Maybe I missed something but he never actually addresses how to play by ear. I opened the PDF and started reading, and eventually found myself jumping past the mathematical harmonic mumbo-jumbo and into the good stuff, and unfortunately I was disappointed. He ended up covering the basics of Music Theory found in a lot of books with no real application to ear training.

1) Learning Material

I can sympathize with the author's plight about not having access to good learning materials or schools. For those who want to pick up guitar and want a bit of extra "why things are the way they are" explanations, I have found books such as "Fretboard Logic" and "The Advancing Guitarist" to be useful. There are plenty more, like Ted Greene's "Chord Chemistry" and whatnot that have insight on ear training. Also, you have software today like Ear Master, Rocksmith and others that can help and are fun as well. YouTube is full of backing tracks that you can use to practice and hear how various notes, phrasings, melodies sound on top of the backing chords.

2) Notes

Personally I have not found using "natural" note names to be that useful; using relative note names like root, second, third is to me much more functional, because the third of any key sounds like a third. The whole "returning to _root_" thing he mentions should be hint enough.

3) Ear Training

His story of his dad being a multi-instrumentalist struck a chord with me (sorry). I have friends and teachers who are like this and I've come to believe that it's less about knowing any of what he is talking about (although it is important) and more about their storytelling ability and translating it over into music. I doubt his dad was a shredder, but his dad's ability to pick up and play an instrument wowed him all the same.

To put another way - most people are lucky if they can passably play one instrument. The question is, how is it that there are people that can pick up an instrument and leave an impression on others? I doubt they have a massive IQ or some arcane knowledge of music theory.

Storytelling and Composition are the same thing. All the high brow shit about Principles of Composition and Counterpoint and whatnot does not matter. Being able to evoke imagery in someone's mind or make them feel a certain way with music is something that the first half of the author's paper cannot begin to rationalize.

I believe it's important to really _hear_ the music you are listening to. It doesn't matter if there are no vocals explicitly telling you how to feel (ie. pop music); it is on you to allow yourself to engage with what you're hearing and to eventually formulate an opinion on how something sounds and how it makes you feel. I've heard music teachers expressing this sentiment, and it's no surprise that it goes over most people's heads. It's not a subject for the dabblers. Composers for film, Classical composers etc excel at this. Learning from them is a masterclass in both ear training and storytelling. They've written plenty of books on this subject.

Take a song you know, and start hearing it on a deeper level. Start transcribing it first by singing/humming/whistling along to it and then with your instrument. Take a chunk of it, and alter it in some way. Slow it down, add vibrato if there isn't already, elongate certain passages.

I can't stress enough listening to music that actually tries to tell a story, because that's when you will hear the most _functional_ use of chord changes, progressions, melodies and vocals, which will ultimately aid in your ear training abilities and answer "why" better than the first half of the paper. Yes, you still have to sit down and woodshed (scales, progressions, metronome practice, etc) and whatnot, but that's assumed to be the case.

PS. Lastly, you can find most of his mathematical explanations on Music Theory Stack Exchange. None of it will help you play by ear any better.