First there was Get Carter, the thriller starring Michael Caine.

Then later in the 1970s and on into the 1980s and 1990s, whenever record producers were looking for a saxophonist to add a distinctive colour to a track, the cry went up: "Get Collins."

Mel Collins, for it is he, has played on so many records that even he can't remember them all. Eric Clapton, Dire Straits, 10cc, Marianne Faithfull and Rick Wright of Pink Floyd are just a few of the people he's worked with. But he's been known to walk down a supermarket aisle, hear something that "sounds a bit like me" and have to go home and do some research. The last time this happened, it was a track by the Alan Parsons Project that he'd only a faint recollection of working on.

Some sessions are all too clear in his memory, like the "not very comfortable" two days he spent working with the Rolling Stones in Paris, only to be edited out of the single version of Miss You – he's on the parent album, Some Girls. And there are the ones that got away. George Michael's Careless Whisper? That was Collins's gig, except when the producer called to book him, he was out on tour and Collins's then wife suggested they try Steve Gregory, a good friend who used to deputise for Collins instead. The subsequent divorce wasn't entirely caused by this incident, apparently.

The record that Collins wishes he had played on most of all, although he later played the song on tour, was Gerry Rafferty's Baker Street. He'll be playing it again during the Celtic Connections tribute to Rafferty, Bring it all Home, but how he came to miss out first time round is the sort of story that could inspire a blues song.

"Gerry told me later what happened," says Collins down the line from his storm-buffeted narrow boat on Hertfordshire's canal network. "He'd had this line that he sang to begin with, then they tried various instruments playing it and it still wasn't what he was after. So someone said, 'Let's get Mel Collins to play it on saxophone' and I think it was the producer who said, 'Nah, Mel Collins won't come away out here' – they were recording in Chipping Norton in Oxfordshire – 'he only plays sessions in London,' which wasn't true. So they got Raph Ravenscroft to do it and it became one of the truly iconic saxophone lines in pop music."

Collins went on to work with Rafferty often – you can hear his saxophone playing on the Snakes And Ladders, Sleepwalking and North And South albums – and of all the musicians whose records he appears on, Rafferty was one of his favourites.

"He could be a real task master in the studio," says Collins. "Because he had such a great ear, he would spot the slightest imperfection and have you playing something over and over. I've seen me working on something with him until my lips were starting to bleed. But as well as having this natural talent, he was a lovely fellow and he became a friend. I'm just sorry that he didn't like touring because I loved playing his music: that tour we did in the 1990s was great; he put together a fantastic band – and although I missed out on playing on Baker Street, I can hear why that saxophone line became so popular. It has a charm about it. It somehow manages to be quite grand and yet it's so direct. It's the blues, essentially. It's so simple but so lovely."

Collins was born into the music business. His mum Bebe was a singer and his dad Derek a saxophonist and session musician who also toured with Judy Garland and Shirley Bassey. They were working on a summer season on the Isle of Man with the Roy Fox Band when Mel's arrival interrupted their gig. Growing up in Surrey there were always instruments lying around the house and after taking piano lessons Collins gravitated to the clarinet and then saxophone and flute. He first came to prominence with prog rockers King Crimson but after a court revolution that saw Collins, bass guitarist Boz Burrell and drummer Ian Wallace abandon Crimson's founder Robert Fripp at the end of an American tour, he went on to join the father of British blues, Alexis Korner, in Snape and it was there that Collins's reputation began to grow.

"Boz, Ian and I basically got fed up playing in odd-time signatures and Robert can be quite hard to work with. So there we were in America and there were all these great boogie bands like Little Feat and I thought, that's what I want to do, get back to playing rock'n'roll," he says.

"I'd listened to jazz a lot in my teens. My dad was a jazz musician who knew he couldn't feed his family if he played jazz. So I'd checked out the jazz players, guys like Dick Morrissey who I really liked, then gone off to play like the guys in Little Richard's band, which didn't please my dad so much. Anyway, after Alexis along came Kokomo and the session thing all started from there."

For those too young to remember, Kokomo was a British soul band whose vocal frontline was a match for almost anyone and whose rhythm section grooved like nobody's business. They were briefly handled by Pink Floyd's management, made two albums, Kokomo and Rise And Shine, which were re-issued not so long ago and still sound great, and toured as support to Average White Band. They also had a recording with Bob Dylan aborted, in part thanks to one of their number's unflattering comments about Dylan's voice, and despite recording a strong third album they never quite achieved the fame they deserved, although their occasional reformations still cause excitement around London.

Their saxophonist, meanwhile, became known by word of mouth for being able to spontaneously fashion the kind of sax lines that lifted, or even rescued, a song.

"The recording business was very different back then," says Collins. "You could be in and out in half an hour and away to another session or off to play a gig. My dad would play three or four sessions a day at his peak. I was never quite that busy, more like three or four a week, but if people knew you could turn up and play something they could use, word definitely got round. They would call me up and basically hire me to be me, which was great." Later on they would ask me to play like David Sanborn, whose saxophone sound was everywhere for a while, and although I really admired him, I'd have to say, 'You want him, pay him the money.' But I established relationships with people like Clannad, who I first recorded with in the 1970s and went on to work with for years, and Gerry and a few others, and if you played on the records, often they'd want you to tour with them."

Bullets have been bitten. There was the well-known singer-songwriter who told Collins she was hoping to get Clarence Clemons to play on her album when he was in town with Bruce Springsteen but asked Collins if he wouldn't mind doing it if she couldn't get hold of Clemons. Collins did the session. And trying to play the solo of his life for Mick Jagger while one other Rolling Stone was throwing a party and another pointedly ignoring him wasn't the way Collins imagined that particular chapter unfolding.

He says: "I've had good experiences and not so good experiences. "I've been edited off singles – Tina Turner's Private Dancer is one instance; they kept Jeff Beck's solo and had me on the album track only – but I've got to play on a lot of really good music of all different styles. I loved working with the Stray Cats, for example, playing that authentic rock'n'roll saxophone style, and I've just done a track with the German prog rock band Nektar – remember them? – whose brief was brilliant. They sent me the track by email, because that's the way it often works now, and said, be as creative as you like. I thought, wow, that's my kind of gig."

Mel Collins appears at Bring It All Home – Gerry Rafferty Remembered at Glasgow Royal Concert Hall on January 22 and 23 as part of Celtic Connections.

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