Cadenzas are a concerto soloist’s time to shine: the moments when the rest of the orchestra dramatically drops out and a single musician gets the chance to command the stage.

For about half of Mozart’s piano concertos, cadenzas he wrote have been preserved, and those are what you usually hear in concerts and on recordings. Other composers later filled in the gaps with cadenzas that have also become traditional. Some performers write their own.

But 250 years ago, when Mozart was a star pianist, he wouldn’t have performed prewritten cadenzas — even ones he had composed.

“When Mozart wrote his concertos, they were a vehicle for his skills,” the pianist and scholar Robert Levin said by telephone from Salzburg, Austria — Mozart’s hometown — where he teaches at the Mozarteum University. “He was respected as a composer and lionized as a performer, but it was as an improviser that he was on top of the heap.”

Levin, 76, has long argued that Mozart, as a player, made up new cadenzas and ornaments in the moment. And he has sought to revive that spirit of improvisation in a landmark cycle of the concertos on period instruments, a 13-album project begun more than 30 years ago with the Academy of Ancient Music, led by Christopher Hogwood.