  Scoring systems are an extremely important class of election systems. A
length-$m$ (so-called) scoring vector applies only to $m$-candidate elections.
To handle general elections, one must use a family of vectors, one per length.
The most elegant approach to making sure such families are "family-like" is the
recently introduced notion of (polynomial-time uniform) pure scoring rules
[Betzler and Dorn 2010], where each scoring vector is obtained from its
precursor by adding one new coefficient. We obtain the first dichotomy theorem
for pure scoring rules for a control problem. In particular, for constructive
control by adding voters (CCAV), we show that CCAV is solvable in polynomial
time for $k$-approval with $k \leq 3$, $k$-veto with $k \leq 2$, every pure
scoring rule in which only the two top-rated candidates gain nonzero scores,
and a particular rule that is a "hybrid" of 1-approval and 1-veto. For all
other pure scoring rules, CCAV is NP-complete. We also investigate the
descriptive richness of different models for defining pure scoring rules,
proving how more rule-generation time gives more rules, proving that rationals
give more rules than do the natural numbers, and proving that some restrictions
previously thought to be "w.l.o.g." in fact do lose generality.
