Most times wizarding music is referred to in the HP books, it is being played on the radio. But what about the existence of recorded music which individuals can purchase and play for themselves?

In a way, such a thing would need to exist for the wireless stations to be able to broadcast music. Unless those stations are all broadcasting live performances, they need some way to store recorded music and play it back for broadcast.

There is just one hint in the books as to what form this takes:

This is the only mention in all 7 books of a wizard playing their own private music collection. Now, you might say that this is inconclusive - Slughorn might have a gramophone for all sorts of reasons, and it doesn't necessarily mean that such devices are widely used in the wizarding world. The Weasleys certainly didn't seem to have one.

However, in the absence of any other evidence, I would suggest that gramophones are indeed the form in which wizards listen to private music collections.

This leads to a number of interesting areas for discussion:

1. It limits the Muggle music which wizards can listen to. Gramophones play shellac records, also known as "78s", which were in use in the Muggle world from the early 1900s until they were replaced with vinyl records in the 1950s. So wizards aren't going to be able to easily access post-1950s Muggle music. You might therefore expect that wizards and Muggles to share a common musical history which then diverges around 1950.

2. Wizards will need their own production of shellac records. Given that Muggles no longer produce them, but wizarding bands will still be releasing records, there will need to be a wizard factory somewhere producing these records, and wizarding recording studios for wizarding bands to use.

3. Magical enhancements. Most Muggle devices adopted by wizards have been upgraded with magical properties which improve their utility. I would expect the same with gramophones. In particular, I think wizards would have (i) improved the durability of the records, (ii) increased the length of music you can record to each side, perhaps by making the records bigger "on the inside" than on the outside, and (iii) augmented the clarity and volume of the sound which is produced.

 

It wouldn't really be all that hard for a wizard to modify a gramophone to play modern vinyl records. The biggest difference is that a gramophone is designed to be a purely mechanical player, while modern records and turntables rely on electronic amplification. The issue for us mere mortals is that to get loud enough sound out of a purely mechanical player requires enough force to destroy the vinyl record - something that shouldn't be much of an issue for a wizard with sound amplification magic and unbreakable charms.

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For purely instrumental music I could see wizards using charmed self-playing instruments. For a middle-class family you could have a single piano, and a more upper-class home could have a full orchestra.

 

I'd imagine that any adoption of Muggle culture or technology is almost always delayed. We see this with fashion most clearly, with wizards trying to pass themselves off as Muggles tending to wear rather outdated outfits. So it's possible that wizarding music on shellac records didn't come into play until a couple decades after its introduction to the world.

There's also music boxes, which predate recorded music by at least a century. I could easily see a wizarding music box as having actual music and lyrics "locked" inside, in place of mechanical parts. You could consider the golden egg from the Triwizard Tournament as an example of this, though whether the book version was a song or not I can't remember. There's also a potential for a painting of an orchestra or musician being able to serenade viewers.

 

Wizards clearly have no problem just having parchment record sound, as we see with Howlers. It would probably be trivial to record music in something like a resealable letter that plays whenever you have it open.

Likwise Portraits are clearly able to hear and be able to be heard, so animated pictures of musicians performing should also be possible. Likewise having animated photographs with audio sound like it could be plausible as well.