Celebrity biographical dramas are ten a penny but it takes audacity for a performer to emulate the famous person in question. What elevates Miles., a tribute to jazz legend Miles Davis, is the role of musician Jay Phelps. Not only does he give a credible imitation of Davis’s spare trumpet style, he also plays along convincingly to backing tapes of Kind of Blue, regarded by many as the definitive jazz album.

Phelps is more than an incidental player. A constant presence in a production written and directed by Oliver Kaderbhai, Phelps plays a Davis acolyte trying to learn from the master, while the pressure of a record company advance looms over him.

What was the secret ingredient, he wants to know, that turned Kind of Blue into a bestselling jazz album? How much did it depend on the collaborators, including John Coltrane and Bill Evans? Where did it fit into the musician’s history of drug abuse and womanising? Now aged 32, the same as Davis at the recording in 1959, could he ever hope to achieve as much?

Answering some of these questions – and evading others – is Benjamin Akintuyosi in the title role. With a raspy post-op voice gurgling up from deep in his throat, he plays Davis as sharp, forthright, hard to impress but passionate in his enthusiasms. His is a tale of musical obsession offset by a lack of money; creative innovation offset by racial prejudice.

With his influences stretching to Stravinsky, Prokofiev and Debussy, as well as Dizzy Gillespie, Charlie Parker and the street rhythms of Afro-Irish tap dance, Davis’s musicality is more deeply felt than his Juilliard education might suggest. Kind of Blue, he says in the play, is “my pain on a 78”, an experiment he thought had failed.

With projections by Colin J Smith adding to the period detail, the show is a fact-packed, reverent and loving testament to the complicated man behind a musical benchmark.

• At Summerhall, Edinburgh, until 25 August

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