NEW YORK, NY.- 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,” 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.

After a two-decade gap caused by record company budget cuts, and with the last installment finally released this summer, the cycle takes an invaluable place as the most complete survey of Mozart’s music for keyboard and orchestra.

Coming on the heels of Levin’s raucous 2022 traversal of Mozart’s sonatas on the composer’s own fortepiano, the concerto series is a capstone to the career of one of the most inspired and creative Mozarteans of our era. The performances are excellent, with the Academy now even freer and more characterful than in the early installments, and Levin’s spirited, daring playing is a consistent joy.

Included in the final release is the Concerto No. 25 (K. 503). Mozart left no cadenza for the majestic first movement, and Levin fills the breach with a dashing yet poised solo, coiling through some adventurous harmonies as it revisits and transforms earlier music. It has the clever confidence of something meticulously planned, as well as the energetic freshness that lets you know it was made up on the spot.

In the last concerto, No. 27 in B flat (K. 595), his ornamented accompaniment goes well beyond what’s charted in the score — but with stylish flair, never obtrusive indulgence. His first-movement cadenza, a dizzying rush of race car virtuosity on an agile, pearlescent fortepiano, makes Mozart’s written-out version seem like a sedate (if elegant) Bentley by comparison, especially played on a modern Steinway.

The story of how Levin came to set down all the concertos — and set them down his way — began around 1990, when filmmaker Jeremy Marre, a prominent music documentarian, was working on a series about improvisation for British public television. He had found copious examples around the world, but few in the classical sphere. He asked Levin for ideas.

Levin had for years been putting his scholarship to work as a kind of Mozart medium, channeling the master as he completed and reconstructed abandoned fragments and improvised in his manner. For the documentary, he proposed a demonstration: He would perform one of the concertos and do an improvised cadenza, then go back and improvise a different one.

Marre was game, and Levin recommended as partners Hogwood and the Academy, who had already produced a pathbreaking period-instrument cycle of Mozart’s symphonies for Decca’s L’Oiseau-Lyre imprint. The results pleased everyone involved, and Hogwood suggested making an album.

“When he went to Decca and said that it would be nice to do a recording,” Levin recalled, “they said, ‘Let’s do them all.’ That’s the way people were inclined in those days” — in the industry’s CD-fueled, cash-rich prime.

The idea was to record everything Mozart wrote for keyboard and orchestra, including double and triple concertos, unfinished fragments and the soprano aria “Ch’io mi scordi di te?” which has a prominent piano accompaniment as part of the ensemble.

The project grew to encompass even Levin’s reconstruction of a piece found in Mozart’s sister’s study book that is believed to have been his first attempt (as a 7- or 8-year-old) at writing a concerto movement. The instruments for the sessions were selected based on the ones Mozart would likely have been using in different phases of his career: harpsichord, organ, tangent piano and copies of fortepianos by Anton Walter and Johann Andreas Stein.

“The overall picture was more varied than other attempts to address the cycle,” Levin said.

Eight discs were released. But the project, conceived during the recording industry’s final boom, was a victim when it went bust. As the 21st century turned, Decca pulled the plug — first temporarily, then permanently.

“It was a shock,” Levin said, “and a real disappointment.” In 2014, a few years after transitioning to emeritus status at the Academy, Hogwood died at 73, which seemed to shut the door on the cycle for good.

“What came to the rescue, strangely enough, was COVID,” Levin said. “During the pandemic disaster, there were no concerts, and orchestras like the Academy were faced with going broke if they didn’t find something to do.”

There was interest in using the time to get the Mozart project back on its feet, and John McMunn, who became the Academy’s chief executive in September 2020, raised the funds to make it happen. (Donors were able to sponsor individual players, movements and even cadenzas.) In August 2021, the group returned to the studio — first under Richard Egarr, Hogwood’s successor, then Laurence Cummings, the Academy’s current music director — and over the next year made the final five albums.

The cycle was never conceived chronologically, so there were still some early concertos to record on harpsichord, and pieces including that reconstructed fragment from Nannerl Mozart’s music book, in which Levin captures all the buoyancy of the boy genius.

But this last round of releases also boasts some of the glories of Mozart’s maturity, starting in the mid-1780s when, Levin said, he “is riding the crest of the wave, on a level of inspiration and experimentation that has no comparison until, of course, you get to Beethoven.”

Levin’s improvisatory ethos and agile touch are well suited to the sparkling outer movements. While some of the middle movements here — for example, the Larghetto of No. 24 in C minor (K. 491) — don’t have the immaculately poetic quality we’ve come to expect in slow Mozart, they have gracefulness and sensitivity, and a sense of breathing life.

“What I worry about in the classical music field is that everything has become standardized, everything is calculated to be polished,” Levin said. “The spirit we have brought to these recordings — of risk-taking, of adventure — is infectious. That’s my hope.”

This article originally appeared in The New York Times.