Thanks for bringing this material together in a logical sequence. I've been interested in maths and music for many years, and still found some nice surprises.
I appreciated your mention of Chinese music theory. In fact, the ancient Greeks and Chinese were both independently aware of the mathematical problems of musical tuning, and Europeans and Chinese also later solved the problem through equal temperament at about the same time, also apparently independently.
From your title, I thought you might also make the connection to chemistry through the periodic table, which also proceeds in octaves of elements with similar physical qualities.
If you like synthesis across subjects, you can also add geometry: the problem of equal temperament is essentially to construct the cube root of 2, which is the same as the famous Delian problem in geometry to double the volume of a cube. Joseph Needham says the Chinese didn't have an equivalent of the Delian problem, but they had the equivalent equal temperament problem in music.
Also, more tenuously but for the pleasure of quadrivium completists, to add astronomy to the synthesis, find the orbit of the planet which will take root 2 years to rotate the sun: it's cube root 2 (Kepler's Third Law of Planetary motion, the real music of the spheres)!