You are right on the first two points...
- Roman numeral indicates the scale degree
- Upper case means major triad, lower case means minor (also include suffix
o means diminished triad and
+ means augmented)
But the important thing is the scale in question is given by the key signature. And there is a sort of matching and alteration that can happen with the symbols depending on the key signature and the intended chord qualities.
Keep in mind there is a system that simply labels scale degrees with all upper case Roman numerals regardless of chord quality.
Also, you should give key signatures with Roman numerals for clarity.
- a sharp, flat, natural prefixing the Roman numeral means a chromatic alteration from the key signature.
C major, just
III could mean simple the third scale degree,
E or any triad rooted on
In case-sensitive style
iii is used because the diatonic triad on the third scale degree is minor. If it were given as
C: III the chord would be
E major. (A small aside: that chord could be considered a secondary dominant to chord
A minor. To clearly convey that secondary dominant sense, you would write
C:V/vi. The secondary dominant is sort of the standard harmony analysis way, changing the letter case to alter diatonic from the key signature is more of a jazz way to label chords.) Finally, you could prefix the number with a flat -
♭III - and that would mean the
E is altered from the
C major key signature to become
E♭. Conventionally this is conceived as "borrowing" from the key signature of
C minor which has three flats
A♭, E♭, B♭, and the diatonic triad in
C minor on the third scale degree is
E♭ major. So...
C:♭III is the chord "borrowed" from
If the music was initially considered to be in
C minor, the key signature already includes an
E♭ and so no alteration or prefix is involved:
Cm:III is the diatonic chord in
Notice the difference in symbols depending on whether you give the key signature before the Roman numeral symbols.