Composing a work of music might seem at first glance to be a deeply untheatrical act. Whereas the lives of politicians or monarchs are often characterised by high drama, where is the action to be found in a working life that involves spending months or years in solitary mental absorption at one’s piano? Yet composers are figures of endless fascination to playwrights, perceived as passionate, colourful personalities, even in an age when classical music has fallen out of the cultural mainstream.

Often the dramatic interest lies in exploring some form of tension. Sometimes this is between the beautiful music and the flawed individual who created it: Peter Shaffer’s Mozart (Amadeus) is a boorish, scatological child; Ronald Harwood’s Strauss (Collaboration) a weak man vacillating about whether to keep in with the Nazis. Or the tension might exist between composers and their contemporaries or patrons, the subject of such recent plays as Mark Ravenhill’s Ben and Imo (about Britten and Imogen Holst) and Oliver Cotton’s The Score (about JS Bach and Frederick the Great).

Amadeus liberally exaggerated the antipathy between Mozart and Salieri, making a sensational conspiracy theory out of a professional association that was almost certainly cordial. But a new play, currently at the Park Theatre in North London, has no need to resort to artistic licence in depicting the fraught relationship between two composers whose rivalry was very real. James Inverne’s That Bastard, Puccini! addresses the race between Giacomo Puccini and Ruggero Leoncavallo to launch their respective versions of La bohème – a dispute that played out in a bitter war of words in national newspapers.

It is a fascinating story. The two musicians were members of the so-called giovane scuola, a generation of composers who came of age in the 1880s and were pitted against each other as contenders to become the successor to Verdi and the voice of a nation. Mascagni and Leoncavallo broke away from the pack early on with Cavalleria rusticana and Pagliacci, which in due course became inseparable stage partners, but neither composer would ever match his early success. Other contemporaries – Alfredo Catalani, Francesco Cilea and Umberto Giordano – were essentially one-hit wonders, and all looked with resentment at the soaring success (with audiences, if not always with critics) of Puccini, who had the backing of the immensely powerful Ricordi publishing house.

That Bastard, Puccini! immerses us in the energetic world of 1890s Milan, a city characterised by, as Inverne puts it, ‘the white heat of inspiration’, in which opera was big business, publishing houses were at loggerheads, and critics had the power to make or break a career. Inverne’s Leoncavallo (Alasdair Buchan) is an insecure yet amiable figure, constantly seeking reassurance from his kindly wife Berthe (Lisa-Anne Wood), who has given up her singing career for a life of what the playwright calls ‘adventure and much hardship’ by her husband’s side. By contrast Puccini (Sebastien Torkia – visually the composer’s double) is cocksure, suave and prone to sticking the knife into former friends as he shins up the ladder to worldwide artistic domination. As the play correctly observes, Puccini was a composer who would spar many times with contemporaries over the rights to various dramas, almost needing competition as a creative spur. And when it came to the two Bohèmes, the snippets of music we hear throughout the play do rather confirm that it was Puccini’s that had that special something, however much we might warm to Leoncavallo.

Inverne has packed a huge amount of authentic historical detail into That Bastard, Puccini!, but wears his learning lightly, creating a script that is witty, fresh, contemporary, and cleverly self-referential. Though the play is a three-hander, both Wood and Torkia are called upon to take on the guise of additional characters in flashbacks. Torkia plays Puccini playing Mahler: an Italian putting on an exaggeratedly strangulated Austrian accent. Donning a fox-fur stole, the multi-talented Wood transforms instantaneously into haughty, intimidating Elvira (Puccini’s mistress, later his wife), and also does amusing turns as Puccini’s publisher Giulio Ricordi (imagined here as a Mafia boss) and Enrico Caruso. Elsewhere, as Berthe, she performs her husband’s operatic arias as he tries them out at the piano. Casting this role well, Inverne explains, was both essential and challenging: Wood, who has performed in Phantom, Wicked and Les Mis, is a rare example of a singer of musicals with the sort of classically trained voice that used to be the norm in the West End 80 years ago.

Musical theatre, of course, dominates today’s theatrical landscape to an ever-greater extent, and getting a traditional play in the manner of Michael Frayn, Alan Ayckbourn or Terence Rattigan staged is verging on the impossible, let alone one about opera, with all the negative perceptions that surround the art form. But occasionally a story cuts through. Artists, Inverne says, capture our imaginations, for their humanity as much as for their genius. Decades as a music journalist (most notably as editor of Gramophone) taught him that even the most dazzling of artistic idols are much like the rest of us beneath the surface. The same backbiting takes place in the rarefied classical music world as in any other profession, albeit sometimes expressed, as Inverne characterises it, in a rather ‘fruitier’ way.

Poor Leoncavallo would turn resenting Puccini into a lifelong project. Reluctant to write another crowd-pleaser, he spent the rest of his career striving to be taken seriously. Forever trying and failing to complete his magnum opus, Crepusculum – an epic, Wagner-inspired trilogy about the Italian Renaissance – he flitted between genres, composing everything from a pompous German national opera commissioned by the Kaiser to cheap operettas. By the 1910s he was in debt, forced to sell off his magnificent home beside Lake Maggiore. He spent his final months ranting about his publisher, the leading critics of the day, and most bitterly about Puccini’s recent success with an operatic trilogy of his own, Il trittico. Yet when Leoncavallo died prematurely in the Tuscan spa town of Montecatini Terme in 1919, who should turn up at the funeral, to mourn the man who used to be a friend, but Puccini. Did he ask himself whether he had behaved like a bit of a bastard? We shall never know.

That Bastard, Puccini! is at the Park Theatre, London until 9 August