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This question may start oozing into the realm of copyright infringement, but let's discuss the theory first and foremost.

I have a song with vocal and orchestral accompaniment from a well-known movie in two different languages. The orchestral accompaniment only of the song has never been published, and I want just the orchestral accompaniment audio to add to my collection of music for studying.

If I have the song in two different languages, is it possible to extract the orchestral soundtrack using the physics of waves?

Here's my idea: I know about constructive and destructive interference of waves, like so: enter image description here

Suppose the audio tracks (waves) in each respective language, $S_1$ and $S_2$ are composed of the orchestral soundtrack, $A$, plus the vocal soundtrack in language 1, $V_1$ and in language 2, $V_2$. So we could create the system of equations $$\left\{\begin{matrix}A + V_1 = S_1 \\ A + V_2 = S_2\end{matrix}\right.$$ But how can we best isolate $A$? I can do something like $$S_1 - S_2 = V_1 - V_2$$ but it is not evident that we may isolate $A$, if at all. Would numerical methods be of benefit here, or do I not have enough information to solve the problem?

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Yes, it is possible but it involves a lot more work. The problem is that the different components you want to separate are not similar enough for a simple subtraction to work. The frequencies are far from exactly the same, and even if they were approximately similar the relative phases would change quite fast. The phase information cannot therefore be used in this way. But you can use methods such as wavelet analysis to extract the orchestral accompaniment.

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