The Edinburgh Quartet is changing.

To say this about Scotland's only permanent, and by far its longest-enduring, string quartet may sound ominous to listeners who like things as they are. But it is not necessarily saying something nobody knows. The players, in their prolonged search for a new leader, a new Edinburgh venue, and a new group attitude, have been for some years in such a constant state of flux that change, however unostentatiously, has become their running theme.

But at least, after a series of trials and errors, they did find their new leader not so long ago in the person of Tristan Gurney. He may not yet be an internationally-famous name in the world of string players, but it's certainly a striking moniker for any musician to possess, with its overtones not only of Wagner but also, more discreetly, of Ivor Gurney, one of those tantalisingly gifted composers of the First World War with a penchant for the poetry of Housman, a life of suffering (he was both wounded and gassed), and a career that ended in madness.

Tristan Gurney, as a former first violinist in the Northern Sinfonia of Newcastle, could be said without too much exaggeration to have ambitions more Wagnerianly assertive than quietly Gurneyesque, but a bit more assertion is surely what, for some time, the Edinburgh Quartet has needed. An alert young string player of already ample experience, Gurney retains a life outside Scotland as head of strings at Newcastle University but it is in Edinburgh and in the rest of his Scottish responsibilities that he is meeting most of his challenges.

This season the problem of finding a satisfying new venue has at last been resolved, or so it seems, through a move from Canongate Kirk – which itself had represented a potentially interesting move from such established surroundings as the Queen's Hall and Stockbridge Church – to the previously untried setting of St George's West, a 19th-century church with a lofty corner tower in Edinburgh's West End. Churches, with their often-excessive resonance and lack of acoustical focus, are not always the best of homes for string quartets, as most players well know. Will this one, in a city manifestly lacking perfect premises, actually prove any better?

The players think yes, in that they can now position themselves almost in the round, with their audience sitting close to them on three sides of the performance area. This means not only better sight lines for the public – something churches tend to lack– but a guarantee that listeners can sit within the ambience of the performance and be less perplexed by the music's tendency to travel everywhere except directly into their ears. "It's a clean, big sound," say the players, who no longer regard delicate intimacy of tone – which at one time made audiences feel themselves to be eavesdropping on their concerts – as their hallmark.

It's for its first season in its new surroundings, moreover, that the quartet has found a forthright new violist, Jessica Beeston, who returned to Edinburgh last week after a pre-arranged American tour performing Beethoven symphonies with the meticulous John Eliot Gardiner's style-conscious Orchestre Revolutionnaire et Romantique. If her name sounds familiar it is partly because her father is Michael Beeston, the quartet's long-serving viola player, who retired last season after 40 dedicated years in his post. Though it is outside Scotland, especially in Salzburg and London, that she has gained most of her expertise, she is an Edinburgh woman whose childhood was spent in a house – her father's in Craigleith – where the quartet frequently rehearsed and she was seldom out of earshot of what was going on.

"Why do you want to play the viola when you could play the violin?" was a question her father, fine violist though he is, reputedly asked her more than once. In reply, she doubtless reminded him that Bach, Mozart, Britten and Dvorak were all violists. But in the end, she says, he had to accept her decision and "things are all right now." Her fellow players, talking to me during a recent coffee break, said they see her as a "fantastic acquisition, with just the right colour of tone."

Nevertheless it was as a violinist that she first studied at the Guildhall School of Music in London before her moment came to go to the Salzburg Mozarteum and work with the distinguished viola teacher Thomas Riebl. Today, as a very positive violist, she has played with the Camerata Salzburg and England's sensationally-versatile young Britten Sinfonia, based in Cambridge.

A guest appearance a few years ago with the Edinburgh Quartet, sitting beside her father for a performance of one of Mozart's great viola quintets at Hopetoun House, South Queensferry, was conspicuously prophetic. Though her appointment as Michael's successor was "not yet on the cards", and it took about a year for everybody to decide what to do, it is clear that they are all, including her father, delighted. The moment she returned she was swept into the complete Beethoven cycle –17 quartets in six concerts – which is the ambitious theme of the current season. This week's concert, with the Grosse Fuge as an abrasive aperitif and the final version of Opus 130 as piece de resistance, will show her ability to plunge confidently into what can only be called the deep end.

Thereafter, from season to season, it will be a matter of all four players – with Philip Burrin (second violinist) and Mark Bailey (cellist and pupil of Paul Tortelier) still happily in situ – achieving the mutual clairvoyance that is the vital attribute of all successful quartets, and of fulfilling all the EQ's other aspirations. These include an expansion of its multiple university work, with a new residency in Aberdeen, further developments in Edinburgh and Glasgow, and increased emphasis on British music, with forthcoming cycles of the quartets of Britten and James MacMillan. There is also the prospect next spring of a fortnight in China, where they will perform the Glasgow composer Edward McGuire's recent work inspired by the Chinese writer Chiang Yee's exquisite book entitled The Silent Traveller in Edinburgh.

The Edinburgh Quartet continues its Beethoven cycle at St George's West, in the city's West End, today.