Suppose that a person does not care to:
learn or play an instrument.
enjoy or learn about music theory (hereafter MT), even if (s)he listens to music.
Except anything concerning music, what does MT cultivate or enrich? What can MT do for someone satisfying 1 and 2 above?
Consider harmonic analysis: what things are consonant, what things aren't, what are cadences. If you look into this you find connections to physics (what are pitched sounds anyway), and psychology/neurology (psychoacoustics), and of course a lot of math (cents, commas and all that).
Also, if you look into the roots of music theory (or its geographic diversity), e.g. how serialism was a reaction to more standard harmonic techniques, which in turn evolved out of contrapuntal techniques... you're doing history, of a sort.
These specific connections and dependencies exist in addition to the general "how to study" aspect. If you study anything you (can) cultivate the tools techniques and strategies for learning. Studying theory outside of practice or appreciation requires "book-learning" techniques. When you start to connect it to more experiential aspects, appreciation or performance, you're exercising additional mental muscles.
An interesting question. What does any field of knowledge teach you outside of its field?
Indulge an analogy for a moment. What does Mathematics teach you, outside of its domain? I think answers like 'it's connected to physics' aren't satisfactory; you haven't left the domain of mathematics yet! I think a better answer is that it teaches you a structured way of thinking. Mathematics requires you to formally define the things you know, develop a mathematical model of that situation, solve it using a set of formal rules, and then relate the result back to some useful real world situation.
Back to music theory. I think it's a great example of an imperfect analytical tool. We seek to describe, classify, reproduce, evaluate an art form that is subjective. We derive rules from existing works. We struggle to adapt those rules to some new style, that is undeniably music, yet doesn't fit our existing models. A significant subset of musicians know these rules (guidelines?) on a purely intuitive level, and in some cases, intentionally avoid knowing anything about them at all. Imagine if mathematics worked like that!
I'm sure there are other fields with similar characteristics; perhaps the study of paintings? I probably shouldn't display my ignorance of that field. But I think the mental gymnastics that we must perform to understand the tangled ball of rules, guidelines and exceptions that we've invented is a useful way of thinking. Humans are messy. Many (most) subjects are less formal, logical, even rational than something like maths. We need to know how to make sense of them, and I think the thinking cultivated by well taught music theory can be applied to these other areas.
That seems very philosophical. Possibly bad philosophy. As may be evident, this is also not an area I have much training in. But I decided to post this anyway. Feel free to disagree completely in the comments. I like learning new things.
Music is forever intertwined with physics and mathematics. The sound that an instrument produces emanates with a wave that can be described by its frequency. Studying frequencies can reveal how certain groups of notes sound better together. When you look at the diagram of a sound wave, and plot another sound wave on top of it (because they are being played together), the resulting amplitudes reveal a single wave. The shape of the resulting wave will indicate evidence as to whether it sounds pleasing or not.
Music theory can be used in programming/computers, mathematics, pattern recognition and literacy (grammar/language acquisition as well!)
now of course MT isn't rule rigid or strict but are guidelines instead.
I think music theory has some overlap with language study (syntax, grammar, poetry) and I personally find those similarities interesting. There are other mathematical comparisons: sets, proportions, symmetries, etc. A computer programmer friend looked at my harmony flow chart and said it was a 'finite state automaton.' I think there are many ways music theory can be appreciated as a system of thought beyond its immediate application to music.
Many years ago, when I worked for a computer company, we always liked music majors as programmers. Writing a complicated program or even designing a processor is a lot like composing a piece of music; one has to make a large numbers of parts work together in an synchronized manner.
Knowing a bit about music theory dos make listening more enjoyable. One simply sees more deeply into a piece. (It doesn't interfere with enjoyment; knowledge is generally additive.)
