The premise of “Originals,” the third posthumous release from Prince’s estate, is to collect the artist’s original versions of his compositions recorded by others, whether performers outside his camp (Sinéad O’Connor with “Nothing Compares 2 U,” the Bangles with “Manic Monday”) or ones he mentored and/or produced (Sheila E’s “Glamorous Life,” The Time’s “Jungle Love”).

Amid such company, it’s a little surprising that this album opens with Prince’s version of “Sex Shooter” — a hit for Apollonia 6 and a song featured in the blockbuster 1984 “Purple Rain” film, but hardly a smash or a stone-cold classic like the above. And yet, within 15 seconds, it all makes sense: Prince’s falsetto, the buzzing synthesizers, the Roland TR-808 drum machine, the electronic handclaps, make it sound like a long-lost song from “1999” or “Controversy,” which for some is the equivalent of finding a previously unknown snapshot from a pivotal time of their life.

While “Originals” is indisputably the best and the most accessible album to come from Prince’s vault yet — last year’s “Piano and a Microphone” and 2017’s expanded “Purple Rain” are more for the faithful — the degree to which listeners will enjoy this 15-song collection depends on the depth of their fandom. It’s a cohesive and extremely well-curated compilation, but several of the songs will be unfamiliar even to serious followers, and at the end of the day, it’s a collection of recordings that Prince never intended for the world to hear.

Regardless, there’s plenty to get excited about. Nearly all of the entries date from Prince’s golden age — the ’80s, duh — and where some are very different from their released versions, others are hardly different at all. Many of these songs are technically demos — sketches or references for others to use while recording more fully realized versions — but with Prince, that was a relative term: He frequently recorded completely solo, handling most if not all of the vocals and instruments, so his “demos” usually evolved into completed tracks. Thus, several of the songs here are virtually identical to their officially released versions, but with Prince’s original vocal instead of, say, Sheila E’s.

That’s the case with “Make-Up,” recorded by Vanity-6, a throbbing slab of electronic new wave that evokes 1981 as vividly as “An American Werewolf in London,” and “Manic Monday,” originally intended for Apollonia 6’s album, which sounds like an outtake from 1985’s “Around the World in a Day,” (but not much like the Bangles’ version).  “Baby You’re a Trip,” recorded by Jill Jones, includes a luscious, shrieking, whooping vocal ad-lib at the end.

Several songs here were written, or at least earmarked, for specific artists, which makes for some odd double-take moments. “Jungle Love” illustrates how the Time had its own sound — a beefier, more masculine funk than Prince’s — even though he wrote, produced and performed the bulk of the group’s output. “Holly Rock,” recorded by Sheila E. for the soundtrack of the galvanizing 1984 hip-hop film “Krush Groove,” has an almost rap-ish style suited to her voice (and even features Prince singing the lyric “Sheila E. is my name”); he croons “You’re My Love,” recorded by Kenny Rogers, in a deep, gravelly register. “Love, Thy Will Be Done” is delivered in a low falsetto that almost resembles late-period Michael Jackson and features the sort of stunning stacked harmonies Prince employed on early ’90s songs like “Thieves in the Temple.” And we won’t even get into the subtext of his comically clumsy female-POV lyrics on “Make Up” (“If I wear a dress, he will never call / So I’ll wear much less, I guess I’ll wear my camisole”), which, 35-plus years after it was recorded, could have been made by any number of gender-fluid artists.

Not surprisingly, the crown jewel here is “Nothing Compares 2 U.” One of his most indelible compositions, ironically it wasn’t fully realized until O’Connor recorded it, five years after its 1985 debut with Prince-mentored group the Family. His rendition is haunting, lonely and beautiful, but also a little overblown (his definitive take is arguably the 1992 live version on “The Hits/The B-Sides”).

Music poured out of Prince — he recorded thousands of songs over his 40-odd-year career, and a major reason the side projects existed was that he made too much music for a single artist; his very public dispute with Warner Bros. Records during the 1990s started when the label wouldn’t allow him to release as many albums as he wanted. Now that the lid is off the vault, projects like this one give insight into both his working methods and his imagination during the decade when he was at the peak of his creative powers.

[First Released by (Artist: Album – year) followed by the year of Prince’s recording included on “Originals”]
1. “Sex Shooter” Apollonia 6: Apollonia 6 – 1984 / 1983
2. “Jungle Love” The Time: Ice Cream Castles – 1984 / 1983
3. “Manic Monday” The Bangles: Different Light – 1985 / 1984
4. “Noon Rendezvous” Sheila E.: The Glamorous Life – 1984 / 1984
5. “Make-Up” Vanity 6: Vanity 6 – 1982 / 1981
6. “100 MPH” MazaratiMazarati – 1986 / 1984
7. “You’re My Love” Kenny Rogers: They Don’t Make Them Like They Used To – 1986 / 1982
8. “Holly Rock” Sheila E.: Krush Groove (OST) – 1985 / 1985
9. “Baby, You’re a Trip” Jill Jones: Jill Jones – 1987 / 1982
10. “The Glamorous Life” Sheila E.: The Glamorous Life – 1984 / 1983
11. “Gigolos Get Lonely Too” The Time: What Time Is It? – 1982 / 1982
12. “Love… Thy Will Be Done” MartikaMartika’s Kitchen – 1991 / 1991
13. “Dear Michaelangelo” Sheila E.: Romance 1600 – 1985 / 1985
14. “Wouldn’t You Love to Love Me?” Taja SevelleTaja Sevelle – 1987 / 1981
15. “Nothing Compares 2 U” The Family: The Family – 1985 / 1984