We consider a general framework for voting systems with arbitrary types of ballots such as orders of preference, grades, etc. We investigate their manipulability: in what states of the population may a coalition of electors, by casting an insincere ballot, secure a result that is better from their point of view?We define a straightforward way to make a voting system respect the Condorcet criterion and we prove that, for a large class of voting systems, this Condorcification is at most as manipulable as the original system. Then we exhibit a sufficient condition, which is frequently met, that ensures this improvement to be strict.These results show that the investigation for a reasonable voting system with minimal manipulability can be restricted to systems that meet the Condorcet criterion.