Could you clarify a bit what you're looking for? If the distribution is normal, then you are reasonably limited in what you can do with the shape of the curve.
You might consider the 5th moment, which is a measure of tail asymmetry--2nd moment is variance, 3rd is skew, and 4th is kurtosis. I played around with generating distributions in R for the statistics class I teach, to use as examples, and found the code here to be incredibly helpful:
How to generate distributions given, mean, SD, skew and kurtosis in R?):
The following link has some useful thoughts.
https://stats.stackexchange.com/questions/2893/moments-of-a-distribution-any-use-for-partial-or-higher-moments