Here is a still from my spiral spectrogram for a synthetic 220-hz tone with l4 harmonics (of progressively reduced strength) stuck in, so that we can see where the harmonics wind up:
I have superimposed an A-major scale (in red). (Note that my note indications here, and always, are for tempered scales.)
Observe that most of the harmonics are at the tonic (6 o'clock) or near the dominant (1 o'clock) or mediant (10 o'clock).
The intervals between these points with most of the energy are major third (considered imperfectly consonant), minor third (imperfectly consonant), and perfect 4th (consonant). When weaker harmonics are included, there are some dissonant (inexact) intervals as well. Note that this particular synthesized sound is richer in harmonics than most instruments -- the dissonant components are usually less substantial in real instruments.
You cannot tell visually, but the harmonics near the dominant, though quite near, are not exactly at the dominant in the tempered scale.
Here is the same still except with A-minor superimposed:
Observe the harmonics at the tonic and near the dominant are still present, but at those near the mediant in the major scale are no longer near any note of the minor scale.
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