The k-spectrum of a word is the multiset of its non-contiguous subwords of length k. For given k, how small can n be for a pair of different words of length n to exist, with equal k- spectra? From the Thue-Morse word we find that n is at most 2k. The construction of this paper decreases this upper bound to θk, where $\bumpeq$ is the golden ratio; the construction was found, though not published, over thirty years ago. Recently the bound has been further reduced, but remains considerably greater than the greatest known lower bound.