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
Tensegrity, nice. Wonder if there is an elegant way to visualize this tension (in addition to the position on the lattice).
You might like the article Tonal Gravity and the Major Scale.