At the far end of Chris May's office are photographs of Phil Mulford and Carlos Henriquez. Chris points to the photo of Phil. "That was taken on the set of The X Factor ," he says. "He's in the house band.

"Carlos is based in New York. He's primarily a virtuoso double bass player. He plays with [trumpeter] Wynton Marsalis."

Phil and Carlos are hugely accomplished musicians, and Chris is aware that most people have never heard of them.

But these men are among the customers he is happy to discuss. Other Overwater Basses' clients, such as Dave Gilmour of Pink Floyd and the late John Entwistle of The Who, are treated with more reticence.

"I don't want to go there," says Chris when I mention Dave Gilmour. It seems that some media coverage about Overwater has focused on the handful of rock stars to have played the company's instruments.

Chris feels this has drowned out the real story: four decades of crafting high-end bass guitars, played by leading session musicians on stage and in studio, as well as by skilled amateurs.

"People like Phil and Carlos are archetypal customers, and the lynchpins around which the business flows," says Chris. "They're musicians' musicians. That's really how it works. I've never chased after rock stars. I've almost tried to avoid that."

Most businesses with products used by famous people are delighted to publicise the fact. But Overwater is understated, like the bass guitar itself. In the background, holding things together.

"Bass players are fundamental to everything," says Chris. "Without the bass, it all falls to bits. In terms of pop music, bass players' egos are slightly less. Although there are those that belie that statement!"

A conversation with Chris and his partner, and fellow Overwater director, Kirsty Cowe features not so much name-dropping as gentle name-wafting.

Their clients include: Trevor Barry, who plays in the Strictly Come Dancing house band. Steve McManus, who played on The Lord of the Rings soundtrack. Sandy Beales, who played live with One Direction. Marcus Cliffe, who has played with Eric Clapton and Rod Stewart.

And our old friend Phil Mulford: Phil is likely to share the stage at Brunton Park this summer with Lionel Richie. Beyoncé is another star who sings while Phil's Overwater drives the rhythm section.

Chris declines to name any of the hits on which his instruments can be heard, saying only: "If you've watched the telly, if you've listened to the radio, you'll have heard one of our basses."

Kirsty mentions the weekend last November when she and Chris saw The X Factor , Children in Need and Strictly Come Dancing . A friend pointed out that the 32 million people watching those programmes will have heard Overwater instruments being played.

The 'quietly going about its business' theme is reflected in the company's location. Its workshops and offices are tucked away in Atlas Works: the Victorian cotton mill in the Denton Holme area of Carlisle. This interview is conducted not to a soundtrack of bass guitar but of whirring and buzzing from other rooms.

Overwater has been here since the early 1990s. The story really started 50 years ago, when Chris built his first guitar at the age of 16.

In 1977 he and his then business partner Andy Preston began working from London's Denmark Street, which was a centre for music publishers, recording studios and music shops.

They made and repaired guitars and basses. A couple of years later, having run out of space, they established an additional workshop in Newcastle.

From there to a unit near Alston. The name Overwater comes from a nearby village.

Chris decided to concentrate on bass guitars, earning a reputation for quality and innovation. "I'd always had a bit of a thing about basses," he says. "I'd played bass as well as guitar.

"In the mid to late 70s and early 80s, bass players were starting to be taken more seriously. They were taking the instrument to new levels artistically. They were more open to new developments, like five-string bass for extra low notes.

"That came about by the advent of keyboards. Pop music was going lower, if you like. Bass players wanted to follow it down. We were at the forefront of those developments."

They still are, making new models influenced by musicians' requests and suggestions.

A quick tour reveals instruments in various stages of completion. Necks and bodies wait to be united. Templates carved in MDF will be transferred to ash, maple and the like. Chris points out some olive wood and declares it beautiful.

"The electrics are the conduit," he says. "But these are acoustic instruments first. The sound comes from the wood. Wood has a life to it."

Kirsty adds: "Because we do such an organic thing, it's not until you get the strings on that you hear the voice of that particular bass."

Who pays the £2,000-plus required to bring an Overwater into their life? As well as professional musicians, a big part of the business is what Chris calls "the Harley Davidson market, or the jazz lawyers. People who play bass and they like nice things, like Harley Davidsons. Very often they are good musicians but they can earn more money as a solicitor.

"The internet has changed everything. Our business has become much more international. We now sell much more out of the UK."

Kirsty says: "Last week we sent a shipment of five: two to Hong Kong, one to Japan, one to Canada and one to the US.

"We're not for Joe Bloggs who fancies trying the bass. It would be like someone who's learning to drive deciding to get a Jaguar. These are hand-crafted. Imbued with quality."

Another major market is musicians in the theatre world. This connection was forged during the Denmark Street days, with its proximity to the West End. Chris says: "Because we had a foothold in the West End we always dealt with the session world: the hired guns within the industry.

"Virtually every West End theatre pit has one of our basses in it. One reason we're so ubiquitous in the West End is we're so sound-engineer friendly. I've been told if you're in the pit at any West End theatre without an Overwater bass, the engineer moans at you.

"New York is almost a mirror image of London, with the West End and Broadway. At one stage the bass players in the West End and Broadway versions of The Book of Mormon were both using Overwater basses."

Overwater's small team makes "a hundred and something" instruments a year. Several local companies supply components. "A little craft industry" is how Chris describes the operation.

It sometimes seems to him and Kirsty that the outside world is determined to intrude. Chris brings up Brexit, saying those who voted for it "just haven't got a clue."

"We both feel very strongly about it," says Kirsty. "The UK is made up of many, many thousands of small businesses. We think it will have a big impact on our business. Probably 30 per cent of what we make is going into Europe."

Many countries outside the EU have prohibitive tariffs and strangling lengths of red tape. Overwater finds itself at the mercy of all kinds of complications. International restrictions on rosewood, intended to stop forests being plundered to feed China's luxury furniture market, mean it can no longer be used to make bass guitars in Carlisle.

"We're now using Indian laurel," says Chris. "And we're having to send paperwork stating that it's not rosewood."

Still, the positives weigh much heavier on the scales. These include out-of-the-blue phone calls from the likes of Simon Franglen.

Simon who?

Kirsty says: "He rang up and spoke to Chris. Chris was asking him 'What sort of thing do you do?' He tried to explain, then he said 'It's probably easier if you just Google me.'"

A quick search revealed that Simon was awarded a Grammy for producing Celine Dion's My Heart Will Go On , from the Titanic soundtrack. He arranged the music for the James Bond films Skyfall and Spectre and has worked with Michael Jackson and Madonna.

Shortly after the phone call Simon turned up at Atlas Works, and became another satisfied customer.