Its players and their circle included the Bancroft twins, guitarist Kevin Mackenzie, trumpeter Colin Steele, saxophonist Tommy Smith and pianist Brian Kellock among many others.
Having spawned many other groups, including Rae's own groundbreaking jazz/traditional music amalgam Celtic Feet, that generation of players has mentored a succession of younger musicians on the burgeoning Scottish jazz scene, most recently showcased in the triumphant short tour by the National Youth Jazz Orchestra of Scotland.
Rae himself, however, upped sticks the best part of a decade ago and moved to New Zealand, where his children Maisie. 18, and Alex, 14, have grown up. When he appears at Edinburgh Jazz Festival, starting with a gig by the basic sax, bass and drums trio at the heart of his New Zealand group The Troubles at Bill Kyle's Jazz Bar tomorrow evening, it will be the first time he's been back in Scotland for five years and also in the vanguard of a showcase of New Zealand artists across all of Edinburgh's festivals this summer.
NZ at Edinburgh includes the return of choreographer Lemi Ponfasio at the Playhouse in the EIF programme and Haka at the Assembly Hall heading a whole list of shows in the Assembly Theatre programme, a big NZ contingent in the Tattoo, and a musical showcase with performances in the Queen's Hall. John Rae, however, as befits a man whose last band-leading appearance in Edinburgh at festivaltime earned him a Herald Angel (in 2005), is the first to arrive, bringing with him saxophonist Lucien Johnson and bassist Patrick Bleakley.
"The Troubles is usually a string quartet with horn, bass and drums, so unlike Celtic Feet and the Big Feet big band it is more jazz-classical in instrumentation. It is still my writing, but not accordian and fiddles, " says Rae.
"I'd always wanted to do music with strings, but in Scotland that was more difficult. Here [in New Zealand] there is more cross-fertilisation with classical musicians than with folk players. Brian [Kellock] and I talked about doing work with strings and knew some classical musicians but we didn't get to play with them.
"I was fortunate to be appointed composer-in-residence at Victoria University in Wellington, the first ever jazz guy to the post, so I had access to an orchestra and string players. I found four or five that were compatible and the band has had a Sunday night residency at a club in Wellington ever since."
The Troubles has become one of the most celebrated contemporary jazz outfits in New Zealand and Rae is spending much of this year on the road throughout the country thanks to Chamber Music New Zealand, an antipodean equivalent of Enterprise Music Scotland here.
"It is not a jazz touring set-up, but we applied to them for a tour at a time when they were looking for something with a bit more anarchy and improvisation - and that wouldn't have happened in Scotland."
Whether or not Rae is correct about that - and I think things have perhaps changed since Rae departed these shores - he is making no big claims about the jazz infrastructure in his new home.
"The jazz scene here is underdeveloped, but there are three universities doing jazz degrees to doctorate level. A lot of the musicians leave and go all over the world, but perhaps 80 per cent of them never earn a living as jazz musicians."
Rae has made that trade work in New Zealand though, now teaching at the university two days a week and at a high school a further day, with the rest of his time devoted to playing and composition.
"In my writing I hardly ever use piano, perhaps as Brian Kellock is such a hard act to follow: you'd give him a chord to play and he'd turn it in to Rachmaninov prelude. In this band I deliberately work without the harmonic stability of a chord-playing instrument like piano or guitar."
What the level of jazz education in New Zealand does seem to have provided is an audience for John Rae's continuing musical experimentation, building on his youthful Scottish success. Certainly the man himself, while looking forward to playing in Edinburgh again, has no regrets about the move. "It was the best thing I ever did," he says simply.
The Troubles play the Jazz Bar tomorrow evening as part of Edinburgh Jazz and Blues Festival.