Mozart did it. Liszt, famously, too. You could hardly stop Bach and Messiaen – even Boulez dabbled. But at some stage improvisation disappeared from the concert platform; experimentation became something to do privately and in advance rather than in public and in real time. Unless you’re Venezuelan pianist Gabriela Montero, who has spent a career reinstating the art on the concert platform.

So far Montero’s three-concert residency at the Barbican hasn’t yielded an opportunity – not so much as a cadenza – so I suspect many of the substantial audience for her solo recital were there in hopes of hearing more than just the advertised programme.

They weren’t disappointed. But first there were the not-insignificant formalities of Iberia, a Spanish-themed recital that embraced 250 years of musical visitors and professional blow-ins – Chopin, Domenico Scarlatti, Liszt – alongside locals Granados, Albéniz and Mompou.

Montero is such a lucid pianist; there’s a clarity and fluidity to her playing that finds the internal logic of Scarlatti or Liszt with the same unforced ease. The former’s Keyboard Sonata in C sharp minor (woven into a continuous larger-scale work with his own G major Sonata and a further two by his pupil Antonio Soler) had the precise radiance of a Zurbarán painting: a clean, architectural starting point for the touristic clutter of mantillas, flamenco fans and bull-rings that we heard in works by Albéniz (his tempestuous “Triana” from Iberia and a swaggering “Sevilla” from the Suite Española) and Liszt’s Rhapsodie Espagnole – a bravura closer, mercurial and dazzling in Montero’s expansive delivery.

But with so many short works it was hard to get beyond postcard-Spain. Granados’s Eight Valses Poéticos was as close to an off-road excursion as we came, its sequence of elegant miniatures not the meat this technically impressive programme needed. Instead, Montero supplied fun: a trio of improvisations, two to themes suggested – and sung – by the audience.

Unfazed when a little girl demanded ABBA’s Mamma Mia (“I just get out of the way and let it happen …”), Montero transformed it into a lushly modal rhapsody, while the Beatles’ Here Comes the Sun became first an earnest Purcellian number (that syncopated rhythm in the opening phrase was born to be a Scotch snap) and finally a swinging ragtime finale. It was the unexpected holiday bonus we didn’t know we needed.