Calculations of the relative stability of C-118, C-119, and C-120, pure carbon species with both trivalent and tetravalent atoms, show the interplay between the various forms of molecular carbon and under which conditions they are thermodynamically accessible. The analysis is performed by quantum chemical screening of all the classical fullerenes with isolated pentagons (8148 isomers for C-118 and 10774 isomers for C-120) for comparison with the most stable structure of C-119 and C-60 dimer. While fullerenes show similar trends of stability over a range of temperatures, introduction of sp(3) carbons profoundly affects them. It is concluded that very high synthetic temperatures may favor formation of clusters with tetravalent atoms.