NEW YORK – “‘Hey Jude,'” someone in the audience yells out.

Gabriela Montero sits at the piano, taps out the bare melody line, pauses and stares at the ceiling through her flowing blond hair and then plays the tune, complete with harmony.

But it’s sad, in the minor mode. Soon, she shifts to major, then back and forth – sad, happy, sad, happy – in the best tradition of Schubert, with a hint of Debussy’s “Claire de Lune.”

Later, she crosses a musical bridge and breaks into a tango – it’s the Beatles meets “The Addams Family.” Call it “Hey Gomez”? Finally, she brings it to a rousing honky-tonk ending.

Her 5«-minute version of Lennon-McCartney song was produced spontaneously during a recent appearance at Joe’s Pub.

“You see, each theme has a little of its own world,” she explains at one point. “It wants to go to one place; it doesn’t want to go to others. Can’t control it.”

At 37, Montero is the queen of improvisation. Classically trained, Montero was born with a gift that the classical establishment has largely shunned since the days of Franz Liszt – creating music in front of an audience.

Improvisation fell out of favor because the 19th-century piano literature became so difficult to master, according to David Dubal, a piano faculty member at the Juilliard School.

“You can’t waste your time screwing around on the piano and having fun,” he said. “The two things that you need are discipline and concentration. It can take 15 years to play one piece well, and then it will crumble.”

But Montero, who made her New York Philharmonic debut last year, has been able to master both realms.

“The wonderful part of her gift is not, for me, the improvisation but the interpretive artist,” Dubal said. “She has beautiful sound and fine technique.”

She even improvises in the recording studio. Her new CD, “Baroque,” released last month, has her extemporaneous take on a dozen favorites from the 17th and 18th centuries, including Vivaldi’s “Four Seasons” and Handel’s “Messiah.” Her “Hallelujah Chorus,” for example, begins with a “Habenera” bass rhythm made famous in Bizet’s “Carmen,” then arrives at a joyous variation of Handel’s melody, but in a Latin style.

How does she do it?

“It just happens in front of my eyes, with my hands,” she said in an interview. “That’s what I love about it – its unpredictability. It’s the complete surprise and it’s just the magic of, even for me, to see how it develops on its own.”

Montero was introduced to the piano while she was still in her crib in her native Venezuela: for her first Christmas, at 7 months old, she received a two-octave toy piano from her grandmother.

“My parents … were surprised to see that I started to pick out the melodies that my mom sang to me at night,” she said. “And by the time I was 18 months old, I was already playing all of these melodies and I was starting to improvise.”

She began lessons when she was 4 with a teacher who encouraged her to improvise. Four years later, the family moved to Miami, where she studied with someone who insisted on a traditional approach.

“I was told there was no value in it,” Montero recalled. “She really made me feel that I should not expose myself to almost ridicule by doing this.”

After nine years of repressing her urge to improvise, Montero stopped playing entirely at 17. “I had lost the love for music and I had lost the magic,” she said.

But she returned to the keyboard in two years. She went to London on a scholarship and won several awards while she remained on the conventional track of mastering every note as written.

Still, she had doubts. “It took me over a decade to really find my way inside as a musician – why I wanted to do music,” she said.

The inspiration came during an encounter with the firebrand Argentine pianist Martha Argerich, who had met with Montero years earlier.

After attending an Argerich concert in Montreal, Montero asked her for advice.

According to Montero, Argerich replied: “I’m the wrong person to ask any questions to, so why don’t you just play for me?”

Montero went home and practiced, then returned for the appointment the next night and improvised a musical caricature of Argerich.

“There was all the things that I heard about her,” Montero recalled. “There was this romanticism, this playfulness and this very volatile temperament of hers. It was all those things. She got a laugh … It was just one of those moments in life where, I don’t know, something happened … This really changed my whole root in my life.”