How many free parameters are there in a covariance matrix?

I'm trying to calculate BIC for a fairly complicated model which includes several covariance matrices, and it just occurred to me that I don't actually know how many free parameters this represents. If I have a D-by-D covariance matrix, the naive answer is that this represents D(D+1)/2 free parameters, because the matrix must be symmetric -- but the matrix must also be positive definite, which is a stronger condition, so I'm guessing that the actual number is something less than that. Any thoughts?

(x-posted to stat_geeks)