For overtones, think of a trumpet (or any other brass instrument for that matter). If you play with open valves, the notes you can play are as follows: C, G, C, E, G, etc... I don't know all the notes above that, partly because I don't play trumpet... But yes, the first few, and most audible, overtones create a major triad. After these notes, the intervals continue getting smaller and smaller.
But then you have to remember, the intervals don't fit into our modern scale. The intervals get smaller, but not in line with our scale. There's equal temperament tuning, well temperament, just intonation, and quite a few other tuning methods. By actual science, the scale we use today is "out of tune". The reason we use it though is because back before it, if you transposed or modulated to another key, or even if you jumped an octave or two, it would sound very different from your original key. That's why Bach wrote The Well-Tempered Klavier, and that's why Benjamin Britten's Serenade for Tenor, Horn, and Strings, which uses a natural horn, sounds "out of tune".
It's a really interesting concept, all the different tuning methods, that a lot of composers explore at school. I mean, most of us grow up thinking that the scale we always hear is the only way to tune...
