Fourier transformation.

https://youtu.be/spUNpyF58BY

Highly basic answer, let's say the strength of the vocals wave over time is:

5, 4, 3, 2, 3, 4, 5, 4

And drums is:

4, 0, 2, 0, 4, 0, 2, 3

Then you add them together for each time slice and get:

9, 4, 5, 2, 7, 4, 7, 7

And you put that on a record, or out to a speaker, and our ears are able to break that up into the two parts when it hears it. This is the same as when two things are in the room making sound, there may be two sources, but my ear only has one hole, and that hole has one eardrum behind it. The different sounds just add their powers together and hit my ear as one mixed wave.

Alternative answer: magic

It doesn't do it particularly well and that's how you know vinyl enthusiasts are smelling their own farts.

Take it back. How does the vibrating air equate to all that? It's not like there's a drums bit of air and a vocals bit of air - the vibration is all smushed together. Your brain separates it back out again. That's why it can take training to separately hear some bits of music, or why you can't usually pick out individual voices in a choir.

Well if you put multiple waveforms above eachother the form on single waveform.(They all occupie the same space,in this case air, so they can't be "separate"). This waveform is then recorded and remastered and whatnot. But basically the waves you can see on the vinyl are the "schape" they will have in the air.

That's the neat part, the brain does that using some black magic. You just have to add all the sounds individual waves together and the brain deciphers it.

So there's this thing called a Fourier series...

Basically any wave can be created by adding together individual frequencies, and with some fancy math it's possible to go the other way with a Fourier transform and get how loud every frequency is (like is displayed in a spectrogram).

I think the real black magic is in how our ears and brains can decode the mess of information coming in and identify meaningful patterns.

That's because it doesn't, your brain does

Speakers do the simplest thing possible and literally just vibrate. A recording being played literally just recreates a recorded vibration. It's a tiny choreography that your ears are incredibly sensitive for.

All the fancy stuff happens in our brains, after our ears has split up the sound around us into different ranges of frequencies (you can think of the hairs in the inner ears as tuning forks). We learn to recognize which frequencies goes together, and then we learn how the frequencies from multiple sources can overlap, and we learn what it all means

The real crazy part is how something as simple as sound can carry so much information and how reliably our brains can tell it all apart and make sense of it

How speakers work in general is just black magic to me, actually.

An easier way to understand it, without knowing the math, is to know how it's made. You play audio into a very similar device and it's needle scratches the grooves. When you then have a needle pick up the grooves it's moving the exact same way the needle was forced to move by the original.

It's similar to how a speaker and a microphone are basically the same device. If you take a speaker and plug it into a microphone input, it still works (though they're tuned differently so it's not as good). A microphone has a crystal vibrate, which creates an electric signal. If you play that electric signal into a crystal it vibrates and creates the same sound.

There's no math or anything being done for this to work. It's purely mechanical. It's just a copy of what the needle did when sound was played into it, so another needle running through it recreates the same sound. You can use math to represent it, but none is being done by the device (other than just the laws of physics).