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We evaluate the triple correlation of eigenvalues of the Laplacian on genericflat tori in an averaged sense. As two consequence we show that (a) the limitinferior (resp. limit superior) of the triple correlation is almost surely atmost (resp. at least) Poissonian, and that (b) almost all flat tori containinfinitely many gaps in their spectrum that are at least 2.006 times longerthan the average gap. The significance of the constant 2.006 lies in the factthat there exist sequences with Poissonian pair correlation and withconsecutive spacings bounded uniformly from above by 2, as we also prove inthis paper. Thus our result goes beyond what can be deduced solely from thePoissonian behavior of the pair correlation.

Triple correlation and long gaps in the spectrum of flat tori