The Velvet Underground’s eponymous third album is also the third record from the group to be the subject of a “super deluxe” 45th-anniversary reissue. Like its predecessors, this six-CD edition offers upgraded sound, several mixes of the original LP, and some sensational, previously unreleased bonus material.

The Velvets made the original version of their third record at a pivotal moment. Lou Reed had just taken full control of the band, pushed out co-founder John Cale and his viola, replaced him with Doug Yule, and turned the music in a gentler, more melodic direction. That’s not to say that the group tilted toward the mainstream; in fact, neither the lyrics nor the lo-fi pre-punk sound on The Velvet Underground spoke to the masses, and this was the first of the band’s albums to not even crack Billboard’s Top 200 list. Nearly a half-century later, though, it is widely recognized (along with the Velvets’ other early albums) as one of the best and most important records in rock history.

“The Murder Mystery,” which fills more than eight minutes on side two of the original album, is inventive but probably not something you’d want to hear more than a few times. That track aside, though, the record is just about perfect, with hypnotic beats, echoes of the doo-wop music Reed loved, harmonies that rival the Everly Brothers, and eminently quotable, often smile-inducing lyrics about life, love, sex, and desperation on the fringes of society.

Though recorded in a studio (in Hollywood, California, in fact), The Velvet Underground conveys the same homespun flavor as The Basement Tapes, which Bob Dylan and the Band created two years earlier. Rather than sounding as if it were made on some country afternoon around Woodstock, however, this record seems like the product of a 4 a.m. session in a low-rent East Village apartment that was littered with empty wine bottles and full ashtrays.

It has been described as an abrupt turnabout from the guitar feedback and viola frenzy of White Light/White Heat, but while the shift from Cale to Yule is immediately apparent, The Velvet Underground was less of a move into new territory than a return to the side of Reed that had earlier produced songs like “Femme Fatale,” “All Tomorrow’s Parties,” “I’ll Be Your Mirror” and “There She Goes Again.” (Even “The Murder Mystery” had a precedent: White Light/White Heat incorporated “The Gift,” another lengthy and deftly written spoken track that deserves a listen but that I suspect many fans soon began to skip.)

Disc one of the new anthology delivers The Velvet Underground in a fresh stereo mix by Val Valentin, the engineer on the original recording. The second CD contains the so-called “closet mix,” which was on initial copies of the vinyl LP and was previously reissued on the band’s indispensable Peel Slowly and See anthology. I prefer this latter version—the one I grew up with, which emphasizes the vocals and sounds more intimate—but I’m happy to have both. Less noteworthy is disc three’s mono mix; it’s difficult to see what it adds; and it destroys “The Murder Mystery,” which relies on stereo separation to produce the desired effect with its pair of simultaneously spoken parts.

Disc four is terrific, though. Here’s where you’ll find the classic songs the group recorded in 1969 for an aborted follow-up to The Velvet Underground. The 14 tracks later surfaced on bootlegs and eventually on VU (1985) and Another View (1986), but it’s great to have them all in one place and in mixes (some original, some new) that in many cases surpass what we heard in the ’80s.

Even better are the two hours of music on discs five and six, which the band recorded in concert in November 1969 at the Matrix, a small but influential nightclub in San Francisco. Six of these tracks previously appeared on 1969 Live while a seventh—a captivating, nearly 37-minute “Sister Ray”—showed up on The Quine Tapes Box Set. But all of those performances have been remixed from the original four-track tapes and sound significantly better here, and the rest of this concert material represents a bona fide buried treasure. There are strong, previously unheard readings of four songs from The Velvet Underground (“What Goes On,” “Pale Blue Eyes,” “I’m Set Free,” “After Hours”), four from the first album (“I’m Waiting for the Man,” “Venus in Furs,” “There She Goes Again,” “Heroin”) and three from the aborted fourth record (“Over You,” “We’re Gonna Have a Real Good Time Together,” “I Can’t Stand It Anymore”). What it all adds up to is the best live Velvet Underground recording we’ve heard to date.

As amazing as the music on this 45th-anniversary package is, it’s even more astonishing how few people were aware of the group at the time of the original album’s release. I don’t know how many fans attended the shows where the Velvets recorded the tracks that appear on discs five and six, but judging by the sound of the applause at the end of each song, I’d guess a few hundred tops. Listening to this collection, you’ll likely be quite sorry you weren’t one of them. But at least you’ll have these wonderful CDs.