We study edit distances between strings, based of based on operations of character substitutions, insertions, deletions and additionally consolidations and fragmentations. The two latter operations transform a sequence of characters into one character and vice-versa. They correspond to the compression and expansion in Dynamic Time-Warping algorithms for speech recognition and are also used for the formal analysis of written music. Such edit distances are not computable in general. We propose weighted automaton constructions to compute in polynomial time an edit distance taking into account both consolidations and deletions, or both fragmentations and insertions. We finally show that the optimal weight of sequences made of consolidations chained with fragmenta-tions, in that order, is computable in polynomial time, and not computable if we reverse the order of fragmentations and consolidations.