By god finally!
I was actually, nerdy as I am, thinking of it in terms of solutions to indefinite integrals. They all hold the same pattern, but can be shifted by a constant. Awesome, right?Now, if you're really smart, you're thinking "wait a minute, that series of intervals doesn't fully describe the C major scale; it tells me how far I have to jump from each note to the next, but it doesn't tell me what note to start on,"