Tom Waits, Playhouse, Edinburgh

The standing ovation hangs in the air as soon as there are enough people in the venue to engage in one. Tom Waits is one of those infrequently touring artists who inspires such anticipation and more, and has grown men shrieking his name and genuflecting before the stage while the house lights are still up and half the crowd are yet to file in.

This is the first of two dates in Edinburgh, the only UK shows of a sporadic summer tour. Waits' awestruck devotees have travelled from far and wide, all engaged in some black pilgrimage to see the king of Gothic country blues. So when he takes to the stage – or, more precisely, when the lights go down and an indistinct shuffling can be seen before us – there is pandemonium. Then Waits is there in the spotlight, wearing an old grey funeral suit, dusty working man's boots and that dusty working man's face.

Everyone is on their feet now, and Waits simply, silently raises his palms to the ceiling, weighing the applause in each clawed hand, forcing it louder and louder. And then he starts actually singing.

That voice, too, is a dusty working man's voice, if the man in question were an aged grave-robber in recovery from a bronchial infection. The growl is unforgiving, almost oppressive in places, but there's something warm and loving, or at least cynically passionate, at the core of it. It's surely one of the most unique and evocative instruments ever to be heard on a stage.

Encircled by old public-address speakers, the podium Waits stands on is coated in dust that kicks up around his legs as he tramps along to the beat of the opening medley, "Lucinda" and "Ain't Goin' Down". His style is singular, but a repertoire culled from 20 albums over three-and-a-half decades is varied in tone and emotive content.

That unspeaking opening continues through the opening few songs, as evocatively transporting as they are nostalgic for his fans. "Rain Dogs" and "Falling Down", for example, somehow bring to mind the sight of a blasted desert plain or the smell of spilled whiskey in a New Orleans dive bar, Waits taking us closer to the heart of Americana than most other artists in any medium can.

There's also a precise, intuitive sense of comic timing, which emerges during "I'll Shoot the Moon", as this most unlikely exponent of live gimmickry acts out each word of the lyrics in mime format. It is hilarious and, although perhaps not entirely unexpected of the sometime actor, it exposes the precision involved in his seemingly off-the-cuff set.

At one point, Waits dismisses his band, double bassist excluded, and sits at his piano for the short passage that steps the night up from the already sublime to the truly soul-tingling. "Invitation to the Blues" and "Innocent When You Dream" – the latter audience-assisted at Waits' invitation – are songs truly to marvel at.

Praise for the excellence of Waits' band is well deserved, particularly the rich saxophone and harmonica of the great Vincent Henry. Against the strength of Waits' persona, Henry actually manages to steal the show for short, solo bursts, while a note of very human pride escapes Waits' lips when he introduces his son Casey on drums.

So, after the gentler piano section, the foot-stomping Waits returns to finish us off, belligerent but growing somehow ever more charming with each song. He preaches his way through the raw "Way Down in the Hole" and chills all present with the cold fatalism of "Dead in the Ground". Then the chorus of the main set closer, "Make it Rain", sees glitter explode from the ceiling and over Waits' head. This is, after all, the Glitter and Doom Tour, and both are to be found in abundance here.