You do realize that even though u have a 3 in the bass, if it is apart of the 1 chord, u still have a 1 chord, right? The only way u can have a 3 chord is if the 3 is the root of the chord. Anything other than that is an inversion.
Just a word of caution here. The above is reasonably true of triads, once you get in chords of 4 or more notes, the idea of inversion gets a little trickier.
For example, the notes C E G A, is it a C6 chord or an Am7 in first inversion? Or the notes G B D E, is it G6 or Em7 in first inversion? The question actually becomes "which is the true root of the chord?" This actually gets down to how the chord functions (and that to can be a bit gray).
Another one from the old theory days is the "Tonic six-four (I chord in second inversion)/Dominant" progression. Is the Tonic actually a 2nd inversion I (G C E) or is it an incomplete V13 (G [B D F A] C E) with the C and E are embellishments of the B and D? Try this, play the progression as triads:
G / G C E
G / G B D
C / E G C.
Now simply add an F (the 7th) to the left hand:
G F / G C E
G F / G B D
C / E G A C.
It's a little spicier, but otherwise does not affect the overall effect or character of the progression, yet the G - F bass makes it decidedly some form of Gx7 [an interval of a 7th above the bass pretty well establishes the bass as the root] as opposed to some form of C chord.
One other instance that might show that all is not what it seems. Try this:
C E / G B D F# [note, the C - E is best as a 10th (an octave and a 3rd)]
C E / G B C E
B G / F# B D
Now try it again with G D in place of the B G of the last chord.
Is the first chord a CMaj7 #11 (C E G B D F#) or a D13 (D F# [A] C E G B) in 3rd inversion?
Now, Change the L.H. C E to D C (add the A if you like). This time there is no denial that it is some form of Dx7.
Dx7 to G is a natural V - I progression. In the first one above, with C as root we have a IV - I plagal progression (which is entirely valid - the "Amen" progression) but with it containing the same notes as the D13, it is a bit ambiguous, particularly since it has a stronger pull to I than a normal IV - I. It is sort of a cross between the two. (it would also make a great choral "Amen" in 6 part harmony).
I will say, if I were to write it in chord symbols, I would opt for CMaj#11 - C Maj7 - G Maj 7 as opposed to D13/C ... since it is clearer to read. But because it has a bit of the V effect, I realize that it may be possible as a substitute for a V - I cadence in certain instances and for certain effects.
I bring this up to just caution against stating too many "absolutes" in music. Not all is as it sometimes seems. And some "absolutes" become a source of confusion, particularly when often, so little music fits into the "rule". I remember when first being taught about "Sonata Form" and all of the parts that a sonata has and how they are put together. In 40 years, I don't think that I have ever played a sonata that fit the "Sonata Form." In fact, the rules as the had been taught in the past would actually deny most of the Sonatas of Mozart, Haydn, and Beethoven as actually being Sonatas! You mean these guys didn't know how to write a sonata?
And then there were the classes on "Fugue Form" (fugue is not a form -- it's a way of writing music) and if one were to follow those rules, one would have to say that Bach maybe wrote one or two fugues since most of them don't fit the description!
But that is the beauty of music. When you realize that we have only 12 possible roots upon which to build the entire harmonic system and a finite number of possibilities to combine the notes above the roots into harmonies, and only 12 notes upon which to build melodies, it is almost mind boggling to realize how much music of such variety has been created in the western world over nearly 1 1/2 millenniums, and that so much music has been built on the same basic harmonic system that was in practice before Bach, nearly 400 years ago.
