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. As 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 re-sequencing data.