Motivation: Rapid technological progress in DNA sequencing has stimulated interest in compressing the vast datasets that are now routinely produced. Relatively little attention has been paid to compressing the quality scores that are assigned to each sequence, even though these scores may be harder to compress than the sequences themselves. By aggregating a set of reads into a compressed index, we find that the majority of bases can be predicted from the sequence of bases that are adjacent to them and hence are likely to be less informative for variant calling or other applications. The quality scores for such bases are aggressively compressed, leaving a relatively small number at full resolution. Since our approach relies directly on redundancy present in the reads, it does not need a reference sequence and is therefore applicable to data from metagenomics and de novo experiments as well as to resequencing data. Results: We show that a conservative smoothing strategy affecting 75% of the quality scores above Q2 leads to an overall quality score compression of 1 bit per value with a negligible effect on variant calling. A compression of 0.68 bit per quality value is achieved using a more aggressive smoothing strategy, again with a very small effect on variant calling. Availability: Code to construct the BWT and LCP-array on large genomic data sets is part of the BEETL library, available as a github respository at http://git@github.com:BEETL/BEETL.git .