late Old English scoru "twenty," from Old Norse skor "mark, tally," also, in Icelandic, "twenty," from Proto-Germanic *skura-, from PIE root *(s)ker- "to cut" (cf. Old English sceran; see shear).
The connecting notion is perhaps counting large numbers (of sheep, etc.) with a notch in a stick for each 20. This counting notion is the origin of the modern sense in sports (1742, originally in whist). In Old French, "twenty" (vint) or a multiple of it could be used as a base, e.g. vint et doze ("32"), dous vinz et diz ("50"). Meaning "printed piece of music" first recorded 1701, from the practice of connecting related staves by scores of lines.