Few bands arrive with a mission statement/prophecy as grandly ambitious as MGMT's "Time to Pretend." Originally released in 2005, and then again on the band's 2007 major-label debut, Oracular Spectacular, "Time to Pretend" spun a familiar fable of rock-star success and excess ("Let's make some music, make some money, find some models for wives") that somehow turned choking to death on one's own vomit into a fairly heartfelt existential meditation ("Everything must run its course"), a "Do You Realize?" as played by Spinal Tap, everything only half-winking.

But then, something funny happened on their way to fulfilling prophecy. The popular read on MGMT's sophomore album, Congratulations, is that it's a dud, a twee-folky comedown from Oracular Spectacular's epic electro-rock daydreams, a sophomore slump of embarrassing proportions. (Kiss the Parisian heroin and private islands good-bye, get ready for that morning commute.)

And up until about a week ago, when I began really immersing myself in Congratulations alongside its predecessor for the sake of writing an article, I had the same vaguely disappointed take on the album. But, side by side, the two albums don't sound all that different, the notable exception being that the former had three insanely huge, dance-floor-ready singles ("Time to Pretend," "Kids," and "Electric Feel"), whereas the latter has none. MGMT had merely made a "difficult" second album.

A corollary (and sometimes cause) of the sophomore slump, the "difficult" second album is meant to reaffirm a band's artistic integrity following an unexpected commercial success. It may also result from the truism that every band has a lifetime to make its first record and a year to make its second. The "difficult" album says, "Hey, we may have 'sold out,' but no one owns us, the Man can't rein in our artistic vision, we're raging against the machine from inside the machine," and so on. And it has a fine, storied tradition in popular music; here, then, a brief and necessarily incomplete survey of some great "difficult" second albums.

Wire's 1977 debut, Pink Flag, broke punk rock down to its most formally economical state—scraps of songs that clocked in from 30 seconds to under two minutes, every one only as long as it needed to be to get its point across—while still hinting at how the genre's stylistic boundaries might be expanded. For their follow-up, the band largely jettisoned that shrapnel approach in favor of longer songs treated with synthesizers and expanding into relatively open ambient spaces. It may have been a jolt at the time for those seeking more advert-length pogo and tumult, but both albums are rightfully regarded as classics today.

PiL were basically the "difficult" follow-up in full band form, a vicious (har) left turn from Sex Pistols' manic (and manicured) punk fury. Frontman John Lydon (née Rotten) along with the core crew of no-training bassist Jah Wobble and Clash guitarist Keith Levene chewed up punk, experimental rock, and especially dub, and spewed out a sound all their own that would nevertheless become foundational for the nascent genre of "post-punk." Metal Box found the band taking that combination to both its greatest and its most grating extremes, including the inescapable scrape and plaint of "Careering." (In 1980, preeminent rock critic Robert Christgau wrote of the album, "Metal Box is difficult; some of it fails.") Notably, first pressings of the album came packaged in embossed metal 16 mm film canisters, although there had been some talk of releasing it in a sleeve made of sandpaper (as Durutti Column later did), which would have made it all just that much more abrasive.

The Jesus and Mary Chain's debut, Psychocandy, was two doomy Scots brothers (and one future Primal Screamer), sulking and cool behind great walls of distortion, fucking around brilliantly with classic pop and rock forms from the Ronettes to the Ramones. For their follow-up, they ditched their drummer for a drum machine, dialed down the epic guitar fuzz, occasionally swapped one brother for the other on vocals, and began inching toward twangier, tamer sonic horizons that would be explored further on later albums. Nothing has sounded quite just like their brand of honey still swarming with buzzing drones since.

Digable's 1993 debut, Reachin' (A New Refutation of Time and Space), reached unexpected chart heights largely on the strength of crossover single "Rebirth of Slick (Cool Like Dat)," which won a Grammy for best rap performance by a duo or group and eventually found second life in a detergent commercial ("So Fresh, So Clean" was presumably out of the soap's price range). Come-lately hiphop fans (or execs) looking for another Afro-boho jazz-rap single so friendly that it could woo both the Recording Academy and the color-safe-bleach crowd were confounded by the act's comparatively more opaque opus Blowout Comb (although several critics were sold). While the standard rap on this album is that it's somehow more staunchly Afrocentric than the group's debut, probably the bigger hurdle to radio play was that its jazz-sampling joints were so incredibly laid-back that they frequently roll out to five or seven minutes in length. Still, like many of the supposedly hard-to-digest albums listed here, it's come to be widely recognized as an underrated gem in the years since its release. (Digable Planets man Ishmael Butler has stayed stubbornly, stupendously "difficult" since, with his slept-on Cherrywine project and his currently rising Shabazz Palaces persona.)

With their self-titled debut, Weezer rode to rock stardom on a wave of willfully "nerdy" power pop infatuated on a surface level with sweaters and surfboards, comic books and rock posters, and propelled by undeniably giant guitar hooks. Follow-up Pinkerton was both less slick and less superficial; its hooks were blown-out more than buffed-up, but perhaps more to the point of its "difficulty," its lyrics were less about the stuff that Rivers Cuomo liked (to borrow a High Fidelity–­ism here) and more about what he was like. And he turned out to be an emotionally messy, complicated character, rather than the caricature of Buddy Holly that some fans may have imagined. In subsequent years, as Weezer cranked out album after dull album of painfully formulaic, easy, and impersonal songs, Pinkerton's supposedly difficult aspects would come to be hailed as its greatest virtues and it would be canonized by many (this writer included) as the band's greatest achievement. Related: Ex-Weezer bassist Matt Sharp's band the Rentals followed their debut, Return of the Rentals, a compact album of wall-to-wall analog-­synth anthems, with the more sprawling and stylistically diverse, but easily as rewarding, Seven More Minutes.

Liars' first album wasn't exactly easy listening, but it did storm out the gate with the unstoppable (and then quite au courant) dance-punk convulsion of "Mr. You're On Fire Mr.," as well as a track that Trojan-horsed their own trance into ESG's eternally sample-able (though they wish you wouldn't) "UFO." So, naturally, they kicked out their foundational rhythm section and recorded an eerie, even more sample-heavy song cycle about German witchcraft. They Were Wrong did have a couple of electrically aggressive songs, but you'd have to be pretty demented or dedicated to dance to them (I have seen it happen, though). Things would get weirder with these guys, of course, peaking with the fine ambient art-punk suite of Drum's Not Dead.