This week, two of New York City’s most intriguing (and sometimes exasperating) bands, TV on the Radio and the Fiery Furnaces, will play local shows to debut new material.

TV on the Radio exploded out of the New York underground in 2004 with “Desperate Youth, Blood Thirsty Babes,” an album that situated soul, R&B, and doo wop elements within a haze of shoe-gazer-rock guitars and paper-thin drum machine beats. Having recently signed with Interscope, the group is preparing its follow-up, “Return to Cookie Mountain.” Though it doesn’t come out until June, the songs are widely available on the Internet, offering a preview of what we may expect tomorrow night.

The best song of the batch, “I Was Your Lover,” is also the biggest departure. It’s built on the kind of lazy, chopped-and-screwed beat that Dirty South rappers dream about. Stuttering percussion alternates with a moaning horn sample and sheets of guitar noise. One can only imagine what an MC like T.I. would do with it. (Here’s hoping it finds its way onto the mix-tape circuit.)

TV on the Radio’s Tunde Adibempe doesn’t do half-bad himself, trading between his husky natural voice and a Prince-like falsetto as he sings about the awkwardness of seeing a former lover. “It’s been a while since we went wild and that’s just fine / but we’re sleepwalking through this trial and it’s really a crime, really a crime, really a crime,” he sings.

David Bowie, an outspoken fan and informal adviser to the band, makes a quavering but subtle appearance (easily missed if you aren’t listening for it) on “Province,” a song of washed-out My Bloody Valentine guitar, plinking piano, and handclaps. On “Blues From Down Here,” hirsute guitarist-vocalist Kyp Malone does his own best Bowie impression, which morphs into an even more spot-on Mick Jagger one by song’s end. Over a driving drum-and-synth track punctuated by gusts of saxophone, the band breaks into a schoolyard chant: “pull the pin, drop it in, let it wash away all …”

The sound here is denser than on “Desperate Youth” but also less exciting. Take the song “Let the Devil In,” for instance. It starts out as a controlled military march and builds toward a polyrhythm like something out of “Stomp.” Still, Adibempe’s vocals remain weirdly rigid; he sounds something like I’ve always imagined a singing telegram would. “Wash the Day Away” likewise establishes a tense, exciting sound – boiling guitars and tribal drums – only to squander it. It just drags on, unchanging, for six minutes.

Sibling duo the Fiery Furnaces have the opposite problem on their new album, “Bitter Tea”(Fat Possum Records). The songs volley between 1980s video-game sounds, sections of erratic harpsichord, and voices and instruments played backwards. They’re filled with so many abrupt changes – whole melodies appear once, briefly, and are never heard from again – that the track breaks begin to feel arbitrary. Several times, entire songs dissolve into a toilet bowl swirl of sound.

Every Fiery Furnaces album feels like a quirky one-off. Their debut, “Gallowsbird Park” (2003), was a dizzying mix of garage rock, folk, ragtime, and keyboard-driven art punk that inspired comparisons to retro-rockers the White Stripes not so much because of a similarity as the need to compare it to something, anything. And subsequent albums have been even weirder. “Blueberry Boat” (2004) was an epic fantasia about (among other things) a little girl losing her locket and facing off against pirates. Last year’s “Rehearsing My Choir,” recorded with the duo’s croaking grandmother, was a nightmarish, nonlinear oral history of a woman living in mid-century Chicago. The only thing I can think to compare it to is one of William S. Burroughs’s freaky spoken-word albums.

“Bitter Tea” continues this determinedly anti-pop campaign, and frankly, it’s exhausting, especially through the first four songs. But perseverance pays off. The fifth song, “Teach Me Sweetheart,” is a lovely little ditty about murderous parents and in-laws set to a backward effect that sounds like water dripping upward. It has only a few really jarring moments.

Eleanor Friedberger peppers the songs with her customary offbeat, offhand observations: “I’m in no mood to comb my hair / there’s a chill in the air / and it’s catchin’, catchin'”; “vibrate, buzz, buzz, ring, and beat / tell me babe what time is it now.”

There are a few other moments worth sticking around for. “Waiting To Know You,” with its prancing harpsichord melody and smeared keyboard, sounds like the Flaming Lips covering a 1950s love song. “Police Sweater Blood Vow” is a bouncy tune that accumulates sounds as it goes along: now apocalyptic jam guitar, now a piping organ, now a steel guitar played blues-style with a slide. “Nevers (remix)” charmingly splices Matt’s and Eleanor’s voices together so they alternate syllables. It’s impossible to tell what they’re saying, but it’s a neat trick nonetheless.

“Bitter Tea” ends up reinforcing a lesson Fiery Furnaces fans have surely learned by now: You’re either going to enjoy the band on its own terms or not at all.

TV on the Radio on Tuesday night, the Fiery Furnaces on Wednesday night, at Bowery Ballroom (6 Delancey Street at Bowery, 212-533-2111).