The first doctorate (for really writing a thesis that is) in Carnatic music was Harikesanallur L Muthiah Bhagavatar. This was awarded in 1942 by the Travancore University for his Sangita Kalpadrumam, a work of which only one part was ever printed. The oldest surviving published book (in the modern sense) on Carnatic music is the Sangita Sarvartha Sara Sangrahamu of Vina Ramanujayya. This was released on April 15, 1859.
Margazhi Moment - Ph.D Bhagavatar