So, the question is: how far was Lord Burlington interested in issues of harmonic composition ?

I think the answer is likely to be, quite a lot.

His mother, Lady Juliana, was a supporter of opera, is likely to have attended the performance of Scarlatti’s Pyrrho & Demetrio at the Haymarket Theatre and is assumed to have been responsible for the commission to Pellegrini and Marco Ricci to do paintings for the walls of Burlington House (Burlington himself was only 15 in 1709). In 1712, Handel was given lodgings in Burlington House, so Burlington would have known him when he was at an impressionable age. On 17 May 1714, Burlington left London on the Grand Tour and, on his first Grand Tour, he was at least as interested in music as architecture – renting harpsichords for musical performances in Rome, Florence, Venice, and Paris, and attending the opera in Florence and Rome. He came back with three musicians, Filippo Amadei and the brothers Pietro and Prospero Castrucci, as part of his entourage. He also brought back 878 trunks and crates containing works of art.

So, when he later developed an interest in architecture, it is more than likely that he would have been interested in the relationship between music and architecture