# Subtracting Audio from a Song with Multiple Forms

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: 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?