The so-called celestial jukebox that digital-music dreamers have been talking about since the 1990s got a little closer Tuesday when Amazon launched its Cloud Player.

This service doesn’t provide online access to all the world’s recorded music, as that vision specifies. But it does let you listen to your collection from any Internet-connected computer with a Web browser as well as Android smartphones and tablets, and that’s nothing to scoff at.

Other companies have done this — see, for instance, and SugarSync — but Amazon is on far more people’s browsers and runs the second-most-popular music download store in the United States.

That makes Cloud Player and its Cloud Drive Web storage a challenge to Apple’s iTunes — and a major hint about where digital music is headed.

The easiest way to think of Cloud Player and Cloud Drive is as a replacement for the USB connection used to sync a phone to a computer. In other words: The network is not just the computer but the cable as well.

You just need enough bandwidth — figure at least 750 kilobits per second, or about three times the 256 kbps “bit rate” of an Amazon MP3 or iTunes download — and an Amazon account with a valid U.S. billing address.

The Seattle retailer provides 5 gigabytes of storage free; buying an MP3 album from it ups that quota to 20 GB for the next year.

You can sync new Amazon MP3 purchases to your Cloud Drive automatically and can transfer songs from a Mac or Windows computer with Amazon’s MP3 Uploader. Contrary to what its name suggests, it will also upload AAC files bought from iTunes, provided they’re not locked with the “digital-rights-management” system Apple retired in April 2009.

That program does not handle Microsoft’s Windows Media Audio format, podcasts or audiobooks (though you can upload files of any kind to your Cloud Drive through a browser).

The MP3 Uploader could be quicker to install — it earns a demerit for requiring an installation at all on a Mac, courtesy of the Adobe AIR software it runs on. But it quickly identified one computer’s iTunes library and another’s Windows Media Player collection and uploaded those files as fast as a Fios connection would allow: about half an hour for 306 songs in the latter test.

(Cloud Player’s terms of use let Amazon “disclose your account information and Your Files.” Spokeswoman Cat Griffin wrote that “we work very hard to establish customer trust” but did not elaborate on that clause. I wouldn’t store anything too personal until it’s clearer.)

Amazon’s Cloud Player browser interface worked as advertised in Safari on Mac OS X, Firefox in Windows and Linux, and Internet Explorer 8 in Windows. Entering my Amazon username and password yielded all the songs I’d uploaded, complete with the correct album art and sortable by song, artist, album, genre or playlist.

Pausing, resuming playback and skipping to the next song happened almost as fast as if these MP3s and AACs had been on each computer. And by selecting them and clicking a “Download” button, I could arrange that, too.

Amazon’s Android program works like the Web app but can’t sort by genre. It can apparently ride out bandwidth hiccups — it played back uninterrupted on a Metro run from Rosslyn to McPherson Square that usually has calls drop — but also crashed several times.

A different playback issue emerged Wednesday afternoon, when I hit a poorly disclosed limit of five devices in 24 hours.

An iPad, iPhone or iPod touch won’t push you over that limit. Amazon hasn’t even shipped its MP3 store app for Apple’s mobile devices, where it would compete with the iTunes Store and, presumably, get vetoed by Apple.

Browser playback doesn’t work in those gadgets either, although you can select an individual track for download and wait for Safari to play it.

Although Amazon has competition from smaller companies, it faces none from Apple. The company behind iTunes bought a popular cloud service, Lala, in late 2009 but shut it down months later and has yet to reintroduce anything like it.

Major record labels — which sued one of the first cloud music services,, into oblivion a decade ago — might not appreciate Cloud Player. (The Recording Industry Association of America declined to comment.) But this service’s requirement of an Amazon account login, which no sane user would want to share, should help insulate it from legal challenges.

Cloud Player could, however, face a much tougher problem: The bandwidth caps imposed by wireless carriers and even some ground-bound broadband providers. They could make this proposition too expensive for even avid music fans.