@ The Caley Bar, Edinburgh, until Sun 30 Aug 2015 @ 20:00

Maybe one for connoisseurs rather than your casual music fan, but this residency in the plush surroundings of the Caley Bar at the Caledonian Hotel offers exquisite musicianship, a mellow atmosphere and, as an added bonus, table service. When the duo open with the jazz standard after which this residency and accompanying EP are named, you’re transported away from the frantic flyer-filled streets of Edinburgh to some filmic late-night club, a touch of class amongst the Fringe chaos.

Multi-instrumentalist Edwards has some CV as a session musician. Originally a member of The Higsons, he’s played with PJ Harvey, Nick Cave, Madness, The Jesus and Mary Chain, Tom Waits and many more. He’s also worked with Tindersticks, whose lead guitarist Fraser is his foil in this duo.

The pair work well together, skipping through genres from jazz to fuzzy indie to surf rock, the Peel show in concert form that they are aiming for, with Edwards doing a little context-setting in between each song. Fraser’s effects-heavy guitar plays perfectly as a launchpad for Edwards’ sax, which runs the gamut from soulfully smooth to cathartically discordant. Sometimes it’s two saxes at the same time, a trick Edwards learnt playing in the Blockheads. Sometimes it’s swapped for another instrument – a horn or a guitar. His flute on Lullaby for Ian (the late Ian Dury) is gorgeously tender.

Edwards’ singing is functional in that thin, post-punky way, which at least allows them to step away from the instrumentals, and they’ve made some great choices of vocal covers too – Tom Waits, John Cale – even if his voice doesn’t have that same grizzled, croaky gravitas. It also feels a pale replacement for Tindersticks’ Stuart Staples, but better that they throw in a couple of Tindersticks numbers, than pass up the opportunity.

Tonight, Mike Heron of the Incredible Sting Band and Pete Wylie are in the house. “Noddy Holder was sat in that chair the other night,” proclaims Edwards at one point. And, according to press releases, Stewart Lee, Bridget Christie, Phill Jupitus and Robin Ince have all been in to pay homage. As far as endorsements from fellow cultural figures are concerned, Edwards and Fraser are doing very well thank you very much.

Audiences will inevitably skew to the completist, vinyl junkie end of the musical spectrum, but they’ll not be disappointed with this wonderfully varied, atmospheric evening of music.