It's not entirely clear where we are. But the drab office environment and sepia tone suggest somewhere in deepest provincial west Germany. It looks like the 1970s, or perhaps the early 80s. There are booths with headphones and a teacher's console. To the side is a piano and a cascade of steps. A professor is off-stage, cursing, having lost his slippers. Seven students troop in and take their seats, and the lesson – if that's what it is – begins.

This is the curious world of Meine Faire Dame: Ein Sprachlabor, a new version of Lerner and Loewe's much-loved musical My Fair Lady. Translated into German and gibberish (with smatterings of English remaining) by the Swiss-German director Christoph Marthaler, it is showing in Edinburgh till Saturday. Translation is very much the point: as the subtitle suggests, the setting is a language lab; Professor Higgins has become Professor Karpathy, not an English phoneticist but a Hungarian one; and there are three Eliza Doolittles, each played by different actors.

What follows is two hours of somewhat eclectic music (Bryan Adams, Wagner, Mozart, Marlene Dietrich) chopped together with fragments translated from the original musical. Frankenstein's monster makes an entrance, too, for reasons that remain a little unclear. Although this is a deliberately absurd reworking, it is nonetheless touching, melancholic and grotesque. Nothing is what is seems. Perhaps that's its message: communication in any language is impossible.

"It's about society and emptiness," says Tora Augestad, a mezzo-soprano who plays one of the students. We speak after a performance in Valence, France, on the European tour. "And language. We are trying to question what language is, and how accurate it is."

Over the past two decades, Marthaler has won just about every award going in the German-speaking musical world; his staging of Wagner's Tristan und Isolde has been performed in Bayreuth for several seasons – sometimes to jeers from the audience. But then his productions are famous for their gentle irony, their ridiculing of theatrical and operatic conventions, and their unheroic protagonists.

The idea for Meine Faire Dame came in 2010, when Marthaler was working as director of Theatre Basel. On the main stage, a traditional production of My Fair Lady was being staged, so Marthaler offered to do a response to it in the smaller space next door. He began with singing and improvisation ("We always start from the music," says Graham Valentine, Scottish actor and long-term Marthaler collaborator playing Karpathy) and ended with something approaching dada.

It premiered in 2010 to a rapturous response from audiences and critics alike, uncertain whether they were watching a play, a comedy, or a pantomime. The Frankfurter Allgemeine's critic wrote "for a full two hours we laughed … until we cried"; the Süddeutsche review knowingly referenced both John Cage and the French composer Erik Satie.

One of the most amusing moments comes when Augestad does a duet of Silent Night with the Swiss pop star Michael von der Heide, who had the distinction of coming last in the 2010 Eurovision song contest. The singing is exquisite. But Augestad and Von der Heide are choreographed to perform a series of painfully awful dance moves – an inverted parody of those X Factor contestants who can dance well but are hopeless at singing. At the end, someone cries: "Thank you. Next."

The show has a multinational cast: Augestad is Norwegian; the Frankenstein actor Mihail Grigoriu Romanian; Valentine, although a Scot, has lived in France for 30 years; and Marthaler lives in Paris and Basel. Situated between Germany, France and Italy, Switzerland is itself a country that knows the problems of linguistic confusion – a theme that may make sense in Edinburgh, too, according to Valentine. "Scottish people have a deeply ingrained sense of alienation from their own language," he says. "Turn on TV. You are bombarded with this English shite. You never feel you have the right to be in your own language."