Next week the members of the Tokyo String Quartet perform their last European concert together – not in London or Paris or Milan, but at the old kirk in Crail, south-east Fife.

As the opening event of this year's East Neuk Festival, it marks the end of an era.

After an incredible 44 years of play, the Tokyos are disbanding, leaving behind one of the classiest chamber music legacies of the past half-century. The sheer length of service is impressive enough; what's perhaps more remarkable is that the Tokyos have weathered more than four decades together without any hint of the personnel histrionics or patchy performance quality that shakes most long-term ensembles over the years.

Theirs is an old-fashioned, rock-steady kind of ensemble spirit: four gentlemen, often photographed wearing matching outfits (black polonecks, white scarves), whose sound is big-boned and burnished, whose interpretations are thoughtful and robust, whose stage etiquette is exquisitely honed. Even the manner in which they're breaking up has a gentle, respectable formality to it.

The quartet's Japanese founding members met as students in the 1960s at Tokyo's Toho School of Music and emigrated en group to study at the Juilliard School in New York. They emerged as the Tokyo String Quartet in 1969, snapped up several top-flight prizes in quick succession (the Coleman Competition, the Munich Competition, the Young Concert Artists International Auditions) and soon signed an exclusive recording contract with Deutsche Grammophon.

Only one of those original members, violist Kazuhide Isomura, remains in today's line-up. Second violinist Kikuei Ikeda joined in 1974; English cellist Clive Greensmith in 2000; Canadian first violinist Martin Beaver in 2002. The members have changed and the sound has unquestionably evolved – largely for the better, especially since the earliest days – but somehow the quartet has retained what Beaver refers to as its "core philosophy".

"The Tokyo approach is and has always been to bring out the composer's intention in the music," he says. "The music comes first. We have never used any piece as a vehicle for ourselves, either as individuals or as a group. That's the cornerstone of the way we make music."

The approach might sound obvious – shouldn't classical musicians always prioritise the composer's intentions? – but it's rarely executed with such integrity. Since the 1980s the quartet has contained at least one non-Japanese member, with violinist Peter Oundjian, current music director of the Royal Scottish National Orchestra, the first to join as leader in 1981.

The two remaining Japanese members have often spoken about the changes that came with the new language: when they rehearsed entirely in Japanese their manner was polite and reserved according to linguistic custom; when English became the working language, the conversation became a little less formal. The freedom was a positive shift, they say, but to my ears that poised courtesy is still inherent in the group's musical manners.

Over the years the Tokyos have become some of the elder statesmen of the string quartet world, winning the respect of audiences and the music community globally with their extraordinary ability to ride out personnel changes without ever denting that precious core philosophy. They seemed like one of classical music's timeless institutions that would somehow be around forever.

Then one day Isomura turned up at rehearsal and quietly, carefully explained to his colleagues that he was thinking of quitting the band. "That was almost two years ago, so it's not as if he just jumped it on us at short notice," says Beaver with a laugh. "He sim ply felt that after 42 years on the road it was time to spend more time with his family. Or rather visiting his family, because his kids are all grown up now. Four decades of service seemed like a reasonable innings. Of course we were all very understanding."

A short while later Ikeda announced that he, too, had decided to go. "For his entire career he had played beside Kazu and couldn't imagine playing with anyone else. His decision spoke to the depth of their musical relationship." It also left Beaver and Greensmith with the daunting task of replacing not one but both of the group's long-standing members.

"Finding one new member of a string quartet is an extremely delicate and difficult thing to do, let alone two," says Beaver. "For a while we considered how we might keep things going. Should we try to recruit Japanese members in order to keep the link with the group's heritage? Should we change the name altogether? Ultimately we decided that the best thing would be to bring things to a graceful close."

When we speak Beaver has just arrived in Perth, Australia, for the group's last-ever Australia tour. "Everything we're doing at the moment has a finality to it," he says. "Our last London concert, our last New York recital. Some of it I won't miss a bit. Our last 24-hour flight, for example. The constant time away from home, the airports. Travelling isn't what it was when the quartet first went on the road; back in the 1970s you could just walk on to a plane, but now the whole rigmarole of travelling is constantly stressful, especially when you're trying to get a cello properly looked after and the violins packed in the overhead luggage. We travel all the time but that doesn't mean we get to be relaxed about it."

What it does mean is that the four members of the quartet spend more time together than they do apart. And they play like it. Listen to any recording of the Tokyos – there are more than 40, including the complete quartets of Beethoven, Bartok, Janacek and Schubert – and you'll hear a profound depth of understanding and respect between the musicians that only comes with years upon decades of day-in, day-out ensemble work.

Even the Tokyo's instruments are a family: the Paganini Quartet, so-called because all four instruments were once owned by the Italian virtuoso violinist Niccolo Paganini. It's one of only a handful of complete sets made by Antonio Stradivari and is loaned to the Tokyo's members by the Nippon Music Foundation. After the quartet's final concert, the instruments will have to be returned and reallocated to another group. As yet the lucky recipients have not been announced.

So it's all change for the last generation of Tokyos. "The split will be a massive wrench for each of us," says Beaver. "During the 11 years that I've been with the quartet, it's been an all-consuming commitment. I can't begin to imagine how I will feel afterwards."

Before he joined, Beaver had a busy diary as a soloist; "since then I must have played four or five concertos at most – quartet life has just been too busy." Has he missed the solo work in the meantime? "Not really. The depth and breadth of quartet repertoire has kept me more than fulfilled. A normal quartet programme is like playing three concertos' worth of music all at once, and the repertoire is usually far more meaningful."

After the group say their farewells in July, Greensmith and Beaver both move to Los Angeles to take up teaching positions at the Colburn School of Music, while Isomura and Ikeda will remain on the East Coast to teach at Yale and the Manhattan School. "Clive is calling it a 'soft landing', us heading west together like this. Going cold turkey might have left us each too bereft." There's a weariness in Beaver's voice – the quartet has been touring heavily during this final season and he's tired – but his words are plainly heartfelt.

How, then, to choose the exit music? On the programme of next week's Crail concert is Mozart's Hoffmeister Quartet, Webern's Quartet in E and, to close, Schubert's String Quintet in C with Scottish Chamber Orchestra cellist David Watkin. It's the last piece of chamber music that Schubert wrote, completed just a couple of months before he died – "we thought it an appropriate choice", says Beaver. "Beautiful music tinged with finality. This entire season has been bittersweet."

Three days later the Tokyos play their very last concert at the home of Yale Summer School in Norfolk, Connecticut, with a programme that includes Haydn, Bartok and the shimmering Debussy quartet. "I dare say there might be calls for a couple of encores afterwards, and probably a glass of wine or two. I've got to drive back to New York that night and fly to the west coast early the next day for other concerts. I guess the first day of the rest of my life starts then."

The Tokyo String Quartet performs at the East Neuk Festival on July 3, www.eastneukfestival.com