How can I get the audio of only the dialogue of a movie, without any sound effects or music? I really need it for a fan edit.

This works well to separate sounds which are repetitive, like the background noise on a street from dialog, sounds like a steady electric hum, engine noise like a jet or vacuum cleaner, etc.

Get the original master audio track from the movie production. They will have that for any recently produced movie from the surround sound era. Audio is recorded in many separate tracks, and dialog routinely uses many tracks - sometimes multiple tracks per actor.

But much dialog isn't mixed in the center in stereo. It is offset, to set the sound stage left and right.

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Beyond that, it isn't easy, and won't ever completely eliminate the music and background sounds.

Let's go in reverse order from easiest and most expensive to simple, cheap, and low quality. Dolby Atmos audio has separate tracks for all sound sources. If you can afford a proper Dolby audio and video editing software suite, you can extract each of these sounds from the mix.

The cheapest method uses simple audio frequency filters, to reduce parts of the sound spectrum. This can make the dialog louder relative to the other sounds, but won't eliminate them.

Is there a software or anything else where you could remove a certain instrument sound from a song and replace it with you playing that instrument instead while the rest of the instruments play?

Whether you can afford this, or even get permission, is another matter. But it is guaranteed to work perfectly.

Back to the most perfect, but potentially most expensive method.

This process can also extract only the center. If you're lucky, you'll get mostly vocals and dialog.

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The simpler surround sound formats, 7.1 and 5.1, have a single monophonic center track. In any audio editor that can separate these tracks, you can easily get just the center track. If all of the dialog is mixed there, without any sound effects or background music, this will work well. But not all mixes do that, especially if some dialog is intended to come from the sides of the screen.

That leaves other means to reduce the sounds and extract the dialog. Modern AI audio processing uses filters and pattern recognition to identify elements in the sound.

In a stereo mix, the monophonic elements will often be balanced in the center, equal in both tracks. A trivial vocal removal effect inverts the phase of one track, then adds them together to produce a monophonic version of each track with the monophonic center reduced.

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It's not that great at separation of music, vocals, and random sound effects.