Dick Lee tells a story about how his new band, Dr Lee's Prescription, acquired a drummer.

The saxophonist and master of a veritable orchestra of blowing instruments had been rehearsing with the band for a few weeks when he realised it needed percussion alongside the two guitars and bass that complement his own work. Lee had heard and been impressed by Stu Brown, one of Scotland's leading young drummers and a player whose versatility would make him naturally compatible with Lee's liking for stylistic variation. But would Brown be a) interested and b) available?

A phone call answered both questions. Brown was interested and it turned out that, when he was but a lad going to jazz gigs with his dad, his father would tell him that if he kept practising drums he might get to play with Dick Lee one day So he could hardly turn down the opportunity. "That made me feel old," says Lee, "but also happy." Lee is hardly one of jazz's senior citizens but the band Brown's father had taken him to see was probably one of the instalments of the gypsy swing-style quartet Lee has been fronting – and renaming annually – since its inception as Swing 1980.

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Besides featuring on saxophones and clarinets and adding perhaps unlikely but very fluent swinging recorder to this quartet's Django Reinhardt-inspired music, Lee has appeared in myriad settings over the years. He has composed and played chamber jazz, worked in duos with piper Hamish Moore and accordionist David Vernon, played folk music, klezmer and eastern European metres and can also be heard in the saxophone section of drummer Ken Mathieson's Classic Jazz Orchestra.

"I've always enjoyed playing different genres and I've been moving from one to the other for so long I can do it without thinking," he says. "So when I put this new band together, I wanted it to play lots of styles but also to steer clear of the Django-esque swing style so it would be completely different to Swing 2012, although there's one number we do in that style."

Dr Lee's Prescription actually came about after Lee and acoustic guitarist Marcus Ford had been chatting while driving back from one of the frequent gigs where they provide high-quality background music for corporate functions.

"Marcus said, 'It's great that we get this kind of work but it would be really good if we could put together a band to play concerts and festivals where people listened rather than talked over us.' And I said, 'Yeah, let me think about that,'" says Lee. Getting away from the Swing 2012 format wasn't easy. Apart from writing a tune that would fit into either band's repertoire, Lee found the guys he immediately thought of bringing into the new band gave him the same instrumentation as Swing 2012. Hence Brown's arrival.

"It's funny – there's one tune we do that's not so dissimilar to the kind of thing Stu does with his Raymond Scott tribute band," says Lee, "but when I wrote it, I hadn't heard the Raymond Scott tribute band and hadn't consciously listened to Scott, although I'd probably heard his music since he was prolific as an animated film soundtrack composer. When I chose the other musicians I knew they would be comfortable moving from rock-influenced music to eastern European time signatures to melodies that have a Scottish quality, and we've been lucky because Stu has all that covered. I could gush but I don't want to embarrass him."

Lee is very much a tunes man. He likes to challenge the musicians he's playing with but his music boils down to strong melodies. This makes it, perhaps, all the more surprising that he composes straight from his imagination on to the manuscript paper – or computer screen – without having one of his many instruments to hand.

"Sometimes a chord sequence will come first – sometimes nothing comes at all, which can be a worry – but once I have the basic idea, I'll arrange it to suit the musicians' strengths and personalities," he says. "That's what Duke Ellington did and it's possibly the only similarity between him and me. I've been encouraging the other guys to write for the band too. Phil Adams, our electric guitarist, has written a clawhammer country blues and Marcus has contributed a jazz waltz, so that's another two directions we can go in. We're calling it music to cure all ills and while we don't expect people to take that literally, it's inspiring for us and hopefully interesting for other people to move from style to style."

Dr Lee's Prescription plays the Jazz Bar, Edinburgh tonight and Glasgow Art Club tomorrow.