When to use (add13) or m7th chord first inversion for 6th chords

I am wondering when to use an inverted seventh chord or add13 when you need a 6th chord

Example: When should i use I(add13) or vi65 for C6 in C major?

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I think this mostly comes down to personal preference. People with a classical background tend to explain dissonances with inversions and suspensions, whereas people with a Jazz background tend to explain dissonances with chord extensions.

When analyzing something from the baroque or classical period, I would definitely go with the inversion, since chord extensions just weren’t a thing back then, so an add6 would certainly fail to convey the intention of the composer. Same goes for music that heavily draws from these periods, like a lot of 8 bit video game music.

Also if the chord in question is preceded by a dominant chord, that already tells you which one to use. So G7 to C6 would be V7 to I(add6), but if the progression is E7 to C6, V7/vi to vi65 would make a lot more sense.

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Same goes for other types of progressions too by the way. If we have something like C - G7/B - Bb6, I would definitely go with I - V65 - v65 instead of I - V65 - bVII(add6).

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Agreed with the above, additionally in a descending progression like the Andalusian cadence, I personally would go i-bVII(add6)-bVI-V rather than i-v65-bVI-V. Same goes for ascending sequences

I think it really depends on the context. I agree about the andalusian cadence. It’s such a well known progression that it would be weird to write it differently just because there’s an extension on one of the chords.

I don’t think it’s fair to say that all ascending and descending progressions containing sixth chords should be analyzed using chord extensions though. Just go ahead and listen to am - G6 - F6. Does this sound like i - bVII - bVI to you? It certainly doesn’t to me.

But then again I rarely listen to jazz or other genres that make heavy use of chord extensions, so maybe it’s really just a matter of habituation. As I’ve said before, I think in modern music there’s no definitve right or wrong way to label a sixth chord in most cases. Exceptions mostly occur when the sixth chord is part of a functional progression.