Over the past 20 years or so, no single figure in British music has consistently afforded me more pleasure and entertainment than Gruff Rhys: I loved Super Furry Animals from the moment I first encountered them. It would have felt odd not to, like forming an aversion to a basket of semi-feral puppies who could do magic tricks. His two albums as one half of Neon Neon are marvellous things; I consider myself fortunate to have seen the staging of the second, Praxis Makes Perfect, a concept piece about the Italian publisher and leftist militant Giangiacomo Feltrinelli. Even The Terror Of Cosmic Loneliness – his 2010 collaboration with Tony da Gatorra, Brazilian inventor of a bizarre guitar/drum machine hybrid – exerts an odd fascination despite demanding both considerable patience and reinforced eardrums. For me, Rhys is the musical equivalent of one those actors who are always worth watching, in anything at all.

The Super Furries are a band, of course, and while Rhys is their face and their voice and, in essence, their avatar, he and they are not interchangeable. The Super Furries are organic and multifaceted, with extraordinary range and invention - the ability both to work in pastiche and to create their own musical world, all glued together by the sole common factor of Rhys’s relaxed, faintly psychedelic lilt. On his own Rhys is a singer-songwriter with a band assembled around him - the music feels like a reflection of his voice, rather than a rich and varied confection in which that voice is one ingredient. He is a troubadour, inclined towards the folkish, and his solo work is slighter, and lighter – which is not a shortcoming. The lightness of it makes it feel as if he is floating freely, quite by himself, untethered to anything of equal mass, propelled only by the currents of his own whims.

But where you have whims, you risk whimsy. A colleague whose views I hold in high regard complains of Rhys’s “self-mythologising whimsy and feyness”, a sentiment I don’t share, but which I do understand. It set me to thinking that self-mythologising is fundamental to pop music, and has been from the start. We welcome in it in those artists we enjoy and admire, while condemning it in those we dislike. Perhaps one reason for the insipidity of the present UK pop mainstream – or a consequence of it, it’s impossible to say – is a tendency in our biggest acts to shun such self-mythologising because ‘it’s all about the music’. Or perhaps their putative self-effacement is a form of self-mythologising; if so, it’s a ditchwater-dull one.

I like Rhys, so I like his self-mythologising – and indeed his mythologising in general. Where those who don’t care for him hear feyness, I hear delicacy and charm. I saw his American Interior show, more a piece of miniature theatre than a gig, every chance I got (indeed, I’ve never missed a chance to watch him perform, and never been disappointed). Certainly, Rhys invoked a mischievous sense of kinship with its subject, John Evans, a quixotic Welshman who went chasing after fairy stories masquerading as occult knowledge, yet had real and remarkable adventures and made genuine discoveries. Onto the bare bones of what little is known of this explorer’s history, Rhys layered apocryphal flesh, with real feeling as well as a wink.

Four years on, Babelsberg is his most richly realised pop record, redolent of those baroque, string-laden and frequently country-inflected sounds that somehow insinuated themselves right into the American heartlands in the 1960s when wild cards such as Lee Hazlewood, Jimmy Webb and Van Dyke Parks were permitted near the controls. If Rhys’s earlier albums often had a coltish or even foal-like aspect, teetering here and there on spindly limbs, this one feels much more assured.

He still sings in riddles that are perhaps less cryptic than they sound, images of contemporary life in distinctly un-contemporary musical settings, switching between his baritone croon and his airy tenor as the mood demands. “On the frontier of delusion / I’m your foremost frontier man,” runs his opening line; the song is ‘Frontier Man’, brimming with showband horns and female backing vocals. It recalls that Nashville-meets-the-counterculture sound: faintly cheesy and satisfyingly strange, as straight America absorbed the fumes from the netherworld. Then comes ‘The Club’, which is all Scott Walker-esque melodrama, or at least as close as Rhys’s languid manner can get, and whose grievance – the narrator is evicted from the titular club – might be literal rather than metaphorical, so much does it sound like the indignant, drunken protestations of anyone chucked out from a nightspot.

Babelsberg, you soon realise, is a kind of primer of those pleasingly eccentric, non-youth-oriented mainstream pop styles that flourished in the States from the late 1960s to the early 1970s, somewhere between Glen Campbell’s first flush of success and AOR’s first incursions, via MOR’s heyday. ‘Limited Edition Heart’ has a lovely, smooth Carpenters swirl and depth to it. There is more than a touch of early Steely Dan to ‘Take That Call’. ‘Architecture of Amnesia’ might have Father John Misty checking his wallet, but Rhys has been craftily plundering the same sources for a good while now. A duet with Lily Cole, ‘Selfies In The Sunset’, makes a poignant coda, as though Nancy and Lee replaced their archness and menace with a meditative, wistful air of acceptance.

Throughout it – as throughout his career, with others and solo – flows Rhys’s gift for melody, which seems to rise from him as easily, naturally and endlessly as water from a spring. In the end, you can’t beat a good tune or ten. And when they carry this much imagination with them, it becomes a form of gentle wizardry. Any enchantment is bound to fall on deaf ears, at times; mine remain tuned in to Rhys, and I’m glad of it.