The all-male vocal quartet - known for their icy-pure tone, their piercing attack, their soaring, spacious phrasing in repertoire from Perotin to Arvo Part - celebrated 40 years together in December and are marking their birthday season with a 12-month global retrospective. They are also breaking up.
Their last concert will be on December 20, after which the four members go their separate ways. It will mark the end of one of the most distinct and ground-breaking vocal ensembles of our age.
There has not been a major fall-out or an irreconcilable artistic rift. I have come to meet the group at countertenor David James's house in north London and, judging by the quietly daft banter going on around the kitchen table, they are all still good pals.
The decision to disband has not been commercially-driven, either: Hilliard concerts are wildly popular and their recordings for the German label ECM sell phenomenally well - some of their discs have sold more than a million copies, which in classical music terms is near-mythical.
So the obvious question: why end it?
"And the obvious answer is, well, age," says tenor Rogers Covey-Crump. "We started talking about it a few years ago. Three of us are getting on. Sure, we could pass on the baton to Steven (Harrold, the other tenor and the youngest of the group) and he could keep the Hilliard name going like a kind of franchise. But to replace three quarters of the group all at once? It would not really be the same Hilliard Ensemble afterwards."
Having decided to quit, next they had to figure out how to go about it. The classical industry tends to work a year or two in advance so breaking up takes significant forethought. "The 40th anniversary seemed a natural end point," says James. "By focusing on our birthday we have been able to stay positive rather than gloomy."
But the prospect of the final concert is daunting, says the group's baritone Gordon Jones. "We already know it will be emotional. A lot of the composers and performers whom we have worked with will be there. Do you think we will be able to sing anything much at all?" he asks the others, who all shake their heads. "Maybe we will have to keep the lights down on stage. Maybe we should book a tribute band to be waiting in the wings..." They have already decided to meet up for lunch the following day.
Until then the Hilliards are packing in as many concerts as possible, with worldwide tours and revivals of some of their most popular projects. This week they are in Scotland with the collaboration that shot them to global fame, sold 1.5 million albums and arguably (whisper it) kick-started an entire dubious musical genre.
The story of how the Hilliard Ensemble first paired up with Norwegian saxophonist Jan Garbarek goes back 20 years and involves one of the record industry's most innovative producers, Manfred Eicher.
The quartet had joined Eicher's label ECM in the 1980s to record the music of Estonian composer Arvo Part (they had given first refusal to their previous label EMI whose bosses decided Part would never sell - ha!). Enter Garbarek, also an ECM artist, a few years later.
"Manfred likes to encourage collaborations between his artists," says James. "He likes to think of ECM as a kind of glorified family. So he approached us to ask who we would be interested in working with. John Potter (the Hilliard's then-tenor) thought of Jan.
"Then Manfred came back saying he had had a vision somewhere in the frozen north, that he had been driving his car through the tundra with two cassette tapes playing, one of Garbarek, the other of us singing Gesualdo. He said he knew instantly we had to work together."
Eicher arranged a meeting in the Benedictine monastery St Gerold in Austria, a regular venue for ECM recordings. "John Potter brought along a pile of music we could potentially sing," Jones says.
"He plonked this wodge of manuscripts on the alter. Jan was hanging about... nobody knew quite what to do, so John handed out the piece that was on the top of the pile. It happened to be Parce mihi Domine by Morales - a long, slow, funerial motet that lasts about seven minutes. We started singing, huddled around the alter while Jan was off wandering around the church. After about five minutes we became aware of another sound."
Garbarek had gradually joined in, using his saxophone to add a fifth voice to their four. "We finished the piece and there was stunned silence. Manfred leapt from his seat, ran to the altar and said we had to record it as soon as possible. We didn't sing another thing with Jan that day."
The overwhelming success of the resulting album Officium stunned everyone, not least the Hilliards themselves. Hundreds of concerts and two further albums with Garbarek followed (1999's Mnemosyne and 2010's Officium Novum). "It was completely unexpected," says James of the snippets of rock star treatment they experienced along the way. "It does not happen to people like us. Once in Greece we were met on the airport tarmac by a black limo. There was an element of panto about that kind of thing."
Jones says: "It was also funny watching ourselves being copied. I remember seeing some Swedish choir on telly - they were on a bus singing Parce mihi Domine with a saxophonist. I thought, well at least they have heard our record!"
The image that stands out, though, is of the Hilliards huddled around a little alter in the Austrian mountains singing Morales as Garbarek's lines emerged out of nowhere. Glasgow Concert Halls describes Officium as a "spiritual ambient classic": for many people, the album's echo-shrouded chant and drifting renaissance serenity heralded a whole line in new-age meditation music.
The Hilliards shudder when I ask what they think of the term 'cross-over'. Jones says he used to go into music shops and physically move their records from cross-over and new-age sections. "I would take our albums and put them somewhere else in the shop. Classical, jazz - I didn't really care. Anything except cross-over or new-age."
James says: "Because neither Jan nor us were crossing over. If anything, we were standing on the bridge together." What would have been a better term? "I don't know. It's just music."
Jan Garbarek and the Hilliard Ensemble are at Kelvingrove Museum, Glasgow, on Friday and St Mary's Cathedral, Aberdeen, on Saturday