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Suppose I have a chord such as FAC. How do I know whether the notes belong to C (Fa La Do') or F (Do Mi Sol) or perhaps another key? Do music theorists have theories for how a key gets established?

In the artificial neural network community, I know connectionists have modeled key recognition using neural networks, but I am not familiar with how music theorists think about this problem.

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    Not sure if I understand your question, but the F major chord belongs to any key which contains the notes F, A, and C, i.e. C major (A minor), F major (D minor), Bb major (G minor). – Matt L. Jan 5 '15 at 8:04
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In isolation, FAC doesn't belong to any key. It's just an F major . When it's put into a piece it goes into context. Usually, a piece IN F will end ON F, and feel that it has ended, without the need for any more to follow. Often, that piece will start ON an F (usually in the first full bar if there's an anacrucis) as well. The feeling of 'home'.

At various other points in the tune, cadences will appear. They will give clues as to whether the song is IN F. If they are Fs themselves, and the song is indeed in F, those points will feel that the song could finish there. If they are, for example, imperfect cadences, they will give the feeling that there is more to come. For example, a middle 8 ending on CEG makes it feel the need for the next part to continue back on FAC. This is known as V-I, the 'I' being F in this case.

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    +1 For discussing context, which extends beyond music with keys or functional harmony. I can create tone regions, pitch collections, or sonic pallets simply by emphasizing a tone or tones more than others. To that end, how you manipulate the sound is dependent on what you mean to do with it. – jjmusicnotes Jan 6 '15 at 4:53
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Chords are not bound by any specific key. They stand independent of Key. Any key with the notes F/A/C can have that F Major chord in it. There is also some styles / composers who may write music that has a very loose adherence to key. This chord may come in music if this is the case also.

There is also at times places in music where you move away from the key and us notes that are foreign to the key and this chord may very easily be used then as well.

So in closing chord usage often relies on the notes of the scale for the notes but can at times move away from such an idea. So don't get hang up on the chords = scale idea.

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