Graeme Thomson

Adam Cohen knows the score. "For years I came to accept that every interview I did would show at worst a morbid curiosity and at best a healthy interest in my father's work," says the 42-year-old Canadian singer-songwriter with a dry half-laugh. "We have a very close, loving relationship but, you know, it's not something I want to talk about all the time."

When your father is legendary poet and songwriter Leonard Cohen, the shadow looms large and long. The son of Cohen and American artist Suzanne Elrod, Adam was brought up steeped in what he terms "the family business". On the one hand, there could hardly be a better environment for an aspiring songwriter. "I couldn't look at a spreadsheet and tell you if someone is stealing from me," he says, "But I can listen to a song, or read a book, and tell you whether the writer, or singer, or author is telling the truth."

On the other hand, it has taken a while to find a position of usefulness within the firm. Cohen signed to Columbia in his mid-20s and released his first album in 1998, but realises now that his priorities when he started were all askew. "When I began making records, I was more worried about having the right kind of tight jeans and silly hair," he says. "I had the balance between the artistic and the commercial all wrong, I was more focused on what would work on radio. It took me a long time to find my voice."

Between 2004 and 2011 Cohen didn't release any records at all, and even contemplated giving up music entirely. When he did eventually return, it was with Like a Man, a more considered reflection on his life and art which marked a bold leap forwards. His fifth album, We Go Home, is even more impressive. Rather than seek to distance itself from Cohen Sr., the album boldly invites comparison. Not only was it recorded at Adam's childhood homes on the Greek island of Hydra and in Montreal, both locations ingrained in the folklore surrounding his dad's work, but the lyrics make reference to "hallelujah" and "taking Manhattan". The music has a familiar burnished elegance, while Fall Apart begins "they will speak of my father." Indeed they will. How could they not?

"Yes, those references are deliberate," says Cohen, who is either blessed or cursed, depending on your perspective, to share his father's wry eloquence, and to sing with a similar conversational intimacy. "There are lots of them, some are obvious and some more private. I longed for years to have my work be deserving of being in dialogue with his, and now I feel it is. It is partly me feeling more confident, but it's also part of what the album is about. It definitely has a theme. The last album was about being a man, becoming who I wanted to be, finding my voice, and this record is about raising my voice and talking to my friends, my family."

Despite the complications of being a Cohen, he regards himself as lucky beyond words to receive advice beamed direct from the tower of song. "Because I went into the same shabby line of work, like an apprentice in the family business, I've always recognised what a privilege it is to have my father's ear and have a master lean over my simple notebooks," he says. "Since the very beginning he's been vetting my adjectives, prepositions, nouns, similes, metaphors - and he plays me his new music, always. It's a pleasure to talk shop with my old man and tell him the sections I think work best, or maybe suggest stuff when he's playing demos. Sometimes he actually incorporates some of my thoughts, which is beautiful. It's incredibly gratifying to find that I'm a reliable interlocutor."

Cohen views We Go Home as the second part of a trilogy which began with Like a Man. "The next album will be more perverse," he promises. "This is like a sturdy stem that grew from a healthy seed; the next will be a beautiful flower." He laughs. "I just need the terrain to be more welcoming." By which he means, having found his voice, he's finally ripe for some commercial recognition. Basically, he's ready to do battle with Bieber? "Oh yeah," he says laconically. "Really, I just want to be fighting over women in a bar with Orlando Bloom."

For his Scottish dates this week, Cohen will be joined on stage by some stellar string players - and, of course, a whole heap of baggage which appears to weigh a little less than it used to. "For anyone coming along because they have a morbid curiosity about my name, that's fine," he concludes. "And for anyone who wants to hear great, modern folk music, I promise they won't be disappointed."

Adam Cohen plays Voodoo Rooms, Edinburgh, on Wednesday, and Oran Mor, Glasgow, on Thursday. We Go Home is out now.