In this case, your guess did not pan out :-) In Indian languages (from which both sanjit and sangeeta originate), j and hard g are very distinct sounds. The th/t alteration is alternate renditions of the same sound into the Roman script: the th here would be preferred in South India, and t in the north.
The Sanskrit prefix sam- (cognate with English same) `means' together, and the root gai (with Indoeuropean cognates) means to sing. sangIta is the common word for music, the art of dance and music, person providing musical accompaniment, and other similar meanings. sangItA is the feminine form.