All the notes I’ve discussed so far are found above and to the right of the tonic, in the northeast quadrant of the map. These notes are generated by multiplication alone.
What about the notes that are generated by division? These are found to the left and down on the lattice.
The closer a note is to the tonic, the smaller the numbers are, and the easier it is for the ear to tell where it is. I’ll cover this much more in later posts, but I think the character of an individual note, its unique harmonic color, is largely determined by two signals it sends to the ear and mind:
1) How far away is home (the tonic)?
2) What direction is it?
And the closer the note is to home, the clearer the signal is.
The perfect fourth is the same distance from the center as the 5 is, but in the exact opposite direction: divide-by-3 instead of multiply-by-3. So it sends a signal of equal strength and opposite direction. How does this mirror-fifth sound?
I hear beauty, and tremendous tension. Something has to happen here, and soon — it feels like a pencil balancing on its point, unstable equilibrium.
Here are two of the most powerful phenomena in music: tension and resolution.
There’s another way to resolve the tension, and that is to move the playing field so that the fourth is in a stable spot. This is done by changing the root. It’s a chord change. Check this out:
The ratio of the perfect fourth is 1/3. This can be octave-reduced (octave-expanded) by moving it up two octaves, to 4/3.
The energy of the fourth, the division energy, has had a number of names. Harry Partch, a major composer and explorer of music in just intonation, called the quality of the right-and-up harmonies (the ones you get to by multiplication) otonality, from overtone. He called the energy of division-based harmony utonality, for undertone.
Once again I’m going with Mathieu on this one. In Harmonic Experience, he gives an excellent rationale for calling this energy reciprocal. I think he’s right. Each overtonal note has its mirror twin, and the twins are identical, just upside down from each other — reciprocals. The fourth is the reciprocal of the fifth.
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