GIVE THEM LIBERTY: They can talk about concept, but on Amputechture they cut loose.

The Mars Volta’s new Amputechture (Universal) is the first album the Los Angeles–based prog-punk ensemble have made that doesn’t have a unifying story line. Their full-length debut, De-Loused in the Comatorium (2003), commemorated the life of Julio Venegas, mentor to singer Cedric Bixler-Zavala and guitarist Omar Rodriguez-Lopez; Frances the Mute (2005) told of an adopted kid whose diary former band member Jeremy Ward (who died in 2003) found in the back of a used car. (Scab Dates, a live album the Mars Volta released last year, offered the drama of an audience withstanding inhuman amounts of free-jazz noise terrorism.)

Yet since forming in 2001 out of the ashes of the late, great El Paso emo-punk band At the Drive-In, the Mars Volta (who open for the Red Hot Chili Peppers on October 2 at the Garden) have told just one story, and that’s the flying of their collective freak flag after spending years inhabiting a regimented, neo-puritanical hardcore underground. At the Drive-In were plenty arty (especially considered in the context of the subsequent emo acts they inadvertently inspired), but on Amputechture Bixler-Zavala and Rodriguez-Lopez cut loose, doling out 13-minute songs, guitar solos that sound like a fax machine dying, and lyrical tone poetry stuffed with invented words and phrases seemingly cribbed from extraterrestrial medical textbooks. Bixler-Zavala can talk about concept, but this is music about freedom.

And how lovely that freedom sounds on the band’s least aggressive, most nuanced recording yet, as Bixler-Zavala and Rodriguez-Lopez discover that liberty comes in more than one flavor. On the opener, “A Vicarious Atonement,” Bixler-Zavala works his tender falsetto over a woozy space-blues lullaby that could be Air or Portishead or Pink Floyd; near the end, a hotel-bar piano starts tinkling away while Rodriguez-Lopez’s gentle-weep guitar impersonates a saxophone (or vice versa). “Vermicide” is another gorgeous slow jam complete with freak-folk flute whisper and brushed-drum shuffle; like Frances the Mute’s big radio single, “The Widow,” it’s what I always wished Tool frontman Maynard James Keenan’s radio-friendly side project A Perfect Circle would sound like. “Asilos Magdalena” is an acoustic ballad sung in Spanish that has nothing to do with emo or punk — or prog, for that matter.

Equally telling are the smaller changes you can hear in the loud rock material. “Viscera Eyes” sports greasy organ and a wailing brass section that gives the grungy guitars the funky tang of classic Frank Zappa; in another dimension, it’s killer roadhouse R&B. Bixler-Zavala overdubs his vocals in “Meccamputechture” into little bundles of glam-pop glitter, as if Freddy Mercury were fronting Santana. And for “El Ciervo Vulnerado,” Rodriguez-Lopez surrounds his six-string sorcery with a droning sitar that gives the track a smoky, hallucinatory vibe. On these cuts, the band’s narrative starts to look pretty conventional: maturity happens.

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THE MARS VOLTA + THE RED HOT CHILI PEPPERS | TD Banknorth Garden, 1 Accolyn Way, Boston | October 2 | 617.228.6000

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