First is the basic 12-semitone "chromatic" scale common to most Western music:
However, it turns out closeness isn't such a simple concept in music. Melodies (the "foreground" tune) and harmonies (the "background" layers) work with different definitions, as Gary Garrett explained here:
Melodies “like” to move up and down on a linear scale. They want to go to a nearby note when they move — that is, near by in pitch. We hear, and sing, small movements in pitch better than we hear leaps.
Harmonies “like” to go to nearby notes too, but harmonic space is different than linear, melodic space. The 1 and the 5 are harmonic neighbors. In fact, they are as close together as notes can be, harmonically, without being the same note .... But they are far apart melodically — the 5 is almost at the midpoint of the scale.There's been a long history of people trying to visualize "harmonic closeness" using graphical techniques. Garrett summarized this history, and explained his own graphical "Lattice" technique, in this post. Below is my own attempt to visualize his Lattice:
I should make clear that my version of the lattice is much simpler than Garrett's in one notable way. Mine uses the standard Western approximation of equal temperament. It just shows the notes you can reach with the white and black keys of a piano. Garrett is a fan of just intonation (which also intrigues me, as a wannabe Pythagorean), so there's much less repetition in his version of the Lattice than in mine. In my version, say, C sharp is the same note as D flat. In Garrett's, it's not! :-)
But, you may ask, where's that Glass Bead Game-ish link between far-away ideas? It comes in Garrett's recent post about "tonal gravity:"
I believe that the great driving force in tonal music, that creates the drama and story of the music itself (independently of any lyrics), is the longing for home. Home is the tonic. If a song is in the key of A, all the A’s in their various octaves will sound like home.
It’s as though the tonic creates a sort of gravitational field around itself. It acts a lot like real gravity. The chords and notes move in this gravitational field, like planets and moons around a sun. The gravitational field follows a few basic rules:(Note: his phrase "east on the lattice," in my version, is "up.")
Roots generate local gravitational fields. I think of them as Jupiter to the tonic’s Sun. When the root is on the 5, for example, it shifts the gravity field to the east on the lattice, and the 2 and 7 become harmonious, consonant notes, rather than dissonant ones. The tonic still has great influence, so the entire chord feels unresolved — a 5 chord pulls very strongly toward the 1 chord, a property that is heavily relied upon in Western music.
- Movement away from the center creates tension; movement toward the center gives a sense of resolution.
- The closer you are to the center in your journey, the stronger the sensations of tension and resolution are. The field is stronger closer in, just like real gravity.
- The closer together two notes are, the more consonant, or harmonious, they will be when sounded together. The farther apart they are, the more dissonant they will be, the more they will clash.
I love this analogy. It falls right into place with my other thoughts about how tension and release must be core elements of the Glass Bead Game, since they're so universal across many different domains. Now, what will I actually do with these beautiful ideas? That, I'm not so sure about... :-)