"We got ourselves on the David Bowie fan website, which I thought was quite an achievement," he says.
He's pleased with his group's name, too. The original red priest was Antonio Vivaldi, so titled for the colour of his hair and his brief tenure in the clergy. Like Red Priest's album titles, the name serves a dual purpose, acknowledging tradition - and the quality of Vivaldi's recorder music ("he wrote the absolute virtuoso fireworks," says Adams) - while also tipping its hat to modernity.
"It takes a bit of explaining," says Adams, "but those in the know immediately recognise the reference to Vivaldi. For everyone else, it sounds like a prog rock band."
Adams reckons that's a nice fit with Red Priest's ethos: "The group is about being creative, having fun and doing whatever comes to us and not being bound by a lot of the rules."
Period instrument groups have more rules than most. In the 1960s and 70s, the leading figures of the authentic performance movement could be painfully dogmatic about how Baroque music should be played; the method became madness.
Adams claims he's not much of a music scholar and while he studied the relevant treatises and what he calls the "this-is-how-you're-supposed-to-do-it" stuff, Red Priest has moved away from the strictures of the authentic performance adherents.
"What we do is based in historical music performance but then we've added to that and reinterpreted it and tried to get away from the idea of just reproducing something that might have been done back then. It's all about creating music as speech, music as theatre and communication and genuine emotion. So much of that gets lost if you're too bound up in the detail."
Besides, Adams reasons, there's a paradox in the very idea of copying how music was performed in the 17th and 18th centuries.
"Over the years the authentic music world has tried to look back and thought, 'How did these people do it?' We're saying that's not very authentic, because in Baroque times no one was trying to play music in the style of somebody else. People were experimenting and being themselves."
Which is why Red Priest plays Bach's fifth Brandenburg Concerto as a rambunctious recorder quartet, rather than the work for three soloists and orchestra the composer envisaged.
The group brings that piece on its New Zealand tour, where audiences will be offered two programmes. The Brandenburg appears in Truly Madly Baroque, a new show Adams describes as a "back-to-the-roots programme" of works by Vivaldi, Albinoni and other well-known composers. The other programme follows the theme of the group's latest recording, The Baroque Bohemians, and features composers influenced by gypsy music, including Telemann.
Less famous is a work by 17th century Polish composer Mielczewski, who was fêted in his time but quickly fell into semi-obscurity. Non-scholar Adams found the piece in a museum library and Red Priest gave the first known performance in modern times.
"That's pretty exciting, particularly when it's an obscure composer nobody's heard of, and it's in a kind of obscure musical style. It's quirkier than the high Baroque stuff of the 18th century but if you play it with real balls, it's quite a jaw dropper."
Lowdown
What: Red Priest nationwide tour for Chamber Music New Zealand
Where and When: Auckland Town Hall Concert Chamber, March 19. For other concerts see chambermusic.co.nz