Born in 2005, Ms. Deutscher spent her first five years in Oxford, England, where her mother was teaching. While her parents were academics, they also loved music and would play together as a family, with her mother, Janie, on the piano and her father, Guy, on the flute. But Ms. Deutscher’s ability rapidly surpassed that of her parents.

She received her first violin at age 3, and left her first teacher on the instrument begging to take a break after more than an hour. At 4 she would sit for hours at the piano, working out melodies she said were songs from the imaginary world she called Transylvanian. Recognizing his daughter’s extraordinary abilities, Mr. Deutscher tried to find a teacher willing to work with a preschool composer, but most turned him down. “They would say to call back in 10 years — that was those who were being polite,” he said.

As Alma’s renown spread, the Deutschers gave up their teaching jobs and moved to Dorking, in Surrey, and devoted themselves to managing her schedule. These days, Mr. Deutscher said, there are so many people who want to work with his daughter that he spends a lot of his time fending them off. Having realized that no school would be able to meet Alma’s special needs as a budding musician and composer, her mother began home schooling her and her sister Helen, now 11. Mr. Deutscher has managed to keep one foot in the academic world, writing books on his specialty, linguistics.

They are selective with her performance schedule, which last year included concerts in China, Germany and Switzerland, as well as several in Austria, where Ms. Deutscher has been embraced and celebrated for years. After spending months in 2016 in Vienna for rehearsals of her opera, “Cinderella,” the family decided last year to move, so the girls could learn German and Alma could indulge in the wide range of musical opportunities the city offered.

“I lived in England, but I grew up on the music of Mozart, Schubert, Beethoven and Haydn,” she said. “Musically speaking, I think that Vienna’s always been my home.”

Moving to a European capital has also offered both Alma and Helen freedom to move around on their own, riding the subway to the opera house or to the vineyards at the city limits. There, beyond the paths Beethoven roamed, Ms. Deutscher takes long walks, accompanied by her family and the pink, sparkly tasseled jump rope that she has carried with her since she was a child.

“When I was younger, I really thought it was the rope that gave me inspiration,” she said, with a sly smile at her former self. Although she still wears dresses, jumps on the trampoline and climbs trees, the little girl who charmed a TV audience as an 8-year-old on “The Ellen DeGeneres Show” has been replaced by a tall young woman.