The Fiery Furnaces’ “Rehearsing My Choir” is a downright extraordinary album. Matthew and Eleanor Friedberger, the handsome brother-and-sister duo behind the young band, have gone eons beyond the eccentricities of their previous three albums, plum giving up pop-music sensibility. Any semblance of songwriting normalcy is exchanged in favor of a complex extended narrative about Grandma Friedberger’s memories — told straight from her mouth. As a result, each song is monumentally strange, but together the album’s tracks create something genuinely wonderful.

Considering the lovely melodies that fill last year’s “Blueberry Boat,” the band’s peculiar approach here may seem like a pity. But a few listens allow “Rehearsing My Choir” to become catchy in its own right.

The Friedbergers’ grandmother, named Olga Sarantos (what else?), shares vocal duties with the duo, though throughout she speaks instead of sings (excepting brief and surprisingly effective rounds of La-la-las on the first two tracks). Sounding something like Liza Minnelli on last season’s “Arrested Development” crossed with those old recordings of Ginsberg performing “Howl,” she takes a bit of getting used to.

As does, of course, the sheer weirdness of the album. Hard-struck pianos and guitars drop in and out, synthesizers warble atonally, church organs and harpsichords fill the air like some time machine is being accidentally set on and off.

Though the Furnaces have always paid more attention to storytelling than crafting poppy music, these songs sound very little like their past work — let alone anything else. They fall somewhere between the glitchy soundscapes of The Books, Tom Waits’ spoken-word experimentations, and Sufjan Stevens’ dazzling statewide song-cycles.

And then again I can’t help but imagine that this joyous, artful album is exactly the sort of music John Lennon would have made today, had he lived to a grandfatherly age. (Though the early Beatles never sang anything like: “Chatter down the tracks, you thumb tack smiley skull teeth ticking 5 dollar throw away pianos past.”)

It’s difficult to refrain from imagining historic contexts for “Rehearsing My Choir” because of its gloriously nostalgic narrative, but it’s best to stop right there and turn back to the music. After all, there’s enough sonic depth jammed into these eleven songs to keep a close listener busy for weeks.

The album opens with sixteenth-note strikes of a harpsichord, apparently pounding away like the mallets that built our nation’s train-tracks. “Faster hammers!” Olga orders in her geriatric warble, “Faster hammers! Churn and turn into my late train to my late lost love.” Then again, it also seems she could be singing about the hammers of the band’s piano, exorcizing her once-lost memories.

The Fiery Furnaces show a slightly annoying penchant for turning the least likely strings of the narrative into discordant sound effects (“the noise from the construction,” “a gaggle of priests,” “altos out of tune”), but here the device is more engrossing than anything else. The pounding doesn’t drop away until halfway through the song, when it’s replaced by a melancholic piano’s three-note melody that is repeated throughout the album.

“I’d like to tell you a story, kids,” Olga sighs. “But instead I’ll change the subject: Listen to this tune that sounds like a condolence card.”

The next song, “The Wayward Grandaughter,” opens with a jarringly electronic beat that seems profoundly misplaced beneath the song’s memories of long-lost love. But the track goes so many different places in its six minutes — from drum-machine tropicalia to blistering bursts of silence — that it hardly matters.

This unregulated but ironically consistent wildness is what makes “Rehearsing My Choir” so interesting. The exciting “Forty-Eight Twenty-Three Twenty-Second Street” erupts into a two-minute Mariachi breakdown; “Slavin Away” swings suddenly from a refreshingly calm acoustic guitar into a cascade of ’80s synthesizers; the already strange title track has an unannounced ten-second breather of enormously cacophonous electronic wails.

Jazz critic Stanley Crouch once said that he tried listening to Miles Davis’ “Bitches Brew” straight, backwards, drunk and stoned in order to appreciate it more. It’s hard to imagine that the same approach wouldn’t work for “Rehearsing My Choir” — though I fear that chemically induced inattentiveness would ruin this music’s high-speed brilliance. And the album’s jittery dissonance doesn’t exactly lend itself to solo substance abuse, let alone group listening.

It seems unjust to ask anything more of a band that’s now released three quite spectacular albums — plus one excellent EP — in its first three years. Yet no matter how enjoyable the album’s endless soundscapes are, one can’t help but hope that someday the Fiery Furnaces will head back toward the more accessible beauty of their earlier work.

For the time being, it is hard to complain about such an unimaginably creative work.