Six-Four chord: inversion of tonic or dominant with non-chord tone?
Is the triad G-C-E an inversion of C major or a dominant with a non-chord tone?
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The three notes G C and E are 1,3 and 5 of C major - in C, E, G order it's root; in G C E order it's second inversion. Dominant with a non-chord tone? If a note has been changed in a triad chord, it becomes a different chord! It can be understood that it is the dominant triad of key F - that built off the V note - which is C, but any closed or open inversion of a triad still keeps its original root name.
Nobody can tell you what it is (in a given context). Obviously, in isolation without any context it's just an inverted C major triad, but that's a rather uninteresting and obvious fact.
A more reasonable question would concern the way it is analyzed, and indeed, both interpretations are possible. And note that the context is important. When used in an authentic cadence I64-V-I, then the I64 chord can be either interpreted as the second inversion of the tonic triad, or it can be analyzed as a dominant with non-chord tones that resolve to a consonant dominant. The first interpretation is more common in traditional texts, whereas the latter is more often found in modern texts.
I don't think there is an "answer" so much as a historically informed way to understand why it could be labelled in those two ways. Regardless of labels a cadential 6/4 functions one way in the common practice style.
There was a time when Roman numeral analysis and chord inversions were not part of music theory. This is critical to understand. There was a time when there wasn't a theory of chord roots and inversions! For reference points we could mention composers like Corelli and Bach. Their theory and notation used figured bass where a cadential 6/4 would look like this...
Historically, harmony was the result of counterpoint and chords were intervals above the bass. The thinking would have been something like: the bass goes from G to C and the G takes a chord of the sixth. The 4th over the G is dissonant and therefore must be resolved so it moves down a step to the 3rd. Then the G moves to the C.
From this perspective we do not have a form of tonic chord. It would be a form of dominant chord, because the bass is playing the dominant scale degree. Also, strictly speaking, it probably wouldn't have been called 'non-chord tones' as the 6/4 is the chord, but it's a dissonance requiring resolution. Probably I wouldn't have been called an appoggiatura.
To capture this historical view in modern Roman numerals it would look like...
As long as you mark the dissonant tones as appoggiaturas the V symbol under the cadential 6/4 is OK.
This...
...is the fully modern view labeling it as a second inversion tonic chord.
Personally, I like the older approach, because it allows me to think about it as the past composers did. But I also understand the modern way. The two aren't mutually exclusive.
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