A CERTAIN Ratio never got to release the record they were supposed to make with Grace Jones. Island Records boss Chris Blackwell saw to that after he discovered the planned alliance between the brooding Manchester avant-funk band and the iconic Jamaican diva.

One of the proposed tracks, a cover of the Talking Heads’ Houses in Motion, minus a vocal from Jones, was recently released as a trailer of sorts for acr:box, a four-CD collection of singles, remixes and unreleased demos put out by Mute Records spanning the whole of ACR’s 40-year tenure on the frontline of punk-funk. Following a series of archival releases, the band’s anniversary is being celebrated on their current tour that takes in a show at Edinburgh’s Voodoo Rooms this weekend. Bells and whistles, one suspects, will be compulsory. But what of the fabulous Ms Jones?

“Grace Jones had done a version of Joy Division’s She’s Lost Control,” says ACR’s bass player and co-vocalist Jez Kerr, “and it was the A&R man’s idea to do it. We were going to do Houses in Motion and our song, And Then Again, and there was talk about us going out to Compass Point in the Bahamas to record it with Martin Hannett, who was producing all the Factory stuff at the time, but the A&R man had jumped the gun, and when he found out, he didn’t want Martin Hannett anywhere near it, so it never happened.”

Even so, the two rediscovered recordings of the song that did take place (one overseen by Hannett, the other by the band, both featuring Kerr’s original guide vocal) are a low-end joy that sound like they could have been recorded yesterday. In this sense, A Certain Ratio have arguably found their time, reclaiming a sound later defined as punk-funk and taken into the mainstream by the likes of LCD Soundsystem. More recently, Manchester-based band Duds have picked up ACR’s mantle of skewed inner-city multi-culturalism.

Jones isn’t the only pop icon to have crossed paths with ACR. In 1980 on the band’s first visit to New York, as well as picking up a host of street-smart Latin rhythms, they were supported at a club date by an ambitious young singer called Madonna, who would later make her UK television debut with full dance troupe on an edition of The Tube filmed at a pre-Madchester Hacienda, the Factory-owned nightclub that would later be the epicentre of the UK’s dance scene.

Named after a line in The True Wheel, a song on Brian Eno’s 1974 album, Taking Tiger Mountain (By Strategy), A Certain Ratio were formed by original vocalist Simon Topping and guitarist Peter Terrell, who performed their early FX-driven experiments as a duo. Kerr joined after seeing the pair’s first gig at Pip’s disco (behind the cathedral, as the TV ad made clear) in Manchester. A football protégé who had been on Manchester United’s books since he was a child, Kerr was looking for another outlet after a broken ankle forced him out of the game.

“I was at a bit of a loss and joined Manchester Youth Theatre,” says Kerr. “That’s where I met Jilted John and Gordon the Moron, who I ended up sharing a house with. Punk was happening, and I got chatting to Peter, then I bumped into Simon on the street, and he said they were looking for a bass player, and I had a bass, so that was that.”

The next gig saw ACR sharing a bill with a band called Alien Tint, described by their guitarist Martin Moscrop as a “crap glam punk band.” Moscrop was taken with ACR’s approach, and had already adopted their recession chic sartorial sensibility. They asked him to join, and it was as a still drummerless quartet that they were spotted by Joy Division’s manager Rob Gretton, and released their first single, All Night Party, on Factory Records.

Given their emphasis on percussion and dance rhythms after drummer Donald Johnson joined, such a move seems astonishing in retrospect.

“There was always a percussive thing going on,” says Johnson, who joined after Factory Records boss Tony Wilson turned up at his mother’s house unannounced shortly after Johnson had watched Wilson host local news programme, Granada Reports. “They played their guitars as rhythm instruments, and for me that all made sense. They were all quite shy, and they never looked at the audience, but in terms of opening things out, I was just one of the catalysts.”

A Certain Ratio did their growing up in public, with Terrell and Topping’s departure in 1982, marking a more outward-looking approach. Where the early incarnation of ACR sounded broodingly insular, they gradually came blinking into the light to create full-on club classics, from their cover of Banbarra’s Shack Up through to Latin percussion wig-outs and the heart-on-sleeve euphoria of Won’t Stop Loving You. Along the way they flirted with major labels before reaching a natural impasse in 1994 following the death of Gretton, who had released a couple of ACR records on his own label.

As the world caught up with them, ACR regrouped, playing occasional gigs, before releasing an album of new material in 2008. With ACR’s current line-up fleshed out by long-standing sax player Tony Quigley, some-time Primal Scream chanteuse Denise Johnson and recently joined keyboardist Matt Steele, Mute’s recent archive releases have been an important taking stock for the band. Beyond their fortieth birthday, however, their eyes are very much on the future.

“Mute are such a great label,” says Kerr, “and they made us realise how good some of the stuff we’ve done actually is. You listen to the old stuff, and you realise that no-one really knows it, so it’s great to have things out there on a proper footing. What we’re really concerned about now is doing a new album. Now we’re back out there again, this time out I reckon A Certain Ratio will carry on until we drop dead.”

A Certain Ratio plus Sons of the Descent play the Voodoo Rooms, Edinburgh, June 1. acr:box is available now on Mute Records.