score (n.)

late Old English scoru "twenty," from Old Norse skor "mark, notch, incision; a rift in rock," also, in Icelandic, "twenty," from Proto-Germanic *skur-, from PIE root *sker- (1) "to cut."

The connecting notion probably is counting large numbers (of sheep, etc.) with a notch in a stick for each 20. That way of counting, called vigesimalism, also exists in French: In Old French, "twenty" (vint) or a multiple of it could be used as a base, as in vint et doze ("32"), dous vinz et diz ("50"). Vigesimalism was or is a feature of Welsh, Irish, Gaelic and Breton (as well as non-IE Basque), and it is speculated that the English and the French picked it up from the Celts. Compare tally (n.).

The prehistoric sense of the Germanic word, then, likely was "straight mark like a scratch, line drawn by a sharp instrument," but in English this is attested only from c. 1400, along with the sense "mark made (on a chalkboard, etc.) to keep count of a customer's drinks in a tavern." This sense was extended by 1670s to "mark made for purpose of recording a point in a game or match," and thus "aggregate of points made by contestants in certain games and matches" (1742, originally in whist).

From the tavern-keeping sense comes the meaning "amount on an innkeeper's bill" (c. 1600) and thus the figurative verbal expression settle scores (1775). Meaning "printed piece of music" first recorded 1701, said to be from the practice of connecting related staves by scores of lines. Especially "music composed for a film" (1927). Meaning "act of obtaining narcotic drugs" is by 1951.

Scoreboard is from 1826; score-keeping- from 1905; newspaper sports section score line is from 1965; baseball score-card is from 1877.

score (v.)

"to cut with incisions or notches," c. 1400; "to record by means of notches" (late 14c.); see score (n.). Meanings "to keep record of the scores in a game, etc." and "to make or add a point for one's side in a game, etc." both attested from 1742. The slang sense, in reference to men, "achieve intercourse" first recorded 1960. Meaning "to be scorekeeper, to keep the score in a game or contest" is from 1846. In the musical sense from 1839. Related: Scored; scoring.

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Definitions of score from WordNet
score (n.)
a number or letter indicating quality (especially of a student's performance);
what was your score on your homework?
Synonyms: mark / grade
score (n.)
a written form of a musical composition; parts for different instruments appear on separate staves on large pages;
he studied the score of the sonata
Synonyms: musical score
score (n.)
a number that expresses the accomplishment of a team or an individual in a game or contest;
the score was 7 to 0
score (n.)
a set of twenty members;
a score were sent out but only one returned
score (n.)
he tried to blame the victim but his success on that score was doubtful
Synonyms: account
score (n.)
the facts about an actual situation;
he didn't know the score
score (n.)
an amount due (as at a restaurant or bar);
add it to my score and I'll settle later
score (n.)
a slight surface cut (especially a notch that is made to keep a tally);
Synonyms: scotch
score (n.)
a resentment strong enough to justify retaliation;
settling a score
Synonyms: grudge / grievance
score (n.)
the act of scoring in a game or sport;
the winning score came with less than a minute left to play
score (n.)
a seduction culminating in sexual intercourse;
calling his seduction of the girl a `score' was a typical example of male slang
Synonyms: sexual conquest
score (v.)
gain points in a game;
The home team scored many times
Synonyms: hit / tally / rack up
score (v.)
make small marks into the surface of;
score the clay before firing it
Synonyms: nock / mark
score (v.)
make underscoring marks;
Synonyms: mark
score (v.)
write a musical score for;
score (v.)
induce to have sex;
Did you score last night?
Synonyms: seduce / make
score (v.)
get a certain number or letter indicating quality or performance;
He scored a 200
She scored high on the SAT
score (v.)
assign a grade or rank to, according to one's evaluation;
score the SAT essays
Synonyms: grade / mark