: How does tonality work with chords? By tonality, I don't necessarily mean 'tonal music' but rather the actual functionality of tonality in music. If you were to play a chord progression with no melody notes, is a tonal center

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There may well be more emphasis on a note rather than a chord. As mentioned at the end of your question.
Modal interchange and parallel keys are both used regularly in pieces, which says one particular note rather than a chord is given more weight.
Pieces which use only diatonic notes will more readily be recognisable as being in a key - or two. There's always the relative major/minor waiting in the wings.
Given your example of C, G and Em, there's an ambiguity with them. All three could belong to key C, key G, or key Em. A bigger clue could be when G is followed directly by C - the perfect (authentic) cadence is probably the most used - V>I, possibly putting it into key C. But that's by no means an acid test. Until we know what other chords come after, that's only a vague guess.
Then there's the fact that modes of one key will all contain the same notes (and thus chords) as each other - giving seven possible 'answers'. Here, the constant return to either a particular chord or note usually signifies the 'key'. So the answer is either or both. Obviously emphasis caused by where particular notes/chords are used in a piece will also have a bearing on the clues. A simple example may be if every 4th bar, the same note/chord is used woud point to that being 'home', therefore the root note/key.

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If the chords are built strictly on a diatonic scale, then the progression will tend to sound like it's in the key of that scale. How strongly it will tend to sound that way depends on how the chords are used.
In your example, if G was a dominant chord, then the progression would more strongly sound as in the key of C. If your G is just a triad, then those 3 chords may fit in both C major and G major scales, so what feels like "home" will depend on how long and in which position you will place the chords.
Furthermore, still using your example, both G major and E minor (E is the relative minor of G major) could function as the tonality. Again it depends on where you put the emphasis among those chords.
And if the chords are not strictly related to a specific key, things get progressively even more ambiguous and subjective. The final effect can be anywhere from clearly sounding like in a particular tonality, in spite of altered chords etc., to a sequence of different tonalities, to some more ambiguous situations.