Sax

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early 20th century: abbreviation.

wiktionary

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From Middle English sax, sex, from Old English seax(“a knife, hip-knife, an instrument for cutting, a short sword, dirk, dagger”), from Proto-Germanic *sahsą(“stone chip, knife”), from Proto-Indo-European *sek-(“to cut”). Cognate with North Frisian sax(“knife, sword”), Middle Dutch sas(“knife”), Middle Low German sax(“knife”), Middle High German sahs(“a knife”), Danish saks(“a pair of scissors”), Swedish sax(“a pair of scissors”), Icelandic sax(“a short heavy sword”), Latin secō(“cut”). See also Saxon, saw.

Clipping of  saxophone. Distantly related to etymology 1 above, because the “Sax” surname is a cognate. 

etymonline

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sax (n.)

1923, colloquial shortening of saxophone.