John Williams won his first Oscar for music he didn’t compose — adapting the score of the stage musical “Fiddler on the Roof” for its 1971 screen incarnation.

If the same assignment were to come up today, he wouldn’t even be eligible. Similarly, Marvin Hamlisch couldn’t have won for his adaptation of Scott Joplin rags in “The Sting” if it were released recently.

That’s because the category accommodating that work — “original song score or adaptation score,” or variations on that phrase — was abolished in 1985.

Today, the Academy of Motion Picture Arts and Sciences routinely offers two music categories: original score and original song. But for nearly half a century, there was a third category, honoring the adaptation of songs, or sometimes pre-existing musical material, into a movie score.

Some in the film-music biz are arguing that it may be time to revive the adaptation category, especially considering the number of scores that would qualify. This year’s “Les Miserables,” especially, required an inordinate amount of work (primarily by score producer Anne Dudley and orchestrator Stephen Metcalfe) to take the actors’ live on-set vocals and build orchestral tracks around them.

In recent years the adaptation of “Chicago,” “Sweeney Todd,” “Dreamgirls,” “Hairspray,” “Nine,” “Phantom of the Opera” and other Broadway-to-Hollywood musicals has been without an Oscar category. Their only Oscar-music hopes were pinned on the creation of songs for the movies.

Nor is there a place to honor scores adapted from traditional or classical material (such as the use of Tchaikovsky in “Black Swan” and 19th-century hymns in “True Grit,” both in 2010).

Today, films like “The Dark Knight Rises” or “The Hobbit” — which use themes from earlier films — might qualify more in an adaptation category than original score. (Sequel scores like Elmer Bernstein’s “Return of the Seven,” based on his “Magnificent Seven” music, generally wound up there.)

“Between musicals and adapted work, there is clearly material that should be recognized but isn’t,” says one high-ranking studio music exec who asked not to be identified.

“(Adapting a stage musical) takes no less work, no less creativity, and it can sometimes be harder than creating one from scratch,” says another studio music chief. “I would be supportive (of reinstating the category).”

“We are about to see a lot of films that could fill up this category,” says Robert Kraft, former president of Fox Music. He cites recent musical biopics like “Ray” and “Walk the Line” and upcoming projects in development including films about Marvin Gaye, the Beach Boys and Fleetwood Mac that will require extensive musical work in pre- and post-production.

“All the musical films fall between the cracks,” he notes, citing Fox’s own “Moulin Rouge” as a complex musical project and, among original musicals, the animated “Rio,” which boasted seven original songs.

In fact, there is a third Oscar music category that is on the books but has never been activated. It’s now called “original musical” and is essentially the old third category minus the adaptation part. It requires a minimum of five original songs “by the same writer or team of writers” but the paucity of entries means it’s never used.

This year, the original song score of “The Lorax” would have qualified, and in recent years the composer of those songs, John Powell, could conceivably have been considered for his complex adaptation of pop and original songs in the “Happy Feet” films.

“Musical films are coming back,” says Kraft. “This is an arena that’s underserved, both currently and in the future.”

Acad officials indicated that there are no plans to reactivate, or modify, that third category. Says music-branch governor Charles Fox: “There haven’t been enough scores that were eligible in a given year that would fit into that category to make a field of nominees, rather than practically giving an honorary award. We would love to have enough scores eligible to reactivate that category, and hopefully that will happen.”