"Croton says things like "This harmonization is what later became known as the second inversion of the dominant seventh chord", i.e., it's a lot like what we know as harmony, but not quite the same thing."


That isn't what that quote means, all it means is that when figured bass first emerged there was no concept of chords as wwe know them now.

To understand the mindset you have to go back to the evolution of the western harmonic system beginning pretty muh in medieval times, passing through rennaissance times and into the Baroque era.

This music in the west began with monks singing single melodic lines, and then noticing the effect of having more than one note sounding simultaneously.

this developed into the singing of multiple melodic lines in unison.

They found that some melodic lines worked better together than other melodic lines andd set about trying to figure out why, and discovered that some intervals sounded better than others.

They measured these intervals from the bottom voice otherwise known as the bass note.

By the time the Baroque era had arrived, a pretty comlex system of harmony had developed, which when viewed today can easily be broken down into chords as we understand them today, but it wasnt't until a chap called Rameau wrote a book called "la traite d'harmonie" that western civilization began to think in this way.

Rameau basically observed patterns in harmony and developed the idea of chords in inversion.

So he took, for example, a 6/4 chord with its bass note on G and said that functionally this was the same thing as a stack of 3rds on top of C, but with the moved C up to the top voice.

In other words, instead of viewing it as being one of a number of chord possibilities constructed from G, he observed that it was the same chord as C major, but in second inversion.

so he defined the idea of chords existing within a key.

eg . in the Key of C major there are 7 triads, each with their root on a different note of the scale.

They are the chords of C major, D minor, E minor, F major, G major, A minor and B diminished.

These chords are built by stacking note up in intervals of a third each.

So the interval from C to E is a third, and the intervall from E to G is a third, so we can construct the chord of C major.

From D we build D, F, A = D minor

etc.

After Rameau, we would see D minor as chord ii in the key of C major.

If we were to have a chord with F in its bass and then A and D above it, we would then call it chord iib in the key of C (still D minor but in first inversion).

If we had one with A in the bass and then D and F above it, we would call it chord iic in the key of C major (the chord of D minor in 2nd inversion)


Before Rameau, we would have seen these three different inversions of the same chord as being related to the bass note, We wouldn't have recognized them as being the same chord at all, but three different harmonization possibilities in the context of three concurrent melodies working together.

In retrospect we can look at scores predating Rameau and we can analyse them using modern chord analysis. The harmonies aren't really any different, but the way we understand them is more evolved.