Music theory for nerds

This gets pretty complicated and subjective.

Strictly speaking, a "note" cannot define a key. To determine a key, you really need at least 2 intervals, wherein interval a somehow resolves to interval b. Keys are just a mapping of relationships; how does each of the 12 notes relate to each other? If you just play a C major chord, you aren't really playing in the key of CMaj, you're just playing a CMaj chord.

For example: Play B+G, then C+G. This is (arguably) C major. The B note resolves into the C note. That's the 7th moving into the Tonic (#1). This is an example of a V-I resolution, the strongest possible. If you'd like more examples, look up "cadences". There are 'principles' for determining the key, but they should not be understood as proofs.

I think you're also asking if/how to change keys. This is not only possible, but often highly encouraged. There are two extremely common key changes in pop music: 1) Up one step: This is incredibly common and can be heard in "I Will Always Love You" (album version, right after the 3 minute mark when she starts singing the chorus again). Once you start hearing this, you can't stop. It's all over the place. If we're in the key of C, we'll just play a D chord with a lot of confidence, typically after a G chord. 2) Minor to Major (or reverse, moving from sine to cosine in the analogy): This is much more subtle than the previous example. Typically you'll recognize this as a change in 'mood' or 'feeling'. You'll find a lot of Am-Em changes, then the chorus will be a lot of C-G changes. The tonic changes from Am to C, even though all the chords and notes being used are still the same (although the duration of said chords will probably be different).

Most "classical" music moves around keys pretty frequently, in extreme cases multiple times within a measure. And then there are chords/sections in which the key is debatable if not indiscernible (I would argue that the intro to Smashing Pumpkins "1979" is changing keys every 2 measures, but I think there's also a strong argument for it being in a single key)

Thinking about keys in terms of phases is a good basic explanation, but ultimately phase is much easier to measure. Perhaps I see it this way because my understanding of music is much better than my understanding of physics.

Just as it can be said that music requires at least 3 consecutive notes or beats to have a tempo and a rhythm, it can also be said that it requires a "cadence" which at bare minimum is two consecutive pitched notes but usually at least three.

C-G-C is a simple 'harmonic' cadence. It is, for example, the basis of oom-pah music (think military marches which repeat C-G in the bass with a melody on top, and end on C when the melody is done).

There is a form of musical analysis (Schenkerian) that can be used to show that nearly all music has the basic form C-G-C, which is usually expressed as I-V-I (chord I is C major and chord V is the 5th chord of C major, which is G major). It is of course nowhere near as simple as I've expressed here!

So resolution is basically the cadence of the piece's harmony, much like a story has beginning, middle and end, so does harmony.

The reason chord V and chord I are so powerfully related is that the 3rd note of V is the leading note of chord I and the 7th note of V is the 4th of chord I. This is actually significant to understand because it is how one can see the purest relationship between harmony and melody. In early music (eg medieval plainsong) the fundamentals of more complex harmony were first developed from melody.

In the case of C major with two voices, the notes would provide resolution from notes F and B to notes E and C. The FB is a tritone, which is dissonant and wants to be resolved in the human ear. The EC is the major triad of C major, which is harmonious and consonant and resolves the dissonance of FB.

I'm not an expert, but Indian classical music isn't known for having harmony in the same sense [1]. Modern Indian music is influenced by Western music. I know even less about Chinese music, which isn't mentioned in that article. But in all the sources I've read, the concept of deeply developed chordal harmony is pretty unique to the Western music tradition.

[1] https://en.wikipedia.org/wiki/Harmony

C is 0. Each half-tone up is +1.

If you really want to keep it concise you can write it as base-12 numbers.

    C1 C1# D1 D1# E1 F1 F1# G1 G1# A1 A1# B1 C2 C2# D2 D2# E2 F2 F2# G2 G2# A2 A2# B2 ...
    00 01  02 03  04 05 06  07 08  09 0a  0b 10 11  12 13  14 15 16  17 18  19 1a  1b ...

First number is octave. Second number is half-tone in that octave. Translating is just mechanical addition.

EDIT: on second thought making it base-12 just to save some space makes no sense, people are good with base-10, just keep the numbers.

Many (but not all, or even the majority) music classrooms around the world teach students fluency in solfege (do-re-mi), and everyone I've spoken to about it agrees it helps out a lot in the way you describe.