Another important chord would be needed (very important for trad. blues, R&B, funk): seventh chord without the 3rd (1, 5, b7) and seventh ninth chord without the third (1, 5, b7, 9).
The first one is commonly used in traditional blues and R&B ("higher ground" by Stevie Wonder is one example. A chord with a 3rd wouldn't sound right in such song.
The second one is commonly used in Funk (most James Brown tunes) and songs such as "play that funky music" for example.
It could be designated as C7(omit3) or C7(no3rd), C9(omit3) or C9(no3rd)