Figuring out tonic and dominant triads in any key.
Question: I would like to see all the tonic and dominant triad combinations with or without key signatures in clef written down. I can't seem to grasp this.

Answer: Displaying that in staff notation would not, I think, be of much help. It would look more confusing than it is and would require memorizing. Here's all that is really needed:

Starting on any key of the piano, use one hand to play the first five notes of a major scale (if you're not used to playing the scale, count: whole step, whole step, half step, whole step).

Your first finger is on the tonic, 3rd finger is on the mediant (middle note of the tonic triad), 5th finger is the dominant (top note of the tonic triad).

Now start on that 5th note and do the same thing. Your first finger is now on the dominant, 3rd finger is on the leading tone (middle note of the dominant triad), fifth note is on the supertonic (top note of the dominant triad). So you've just played the tonic and dominant triads for the tonic you began with.

That's really all there is to it. As for notation and note-naming, remember that every note in the scale has a different letter name, and no letter name is skipped. So if your first note is F#, the second one must be some kind of G. Since that first step is a whole step it must be G sharp. It could not be Ab, because that would mean you had skipped G. Following that principle, the scale on F# would begin F#, G#, A#, B, C#, and the dominant for that key would be on C#: C#, E#, G# (notice I had to call that middle note E# instead of F, because it was the third note from C and therefore needed to be some kind of E).

In staff notation, the lines and spaces correspond to letter names. So a "third" will always look like a skip from one space to the next one up or down, or one line to the next one up or down, regardless of sharps and flats. A "second" will always be the next line or space, and so on.

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