[but] The author is missing or skipping key pieces of information about why we use the notes we use, how they were derived, and why they sound the way they do to us.
Edit. There was zero sarcasm in my comment. I really love this style of writing.
The missing concept here is the harmonic series. A pitched note is not generally a sine wave, it is properly modeled as a series of sines waves combined. These sine waves go up in mathematical multiples.
So say we have a note at 100hz. Simplistically it is actually 100, 200, 300, 400 etc all at once. So if 100 were A, 200 is A, 400 is A, 300 is … E, so 150 is E. Etc.
None of this is arbitrary because the When the waves combine the result is more ordered or more chaotic, depending on the ratios of the waves, which our ears hear as consonant or dissonant.
Best I can do from a phone.