Making it up as you go along might work well in the worlds of experimental theatre, freeform jazz, or when you’re caught pilfering biros, but in the world of classical music improvising went the way of Beethoven at around the same time Beethoven went. Chopin dabbled, Liszt doodled, but now, even if ornamentation is cherished in Baroque performance or toyed with in cadenzas, most soloists prefer the security of jotting down a few bars before they put a shoe on stage.

Not so Gabriela Montero, the 38-year-old Venezuelan concert pianist who is reviving this neglected art. In concert and on her website she invites increasingly complex demands from audiences. “Give me Happy Birthday by Tchaikovsky!” challenges one man in a London concert broadcast on YouTube; “ Yesterday by Bach!” says a journalist on an American TV station. Soon she arrives in Edinburgh for her Festival debut.

Far from resenting her insatiable public, Montero precipitates their enthusiasm, and has done so from the moment her grandmother astutely dropped a toy piano into her cot when Gabriela was but seven months old. “It’s not that people are curious like they’re going to see a circus act, it’s real music,” she says. “I don’t improvise for the attention, I was born with this gift, but in this world you have to have something special to stand out. If people didn’t want me to improvise, I would get the message, but it’s almost like, no, we don’t need to hear a concert, just improvise."

There are probably not many concert pianists who wouldn’t feel a little put out that their interpretation of Rachmaninov 2 wasn’t the main event, but then there are no other pianists for whom improvisation is so complete a part of their artistic persona. “I’m definitely not the traditional concert pianist with a big career practising eight hours a day. That’s an absolute luxury, forget it!” says Montero shortly before a sold-out concert at the Ruhr Piano Festival, in Essen. “At home in Boston my role is single mother with two daughters, so I barely get any practice time.” She sighs. “I think they’re kind of embarrassed. The piano takes me away from them, so it’s a love/hate thing.”

If this juggling of work and home life sounds familiar to many working mothers, Montero’s virtuosic skills are almost certainly not. “I was always gifted physically on the piano. Nothing was really a problem, but I’ve changed how I think about music in the past two years, completely changed my technique for a deeper, more symbiotic relationship with the instrument. That has made my improvisations more complex.”

It’s an interesting revelation for a pianist who calls improvising “subconscious”, but this comfortable maturity hides a long, tricky relationship with the piano and a personal life that counts two marriages and 34 home moves in 38 years. Brought up in a middle-class family in Caracas, Montero’s child prodigy was documented publicly from the moment her mother first heard nursery rhyme improvisations coming from the 18-month-old’s cot, to a televised concert with the renowned Simon Bolivar Youth Orchestra aged 8. When the Venezuelan government offered her an American scholarship, her parents sold everything and moved to Miami.

“Those ten years turned out to be the worst possible environment to develop musically and emotionally. My new teacher destroyed my musical spirit, negated my improvising,” says Montero, who gave up the piano at 18. “I despised music, and my talent. I didn’t want anything to do with being musical.”

Resuming two years later, she won the Bronze Medal at the 1995 International Chopin Piano competition, but it was only after a tipsy pub improvisation for the legendary pianist – and now her mentor – Martha Argerich in 2002 that Montero regained her musical spirit.

In Essen’s woody concert hall it is obvious that this “spirit” is what has drawn her audience. Montero tosses out frenetic Chopin and Liszt, but the atmosphere livens when, glass of wine in hand, she banters with a group of flag-waving Venezuelans calling for improvisations amid a sea of sedately rowdy Germans. Requests for Elvis and the Pink Panther follow until someone finally responds to Montero’s plea for “a tune from Essen” with a hearty mining song, which Montero elevates into a slow, touching eulogy.

But there’s a hint that this intensive, structured series of concerts might be taking its toll on the pianist, who admits she was aghast when she first saw her schedule meticulously mapped out until 2011. There are surely, after all, only so many times one can improvise afresh on Yesterday – Edinburgh audiences take note – and, while a few ephemeral creations achieve posterity through recordings such as her album Bach and Beyond (EMI Classics), most survive only as echoes in her audiences’ minds.

“Actually,” she says, “what really excites me, what I really want to concentrate on now, is composing.” It’s an obvious if intriguing move given that her style is frequently comprised of very clever deconstructions of other people’s. The first fruits will appear in 2010, involving cross-platform ensemble work (although her manager hastily caps the details).

But short of a lackey transcribing as she plays, how will Montero get her creations on to paper? “I don’t know, I have not tried. It’ll be a different process, but for me composition is improvisation with the elements of time, decision-making and choice in your favour. It will be fun to construct rather than let my subconscious choose for me.”

I suggest we round off by improvising on something a little abstract in deference to this more constructed future. We settle on A Venezuelan in Edinburgh, a tartan American in Paris-type take on her first Edinburgh Festival. Montero’s quicksilver fingers paint a sweeping Chopinesque “Edinburgh” landscape. “Here comes Latin American me,” she interjects, segueing into a raucous, exhilarating but maudlin Venezuelan riff that eventually resolves calmly into “Edinburgh’s” structure. Whether such off-the-cuff brilliance will translate to the more permanent sphere of composition time alone will tell, but with tunes such as this, Montero will doubtless convert a few more traditionalists in Edinburgh.

Gabriela Montero performs in recital on Aug 12, Queens Hall, Edinburgh, (11am) and improvises at at the Hub (10.30pm). Box office: 0131-473 2000

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