“There are some songs I’m half way through that are not bad,” Leonard Cohen said in his final interview, published in the New Yorker on 17 October 2016. “I don’t think I’ll be able to finish those songs.” Three weeks later, on 7 November, having released his 14th album, You Want It Darker, Cohen died in his sleep after a fall in his home in Los Angeles. The task of finishing those songs was passed, at Cohen’s request, to his son, Adam. The results can be heard on Thanks for the Dance, released last Friday.

“Essentially, I wanted to take the listener on an unconscious journey through the sonic signatures of my father’s career, without it sounding like a regurgitation,” says Adam Cohen, who, in his soft Canadian cadences and carefully constructed sentences, can sound uncannily like his father. “My dad always rejected invitations from producers like Rick Rubin and Don Was to make a retro record that sounded like his older stuff. He said he didn’t want to do the nostalgia act.”

What is immediately striking about Thanks for the Dance is how organic these meticulously constructed songs sound, given that five of them were assembled from the barest musical sketches and four others from lyrics Leonard had recited, but not put to music. It sounds, above all, like a labour of love, and one that is entirely faithful to Cohen’s abiding spirit.

“That’s good to hear,” says Adam, smiling, “For me, it is very much a continuation of You Want It Darker, on which I worked with him for the first time, when he was in quite bad shape. This, though, was a more daunting undertaking. We had to ensure that he was with us all the way as we give these songs a musical language that he had already developed.”

Leonard, Adam and Adam’s son, Cassius, photographed at Leonard’s house in Los Angeles, early 2016. Photograph: Courtesy Adam Cohen

While the previous album was produced in Leonard’s living room with, as Adam puts it, “Pro Tools on the table and my father in his medical chair”, the new album initially took on a rough shape in Adam’s converted garage before being finessed in recording studios in Los Angeles and Berlin. It was created with the help of a stellar cast of musicians, including Damien Rice, Daniel Lanois, Feist and Beck, as well as seasoned Cohen collaborators such as vocalist Jennifer Warnes and virtuoso guitarist Javier Mas. “Most records you struggle to make, but this was not hard,” says Adam. “It was, as the Arabs say, maktoob – already written. I was just a steward, a guardian of the project.”

Adam Cohen is a solo artist in his own right, though his career seems to be on hold while he takes care of his father’s legacy. “I’ve had people judge me through the filter of my father’s reputation,” he says, when I mention this, “and I’ve been on the receiving end of some swift and unforgiving judgments. So, as regards my own work, we’ll see.”

Thanks for the Dance opens with a song called Happens to the Heart and the first of several startling lyrics: “I was working steady but I never called it art/ I got my shit together, meeting Christ and reading Marx...” It sounds like a defiant statement of intent, but the mood, as on the previous album, soon turns darker. The Goal, a short, sombre spoken-word piece, begins “I can’t leave my house, or answer the phone” and contains the poignant self-observation: “I sit in my chair, I look at the street, the neighbour returns my smile of defeat.” The infirmities of old age are broached again on The Hills, on which he sings, “I’m living on pills, for which I thank God”, delivering the last three words with heartfelt emphasis.

Throughout, death is a given, its looming presence articulated with a characteristic degree of Buddhist acceptance. Here and there, though, there are glimpses of the sensuality that characterised several of his early songs and poems. On The Night of Santiago he sings: “Her thighs they slipped away from me/ Like schools of startled fish”. The song is actually an adaptation of a poem by Federico García Lorca, one of Leonard’s literary touchstones. “It’s one of my favourite poems and I begged him to read it,” says Adam.

On an album of powerfully personal songs, the first line of the spoken word Puppets is like a punch from a velvet-gloved fist. “German puppets burned the Jews,” Leonard intones, against a brooding backdrop of synthesiser and voices. The poem is a litany of accusation – “Puppet troops command/ Puppet troops to burn the land” – offset by a chorus from the combined voices of the Shaar Hashomayim choir and Cantus Domus from Berlin – “Germans and Jews singing together as if in a classical Greek play,” as Adam puts it.

On the regretful Moving On, it feels like Leonard is addressing someone specific, perhaps a former lover “who broke the heart and made it new”. Adam nods and smiles. “Yes, that is indeed the case. When he was recording that vocal, he had just learned of Marianne’s passing and I remember him adapting a few lines of the song.” As every Cohen devotee knows, Marianne Ihlen was the woman he met on the Greek island of Hydra in 1960 and later immortalised on one of his best-loved songs, So Long, Marianne. In July 2016, he learned that she was terminally ill and sent her an email via a friend. It was read to her just before she died. “I’m just a little behind you,” he wrote, “close enough to take your hand.”

Moving On, then, is a moving postscript to their fitfully intertwined narrative. “There was,” says Adam, “a feeling of communion in the room when he recited the lyrics, a kind of revisiting. We spent an hour or so, going over and over the words. It was as if we were trying to bring the Greek taverna back to life, and conjure the ghostly presence of a woman in the next room.”

It seems a good moment to ask if Adam’s father and his mother, Suzanne Elrod, were on good terms. The two were together for more than 10 years, having reputedly met in a Scientology class in 1969. She is the striking woman to the right on the cover of Cohen’s 1977 album, Death of a Ladies’ Man. My question prompts a wry smile. “I wouldn’t characterise it as good terms exactly,” he says, “but he was always gracious and devoted and supportive of my mother right to the end.”

Adam Cohen is disarmingly honest about the emotional complexity of his creative collaboration with his famously single-minded father. “He was an extraordinarily generous person,” he says. “But if I was playing his guitar or his keyboard, he would say: ‘It’s like you’re using my toothbrush.’” Did working together so closely give him a deeper understanding of his father as a person as well as an artist? “Well, I almost feel that we were closer before we started working together. You have to understand there were other things at stake – this was his career, his art. So there were tensions and demands. I remember he once said: ‘I’ve compromised everything in my life, being a father even, at the service of trying to be a writer and blacken pages’.”

A young Adam with his father.

Did he say that with regret? “No. He was saying, ‘This is the way it has always been for me, and I’m sorry I wasn’t a better father, but my creative mandate was a call of duty I couldn’t ignore, the loudest call of duty of all’. That mandate was to go into the darkness and emerge with something for the rest of us.”

How is he coping with his father’s death? Is he still in grief? “Well, it’s more like disbelief than grief. I still can’t believe that I don’t have that source of wisdom to consult, and that I have to have imaginary conversations sometimes rather than real ones. But mostly it’s just a case of missing one’s old man.”

At times, Thanks for the Dance has an ambience that sounds almost haunted. It’s there on the ghostly chorus of Puppets and, throughout, in the grain of Leonard’s ageing voice. And, it is perhaps most evident on Listen to the Hummingbird, the short, spoken word piece that ends the album. Its serene sense of intimacy is surprising given the circumstances of its creation.

“Hummingbird was actually recorded last,” explains Adam. “We needed another song and I remembered that the last time my father spoke in public was at the press conference for the release of You Want It Darker. As he was prone to do, he recited a poem that he was working on. Impromptu.” Adam managed to track down a recording of the recital from the vaults of Sony Records. “It’s that vocal, recited in a halogen-lit press conference, that I manipulated to make it sound like a studio recording.”

If this is to be Cohen’s final word as a recording artist – and his son insists that “there is no more material” – it offers a wry last laugh from a man whose dark humour often went unappreciated. The last words on the final Leonard Cohen album are: “Listen to the hummingbird/ Don’t listen to me.” It is, says his son, “very modest, very Buddhist and very funny”. Very Leonard Cohen, in fact.

Thanks for the Dance is out now on Columbia Records