It comes as something of a surprise, 22 pages into The Book of ABBA, to find yourself reading about Heinrich Himmler. But there he is, in between a description of how Agnetha Fältskog and Björn Ulvaeus met and an explanation of the youth cult raggare, Sweden’s answer to rockers or greasers (the raggare’s 60s band of choice were the Hep Stars, featuring a pre-Abba Benny Andersson, surely the least likely musician in history to have once provided soundtrack for leather-clad, booze-fuelled gang warfare). There are details of Himmler’s “breeding policy”, lebensborn, and of its influence on both the plot of Hiroshima Mon Amour and a grim sub-genre of porny Nazi-centric pulp fiction that proliferated after the second world war.

There is, it should be noted, a link between all this and Abba – Anni-Frid Lyngstand’s mother was Norwegian and her father a member of the occupying German army – but, nevertheless, notice is served that Melancholy Undercover is not the book you might expect. A series of essays rather than a chronological history, it certainly covers all the bases, from Eurovision to the groundbreaking “virtual concert” Voyage, alongside global success on a scale even more staggering than you might have realised: Abba, it turns out, were huge in 70s Afghanistan, and so big in Vietnam that one journalist suggests their profoundly morose 1980 track Happy New Year is “probably the [country’s] most revered song … after the national anthem”.

But the book also finds room to delve into such unlikely areas as Swedish birdsong (Benny Andersson compiled an album of it in the 90s), an Australian brain surgeon who listens obsessively to Abba while performing impossibly risky operations, and the grim effect of alcohol on the band and their circle: both Andersson and Ulvaeus are recovering alcoholics, their bassist Rutger Gunnarsson was being treated for alcohol addiction before his sudden death, and their manager Stig Andersson was apparently drinking a bottle of a whisky a day when a heart attack killed him, aged 66.

Occasionally, these digressions feel a touch surplus to requirements – you do find yourself wondering if you need to know quite so much about the Swedish herring industry – but more often, they’re fascinating, not least the depiction of the Swedish pop culture from which Abba sprang.

Here is an intriguingly alien landscape of dansbands and schlager, with three different singles charts, only one of which was based on sales.Meanwhile, the pre-eminent musical force was progg, ferociously leftwing folk-rock, whose adherents hated Abba so virulently that the critical opprobrium they attracted in 70s Britain must have come as a relief: one jazz saxophonist was blacklisted from performing live because he’d done a couple of Abba studio sessions. Film buffs might recognise the world depicted in Lukas Moodysson’s film Together, where the children in a progg-loving Stockholm commune sneak out to covertly listen to the forbidden strains of SOS.

For all the confessions of marital discord blurted out in the lyrics of The Winner Takes It All and One of Us, Abba were always rather guarded interviewees, although it’s worth noting that Gradvall, a well-known Swedish critic, has got more out of them than most British journalists ever did, not least, one suspects, because he’s interviewed them in their native language.

That said, you are occasionally aware that this is an artist-approved work. There’s a lot about Abba’s notoriously painstaking process in the studio, but not a great deal of critical scrutiny aimed at their output, which in reality often swerved from the sublime to the ridiculous, albeit with far more of the former than the latter. A chapter on their lyrics approvingly quotes Ulvaeus’s belief that “the sound of the lyrics is very, very important”, but doesn’t explain 1975’s Bang-a-Boomerang, which has an entirely magical tune, but rhymes “dum-be-dum-dum / be-dum-be-dum-dum” with “love is a tune you hum-de-hum-hum”? Or, indeed, the following year’s Dum Dum Diddle, a song about a relationship torn asunder by the male party’s dedication to practising the violin, home to the immortal accusation “You are only smilin’ / when you play your violin”?

But these are minor criticisms, just as the lyrics of Dum Dum Diddle and Bang-a-Boomerang are small blots on a copybook stuffed with pop music of dizzying perfection. The context this book provides makes Abba’s success seem even more extraordinary. Moreover, it’s hard to imagine even the most obsessive fan leaving Melancholy Undercover without discovering something new, even if it is about Himmler or herring.

• The Book of Abba: Melancholy Undercover by Jan Gradvall is published by Faber (£20). To support the Guardian and Observer order your copy at guardianbookshop.com. Delivery charges may apply.