Bill Miller, who died Tuesday night at 91 in Montreal General Hospital, was a pianist and conductor who served as Frank Sinatra’s primary accompanist for more than 40 years. Miller died from complications following a heart attack last week.

Sinatra once introduced Miller at a concert as “my partner at the piano.” Through the most essential part of Sinatra’s career — the glory years of the 1950s and ’60s, his comeback in the ’70s and his frequent touring of the late ’80s and ’90s — Miller was celebrated as an essential part of the Sinatra experience. The singer’s son and, later, conductor, Frank Sinatra Jr., recently described Miller as “the greatest singer’s pianist there ever was.”

On key records, most famously “One For My Baby,” Miller is more than an accompanist — he is a de facto collaborator. Miller’s contribution to the mood and message on the record rivals Sinatra’s.Though best known for his saloonstyle piano on Sinatra’s songs of heartbreak and loss, Miller was equally skilled at the swinging piano interludes on “The Lady Is A Tramp” and “The Lonesome Road.” His contribution is perhaps easiest to appreciate in those few Sinatra concerts where Miller is not present and Sinatra didn’t swing as convincingly or communicate nearly as effectively.

Miller, born in Brooklyn on February 3, 1915, was almost one year older than Sinatra, although in later years he referred to the singer as “the old man.” Like Sinatra, he first came up through the big bands: he started at the age of 18 with Larry Funk and his Band of a Thousand Melodies, and then with arranger Joe Haymes and his Orchestra. Miller played with his first great band and his first great singer when he joined the band led by vibraphonist Red Norvo and jazz vocalist Mildred Bailey. One of his favorite stories, recounted many times over the years, was how he initially ignored Norvo’s offer because he couldn’t believe the vibraphonist could actually get his act together enough to form his own band. When he heard Norvo and Bailey over the radio, he was so impressed that he fibbed and told Norvo that he had never received the telegram.

The Norvo-Bailey band was one of Sinatra’s all-time favorite musical groups, and that was how the singer heard Miller for the first time. Miller first heard Sinatra’s voice at that time too, when he and Bailey heard Sinatra singing with Harry James’s band over the radio. After his stint with Norvo ended, Norvo broke up his band. Miller was heavily featured in saxist Charlie Barnet’s great band of the pre-war period, and after a stint in the service, played with a variety of bands — Tommy Dorsey, Benny Goodman, and Barnet again — and singers, including Martha Raye.

Miller was recommended to Sinatra by a songwriter and Sinatra crony, Jimmy Van Heusen. Miller began working with him in Las Vegas in 1952. He climbed on the singer’s bandwagon at the low point of his career, and he was an essential part of his return to glory. Miller played a prominent role in virtually every Sinatra album and single. “Bill is the best,” Sinatra’s longtime guitarist, Al Viola, said. “If you listen to the few records they did without any orchestra, like ‘Where Or When,’ you can hear the strong structure of his harmonies backing Sinatra — it’s perfect!”

From working with Sinatra at Capitol, Miller became the pianist of choice for Sinatra’s arrangers, Nelson Riddle and Billy May, and became a very busy studio musician. He took great pleasure in accompanying one of his all-time inspirations, a jazz pianist turned pop singer, Nat King Cole. “Whenever I played for Nat, I always asked myself, ‘How would Nat play this?'”

Sinatra and Miller had their differences over the decades — it sometimes seemed a little cruel when the singer would address the pianist as “Sunshine Charlie” on stage, an ironic reference to the way that Miller, like Sinatra himself, was a creature of the night and was almost never awake during the daytime.Yet when Miller lost both his wife and his house in a 1969 mudslide in the Los Angeles hills, it was Sinatra who paid his hospital bills and bought him a new apartment. The two had a serious falling out in 1979. “He took me for granted and I took him for granted,” the pianist told me in an interview in 1992. But Miller was back at the piano bench in 1985 in time for the final phase of Sinatra’s career.

Miller continued to work with Frank Sinatra Jr. after Sinatra’s retirement in 1995 and his death three years later. He played on “That Face!,”a new album released last month by Frank Sinatra Jr., and was on tour with him last week in Montreal. On July 1, he suffered from a broken hip, and then from a heart attack following surgery. “He was sharp as a tack until the moment he died,” as Sinatra Jr. told Variety’s Army Archerd, “When he was being wheeled into the operating room he said, ‘Fly me to the moon.’ This is a very dark hour for us all. We cried together.””

Mr. Friedwald is The New York Sun’s jazz critic.