Haydn is looking well — in fact, he’s positively glowing. The dignified pose; the modest, intelligent smile: it’s only when you squint closely at the portrait that Thomas Hardy painted in London in 1791 that you clock the full peachy-pink smoothness of his complexion. It’s curious, because Haydn suffered disfiguring smallpox as a child, and a contemporary waxwork bust in Vienna is cratered like a moon in a periwig. Hardy’s portrait is a promotional image, commissioned by the music publisher John Bland. This is the Georgian equivalent of a celebrity headshot: a photoshopped, endlessly-reproduceable selling tool, so potent that it’s still being used to shift recordings 230 years later.

Well, of course it is. This is London, and even in 1791 music was big business here. I’m not sure if that’s the exact point that the Royal College of Music intended to make by placing Haydn so centrally in its Museum, but that’s what museums do, isn’t it? They use objects to prompt reflection, and draw connections. The problem being that when an art exists solely as sound waves, it leaves no physical objects to display: just the equipment used to manufacture it, and relics of the ways in which it was exploited. A museum of music is Hamlet without the Prince, but the Royal College of Music Museum – re-opened this month after a £3.6 million redevelopment – manages to pull off a very convincing Rosencrantz and Guildenstern are Dead.

That’s no small feat. This is surely the tiniest museum in Albertopolis, and in its last incarnation it was one of the more frustrating: a chilly 1960s annexe buried deep in the College complex, where Gustav Holst’s trombone glinted drably beneath fluorescent lights. All gone; they’ve scraped decades of congealed architecture from the College courtyards and reconfigured the entrance so you can walk straight into the new museum from Prince Consort Road. It’s still tiny, but it’s elegantly appointed, and with a collection of more than 15,000 artefacts to choose from, they’ve pulled out the big-hitters.

So here’s Haydn, right next to the impresario Salomon — the man who paid to bring him to London, and did very nicely from it. Opposite them hangs a swagger-portrait of the castrato Farinelli, the West End superstar of the 1730s whose dog — possibly contemplating the price his master had paid for fame — gazes sad-eyed from a corner. It’s another publicity tool, and if there’s a definite free-market subtext to the Museum’s current displays, that’s wholly appropriate for London, where commerce has traditionally fertilised art. A 1799 piano by John Broadwood of London sits near the entrance; we’re reminded that Broadwood sent one of these instruments as a promotional freebie to Beethoven, who utilised its innovations in his Hammerklavier Sonata.

How to weigh the significance of objects like these against — say — the oldest guitar in existence or the earliest surviving stringed keyboard instrument, both of which can be seen almost as soon as you enter? The new museum is laid out thematically (‘Creation’, ‘Craft’ and ‘Performance’) but in such close proximity the objects end up talking to each other regardless. The historic instruments are often exquisitely beautiful. A 16th–century Venetian virginal is draped in red velvet, its panels chased with swirling patterns derived from the art of Venice’s Islamic trading partners. Pochettes — tiny rudimentary violins designed for the pockets of 17th-century dancing masters — are pinned on a board like outsize lepidoptera, and while a video display addresses ecological concerns about the ivory trade, a case of carved ivory cornetts and recorders is presented with no less care nearby.

Digital headsets enable you to hear how these instrumental oddities actually sounded — generally, bloody awful. That should be rectified by the series of recitals that’s planned for the museum: ‘Our artefacts were made to be played and heard,’ says the curator, Professor Gabriele Rossi Rognoni. Meanwhile, the Museum’s treasures find their own way to sing out of the silence. John Singer Sargent sketches Percy Grainger as a tousle-haired faun; elsewhere, an unknown craftsman carves a delicate gothic window into the soundboard of a clavicytherium because what else, in 1480, could they possibly copy?

It’s oddly touching, while the manuscript of Mozart’s C minor Piano Concerto — with the solo part, which Mozart would have played himself, scribbled exuberantly between neat orchestral staves — is the sort of object that makes you shiver at the proximity of genius. There’s only that paper, and those pen strokes, between you and Mozart. The plan is to rotate the RCM’s immense manuscript collection through the museum, and that in itself would make this little treasure chamber an ideal place to beguile a spare half-hour before a Prom — sending you back into the concert hall with a headful of questions, and a renewed sense that there’s more to music than meets the ear.