The Minor Second
The last three notes (b6, b3 and b7) are related to each other. They all contain a reciprocal third. There is a family resemblance of sound and function. (They also all happen to be a little flat in equal temperament. On a guitar it’s a nice trick to bend them a little to sweeten them.)
Here is another note in the family, farther out harmonically, the minor second:
That’s a dissonant interval. The b6 is already tense with reciprocal third energy. Now this b2- (The minus is an accidental to show its exact pitch; more later) is another reciprocal fifth beyond (below?) that note. Its ratio is 1/15, which expands to 16/15 — just above 1. See how the ratios show where the pitch of the note is? 1/1 is the tonic, 2/1 is the octave. 16/15 is just a little bit greater than 1, so it’s just a little sharper than the tonic.
It’s not pitch so much that makes consonance and dissonance. It’s harmonic relationship.
Music is all about tension and resolution. Here’s a very tense note. How to resolve it?
One answer is just a half step away, a drop to the tonic.
That’s a move in melodic space. The tonic is right next door and it’s an easy drop.
On the lattice, the 1 is not a next door neighbor. How about going home through harmonic space instead?
Going to the 4 is an interesting experience for me. There’s still reciprocal tension, but I’m much closer to home — I can smell the stables. It’s as though I felt a bit lost at the b2-, the harmonic distance was too great to really get my bearings. But moving to the 4 allows me to figure out where I am, and where the tonic is, so that the final move home sounds really right. The 4 says to me, “There is home, now go.”
Then the melody moves to the 5, and there is resolution. The 5 sends just as strong a signal as the 4, but of opposite polarity. The 5 says, “Here is home. Now stay.”
It’s a little story, a journey on a microcosmic landscape of attraction, repulsion and beauty.
Next: The Augmented Fourth