Marcus Maroney has a blistering reply to Kyle Gann's proposal to do away with figured bass. I completely agree with Marcus, but teaching a brief introduction to Hindemith's harmonic theory today reminded me of another way to teach inversions.
Hindemith divided all possible chords into six categories:
I - chords with no seconds, sevenths, or tritones (only major or minor triads)
II - chords with tritones, but no minor seconds or major sevenths (dominant sevenths and half-diminished sevenths, some secundal chords)
III - chords with no tritones, but does have seconds or sevenths (or both) (lots of options)
IV - chords with tritones and minor seconds or major sevenths (lots of options)
V - chords with no tritones but indetermininate root (augmented triad and quartal chord)
VI - chords with predominant tritones, indeterminate roots (diminished triad, fully diminished seventh chord)
Within each category except V and VI, there are subdivisions for inversion: Root position = 1, any inversion = 2. (Hindemith had a theory of interval roots that he used to identify roots of chords, which is only confusing with very complex chords.)
I think Kyle would like this system, as it has only two categories to remember for inversion, and no pesky figured bass symbols, simply "1" or "2". Unfortunately, it does nothing to show the dissonance of a 4/2 chord (seventh chord in third inversion) as compared to a 6/5 or 4/3 inversion. Kyle gets all worked up that the second inversion triad is significantly different from the first inversion triad, but the second inversion seventh chord is not. I disagree, as a V 4/3 chord can have an upwardly resolving seventh that the V 6/5 never could (in functional tonal music). But more importantly, these inversions are distinctly different from the third inversion, which has very specific rules of resolution, just like the second inversion triad.
As for using figured bass instead of "1", "2" and "3", figured bass allows indications of many things besides chord inversions. It indicates chromatic alterations in a chord, including the ever-important Ti in minor keys that first-years often forget. Figured bass also provides an easy way of indicating all the different types of suspensions -- including double and triple suspensions and retardations -- and can be used to describe types of sequences (see the new Laitz text for this). What other method of chord labelling provides such power and is more memorable (7th inversions are simply counting down from 7: 7, 6/5, 4/3, 2; all figured bass is about describing intervals up from the bass note)?