Given an irrational alpha and an x in the unit interval, the set of balanced
times, for which the same number of (k*alpha+x) (modulo one) are less than or
equal to one half as are larger than one half, is in general infinite, but
sparse in terms of density. We investigate the sparseness of this sequence in
terms of summation over reciprocals. Our results are that for the generic pair
(alpha,x), the resulting sum diverges, but there are certain exceptional alpha
for which the associated sums converge for every x.