligatory.

Unfortunately, the polish and size of the production — with 20 actors, it’s bigger than several current Broadway musicals — decreases the odds of Mr. Kail giving it the ruthless honing and streamlining it still needs. The guy selling piraguas (shaved ice cones) doesn’t need his own song any more than the floozies at the nail salon need two songs, and despite receiving ample stage time, Nina’s parents remain caricatures. “In the Heights” still doesn’t really know whose story it wants to tell and why. But there’s heat and heart and hope in its rambunctious soul. As Usnave admits near the end,

Yeah, I’m a streetlight!
Chillin’ in the heat!
I illuminate the stories of the people in the street.

Open run (450 W. 37th St., between Ninth and Tenth avenues, 212-307-4100).