Is algebraic sound synthesis possible?

Using a technique (and software) like this: Audacity Vocal Removals I bet you can achieve what you need. As Daniel and Paolo said if you can apply the inverse of a soundwave to the original soundwave you are able to cancel it out (muting the sound).

Yes it's possible but yes there can be loss as well since sound waves can additively cancel each other out (destructive interference). For example, two sine waves that are 180 degrees out of phase would produce silence.

1 This isn't quite right... it's a pure subtraction, therefore linear, therefore there is no loss even in the case of destructive interference. The real problem is getting a good enough estimate of the signal you have to subtract... any estimation error will turn into noise or artifacts. – Andrew McGregor May 7 '10 at 22:12 @Andrew - I agree, mostly, but there is loss if the add or subtract causes clipping, no?

– NVRAM May 7 '10 at 23:09 Sure is, but if you're trying to do DSP on a clipped signal, you're already in big trouble... the data is already lost. You can easily make sure the subtraction doesn't clip in itself. – Andrew McGregor May 8 '10 at 0:02.

Ideally, it should be possible. The catch is, "ideal" is pretty restrictive. In order to pull this off properly, you would have to have a song file that was constructed by additive synthesis in the first place, i.e.By adding the vocal track to the exact same instrumental track you have.

Now, if you do have that situation, then it's simple enough; as others have said, you just add the inverse of the instrumental track to the overall song. Unfortunately, there are a lot of things that can get in the way of that. For example, if the additive synthesis was clipped at some points (which means that the sum of the instrumental and vocal tracks was louder than the maximum volume that can be stored), you won't be able to recover the vocals at those points.

More generally, lossy audio compression tends to remove different pieces of the sound depending on what is most/least audible, and that's heavily dependent on whether you have vocals or not, so if any of these sound files have been compressed using a lossy codec like MP3, you've probably lost the information you need to reconstruct the vocal track. The thing is, even minor changes to a signal can sometimes produce a big difference when you add it to or subtract it from another signal (because of wave interference and such things) so the results are kind of unpredictable when you don't have the exact sound to work with.By the way, if you do have the exact signals you need to do this, you can perform the subtraction using Audacity or any other decent audio editor. There are even some mathematical programs you can use (like Matlab, which is able to read/write WAV files IIRC).

Generally, the word 'Synthesis' is used slightly differently, though the dictionary meaning might agree with your question. As pointed out, audacity/vst plugin/pro-tools versions of 'extract vocals' exist. // Is there going to be a loss?

// Of course, there will be some loss. Vocal and Instrumental tracks are 'mixed' and 'mastered'. Panning, adding effects (echo/reverb), and additional shaping (compress etc) take place in these stages.

Besides, there'll be many instrument tracks (keyboards, guitars, bass, drums) in most of the music that's produced these days. I mean to say, even you make your own music - with 1 instrument track and 1 vocal track, if you just pan your tracks, your logic of subtraction is going to be affected. And, if you had recorded those tracks, say by playing guitar and singing, most probably there'll be some 'leak' in both of the tracks, that makes matters worse.

Hypothetically speaking, wonderful things are possible with this idea. Practically, there are too many imperfections in the actual music production process.

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