A number standing for how well a thing did. Also the written form of a piece of music — both are performance set down as marks.
Old Norse skor (a notch, a tally-mark, a set of twenty), related to the idea of cutting or shearing; a score was first a mark cut to keep count, and a musical score kept that same sense of marks set down to be read.
A number standing for how well a thing did, or the written form of a piece of music — both are performance reduced to marks cut for counting, the score's oldest job.