The Three-Body Problem

Whenever I read harmony books I get ticked off at the sophomoric explanations of harmonic phenomena that are ever present. I should correct that and use the past tense, actually, because the harmony books in which I find these tend to be older, apparently before the concept of observation was developed. Hindemith, in particular, talks about combination tones and their overtones to discuss consonance, and I find that rather silly. He also says that intervals have roots.

What is the root of a given interval? For the major third, for instance, it's the bottom member; for the minor sixth, its inversion, the top member. How does he know? See, I disagree with him, because one would have to consider the interval in a vacuum to identify a root, and an interval is never in a vacuum! I hold that context is critical, and that the tonal center forms a third note around which the two notes of the interval gravitate. When trying to figure out the "meaning" of an interval, the (local) key is an inseparable part of the problem. Hence we have a three-body problem, which complicates the simplicity of a two-note interval with necessary ambiguity.

My favorite example is the major sixth, because either note can be the "root". What's worse, if the major sixth is G E and the key of C is well-understood, the root is the C, a note not even in the interval! If an E minor harmony is implied, E is the root; if a dominant function in the key of C is implied, G is the root. If the interval is present by itself, without any context -- or, rather, with itself as its context -- a trick of the ear can change the root, like the spinning dancer or the cube that looks like it's either coming out of the page or into the page. The beautiful simplicity of the analysis of only two tones is, sadly, not to be.

I hope that this helps dispel the notion that the overtone series has much to do with harmony beyond providing the fifth. That notion needs dispelling, and any little bit counts.