I KNOW what the word “jazz” means to me.

But I’d be lying if I said I didn’t get how polarising the term can be, when dropped into the average conversation about music.

It’s a good time to talk about it. We have, this weekend, an Edinburgh Jazz Festival in its last triumphant two days. We have Ethan Hawke in the cinemas playing the doomed trumpeter Chet Baker (and Don Cheadle on the cable movie channels, free-wheeling through the life of Miles Davis).

We even have Beyoncé, the rapper Kendrick Lamar and the late, great David Bowie slathering their recent music with jazz performers and references (though I am slightly worried about the impending Lady Gaga/Tony Bennett record).

So I feel reasonably well-protected by the overall cultural buzz to address the “J” word directly. But I’m aware, from a lifetime of diplomacy between the worlds of rock, pop and jazz, that some problems are what they are.

Let’s get to the nub of it. Many of you Generation X-ers will fondly remember that section of the BBC’s Fast Show known as “Jazz Club”. The giggles are instant, because the signifiers are mercilessly correct from the start.

A theme tune with splashy drums, too many chords, and a meandering trumpet noodles away over black-and-white grainy film. A smoke-wreathed presenter, dressed like Austin Powers, smugly assumes our knowledge of jazz history: “Tonight we’ve got the legendary American singer Art “Arty” Storm – nice! – who played with such jazz luminaries as HornFinger and the Reverend Blow….”

And then, Paul Whitehouse makes an arse of himself doing vocalese (“shabba-doop dweeb-baah”) before falling off the back of the set.

Of course, we laugh. But for the jazz fan, there’s a nervousness here too. Haven’t they nailed as ridiculous and pretentious exactly what we think is precious and fragile about the music we love?

Take its musical complexity and improvisation, which the Fast Show tries to lampoon from the very beginning. I have had so many skirmishes with fellow rock’n’rollers about this, at every level, over the years. A backing singer used to dismiss everything beyond the functional riff as “jazz waffle” (it sounds better in his native Ravenscraig).

Various bearish Scottish rock critics have launched anti-jazz blitzkriegs at me. They can be reduced to this: “If you decide to play nearly every available note, you never just play the right ones.”

All I can say is that, for this jazz fan at least, you don’t start off in the Wild West of total possibility. My brother Greg and I have as one of our earliest family memories our Saturday-night sit-down with Dad, Irn-Bru shandies at the ready, watching The Oscar Peterson Show on BBC2.

What blew my tiny, pre-teen mind was to hear the Tin-Pan-Alley melodies my father would sing around the house, improvised on by Peterson. There are many examples, but just take People (Who Need People), from the pianist’s 1964 We Get Requests album. When you hear that, what clicks in me is the revelation that a world of possibility and expression can suddenly explode from the most banal and simple of themes.

The idea of a “standard” – a straightforward, popular song that lends itself to improvisation – is one of the obvious routes into jazz. (And new standards are being established all the time. Look up the lyrical pianist Brad Meldhau, who puts Radiohead and The Beach Boys in his repertoire.)

Take the Sound of Music tune “My Favourite Things”. It’s schmaltz of the highest order. But when I first heard the saxophonist John Coltrane’s performance of the song, it was like a universe of freedom opening up. Out of the banal, the sparkling, cosmic and wondrous.

No surprise here, of course, given the way that the freedoms of jazz improvisation have matched the civil rights struggles of African-Americans, both implicitly and often explicitly. Indeed, a reverent appreciation of history is the other element of jazz-love that the Fast Show parodies try to skewer. (“Hornfinger” and “The Reverend Blow”, indeed).

But that history is, in fact, really complex. Indeed, one of the profound joys of jazz is the way it shows how an art form can open up a shared space between ethnicities. How musical inspiration and sharing can jump all kinds of colour bars and lines.

NO POINT in idealising this, of course.

The new Chet Baker movie Born to be Blue shows unflinchingly the tension between Miles Davis and Baker, which was to some degree rooted in the latter’s praise by 1950s critics as the “Great White Hope” of jazz (a label the trumpeter was bewildered and embarrassed by).

In these edgy times, I’d claim jazz continues to teach us a valuable social lesson. I bought a copy of this month’s Jazzwise, the UK’s biggest selling jazz magazine, which has a back issues page featuring 160 front covers. On a rough count, the faces on these covers are split equally between black and white performers.

How many arts and culture publications could actually claim that proportion?

One of those covers – #152, May 2011, to be precise – features the handsome fizzog of Tommy Smith, Westerhailes’ own global jazz titan. I have worked with Tommy since my pop heyday, and as the head of the Glasgow Royal Conservatoire’s jazz course, and director of the Scottish National Jazz Orchestra, the man is well-placed to give an overview of the relationship between jazz and Scottish life.

What we both agree on is that, if you chuck a stone in the street of a major Scottish city, you’ll hit a jazz musician who’s as good as those you might have hired in New York for a fortune 25 years ago. But for all the transporting nature of his music, Smith is a practical man. Jazzers can’t just be gloriously raddled consumptives. Eating (and by implication, making a living) is surely allowed.

Smith says we could do better at enabling this in Scotland. “Norway has six jazz schools, affixed to six jazz regions, with their own subsidised systems of venues and promoters. You can play 20 to 30 gigs in a row there,” says Tommy. “Oslo has a 24/7 jazz station. We have one show a week on Radio Scotland.”

One economic element clearly missing in Scotland is a film and TV sector of sufficient density. Most jazzers are also ambitious and flexible composers, given half a chance and a bit of budget.

But Tommy’s most interesting musings were on how Scottish jazz was, slowly but profoundly, beginning to respond to its national folk traditions. (Scandinavian musicians like Jan Gabarek have been doing this for decades.)

Tommy’s forthcoming record with the BBC Scottish Symphony Orchestra, Modern Jacobite, is a momentous example of this (though his SNJO recording of the Loch Tay Boat Song with Chicago singer Kurt Elling is also worth tracking down). As Tommy says in his calm transatlantic drawl: “It can take many years to get to the right blend of elements between jazz and folk. But it can and should be done. The Scottish brand is strong in the world. We might be able to bring something new to audiences.”

His sense of patient, organic development is key here.

No forcing, no hackery.

Musicians will have their conversations with each other as they hang out, whether verbally or musically – and what will emerge, will emerge. Play of any kind is about freedom. The jazzers I am honoured to know and work with are players in the deepest, most sustainable sense.

Yes, the comic with the pudding-bowl wig and golden pendant is funny on TV. But we should value the presence of jazz in our societies. It helps things (and people) to stay open, fluid and fuzzy.

Perhaps it’s important for us to keep in touch with those feelings, these days.

See edinburghjazzfestival.com for this weekend’s gigs. Tommy Smith’s Modern Jacobite is released on Spartacus Records on Friday, August 26.