As a wannabe musician and beginner jazz player, I feel music is explained all wrong. Unfortunately there's a lot of baggage from older eras where music wasn't well understood and was mostly a collection of heuristics. The fact that harmony understanding is usually explained in a chronological fashion probably doesn't help.

The problem (and I guess you know this already) is that although music is analyzable with math, how we experience music isn't entirely formalizable.

Music isn't math, just like physics isn't math. Math is just the tool we use to formalize and study them.

> Music notation is mostly wrong to me, it doesn't explain a lot.

Just like words don't explain a lot by themselves: their relationships with each other, the whole text, and your tacit knowledge and reasoning do explain far more.

Take this chord chart (excerpt from a jazz tune):

C6 | F7 F#º7 | C6/G F9 | Em7 A7

It doesn't explain anything itself, but with a bit of analysis (and tacit knowledge) you can see patterns. E.g., there's a voice which goes in chromatic ascension and then is held (hinted by the C6 with a G bass and the explicit 9th in F9):

C-[E]-G-A |
[F]-A#-C-Eb [F#]-A#-C-Eb |
C-E-[G]-A F-A#-C-Eb-[G] |
E-[G]-B-D A-C-E-[G]

Or only the ascending voice:

E | F F# | G G | G G

But it can also be seen (and interpreted) as a purely ascending voice (though not chromatic):

E | F F# | G A# | B C (though I'm not convinced with that G to A# gap)

That line is found in the chord chart too (check it yourself). Which one is correct? None. And both. And there are many more hidden.

It isn't obvious, but those relationships are there, and they're important, and you're supposed to be able to see them with your tacit knowledge (and highlight any of them in your playing if you want to!).

Of course it could be explicitly remarked and it sometimes is, specially in classical music or when the composer wants to highlight it himself. But then you lose a lot of what's jazz, like how different people view the same standard and express the subtleties as their music.

Also, there are so many hidden subtleties that the small chord chart would explode into a huge collection of remarks and explanations (imagine explaining what each word in a story means!) and those would probably differ depending on who did the analysis (like what exactly metaphors and allegories mean in text).

And that's why you've never found the correct formula: there isn't. If Einstein would've sworn by classical mechanics formulas, we would've never had relativity. Knowing those formulas is good though: each of them strengthen your understanding little by little. Einstein wouldn't have come up with relativity without classical mechanics.

The beauty here is that those relationships work because notes are just multiples of each other. Why do C and C# sound "awful" (unless you intend them to sound like that) played together while C and E (a major 3rd) sound beautiful? Because of their interference pattern:

C+C#: http://www.wolframalpha.com/input/?i=%28sin%28261.63*x%29+%2...

C+E: http://www.wolframalpha.com/input/?i=%28sin%28261.63*x%29+%2...

Is this math? Yes. Is your brain doing the math? Nope. But the math explains why your brain likes C+E more: their interference pattern is less chaotic (i.e. more harmonic) in the short term. Also, C+C# generates a weird long-term beat frequency[1]. Of course this doesn't explain the cognitive part (why does my brain like harmonic interferences?) but you get my point.

Do players/composers think about these interference patterns? Nope, because they use music notation, which highlights their properties without the burden of interpreting the sound wave frequencies. I can instantly recognize a major 3rd in C+E, not so much in sin(261.63 * t) + sin(329.63 * t).

The musical staff seems weird but it is comfortable too: chords formed with stacked thirds (the classic way of making chords) will always have their notes either on lines or spaces but not both, regardless of whether the thirds are major or minor. The oddball (like a chord with a 9th) really stands out. It also makes it easy to see if a chord is major or minor: three consecutive stacked thirds with a D bass and a # on the second third will be D major (D is minor in the key of C, thus why its second third has to be raised).

Sorry for the rambling :P This is what happens to my brain on music.

[1] http://en.wikipedia.org/wiki/Beat_%28acoustics%29

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Did you study jazz theory? Something tells me you'd love it, and breaking with pre-jazz analysis definitely expanded my horizons. I'm slowly learning it (self-teaching jazz isn't easy, 2 years already and I still feel like a beginner) but it's definitely rewarding.

Here's some theory to get started if you're interested, both classical and jazz:

http://www.musictheory.net/

http://www.cs.uml.edu/~stu/JazzTheory.pdf