The Bee Gees’ sound evolved from pop to R&b in what became the disco music era, but it was their falsetto that was their unmistakable trademark… and it came about by accident many years into their career.

In the HBO documentary “How Can You Mend a Broken Heart,” surviving Bee Gee, Barry Gibb, takes viewers inside the studio, showing archival footage performing with his brothers Maurice and Robin and honing their signature sound.

In 1975, the band took the advice of veteran rocker Eric Clapton and went down to Miami to record their first album since the split at Criteria Studios in 1969. “I thought those guys were really an R&b band that hadn’t really worked that out yet, and I thought, man, this would be so good if they could pick up on what’s going on in America,” Clapton said.

It was in Miami that the Bee...