Originally Posted by Supernova
And here I thought we would actually be discussing harmonics.
I'm game....The western system of notes follows from these goals:
1. The ratio between the frequency of any note and the one preceeding it is constant. In other words, they're geometrically spaced.
2. The two smallest whole number ratios, 2/1 and 3/2, are represented as chords (ratio's of note frequencies)
Unfortunately, these two goals are mutually exclusive: '2' requires rational (fractional) spacing, but '1' requires irrational spacing. So always there is disharmony. The system of 12 notes is a good compromise though, without as much disharmony as most other choices. (Five isn't too bad, hence the eastern system. Fifty-three is quite good in terms of capturing all the main harmonics with high accuracy, but of course that comes with a lot of additional notes.)
Why seven notes in a major scale? E is 1.260 times the frequency of C, which is almost 5/4 (though far enough off to sound 'wrong' to a good ear). F is 1.335, not far off of 4/3. G is 1.498, close to 3/2. A is 1.682, not far away from 5/3. Actual tempering may vary slightly, depending on which harmonics are to be optimized at the expense of others. So those are the main harmonics. D and B fall somewhat naturally into the spaces left by the other notes, but don't actually have harmonics in the same key (C major). Altogether they make a collection of seven, though its kind of a kludge.
It appears to me that intelligent people can sometimes become trapped in numerological thinking if they become especially adept at classical math and philosophy early. Where there are difficult to resolve contradictions between those ways of thinking and modern science, the tendency is to go with what is known and loved and reject other ways of looking at things without understanding them deeply. This results in a brilliant person with an impoverished mental toolset, who also can't recognize the limitations of mental toolsets in general. Of course everyone thinks they understand that, any intelligent person knows that the thoughts through which they filter their experience are limited, and qualify what they experience. But its something else to become deeply trained in one area for many years, then again in a completely different area, and experience the difference on that scale. To be able to see how the wondrous, civilization defining systems of thought can become enormous toy-like prisons. So I'd recommend considering study towards a pure math PhD, or at least theoretical physics, to a young person who is really good at this other stuff. Of course this seems like a waste of time at the outset, because it seems like getting lost in minutiae, and doesn't satisfy the way deep psychological stuff does. But in the end it pays off.
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