Friday, July 12, 2013

Musical Gravity

I'm always on the lookout for interesting Glass Bead Game-ish analogies that link together far-flung ideas.  On the blog of musician Gary Garrett, I just found a doozy!  I've attempted to boil it down to a couple of eye-catching images, which I'll do my best to explain below.

First is the basic 12-semitone "chromatic" scale common to most Western music:

Melodic order
These 12 notes are also 12 intervals between notes, and I talked more in this old post about how human brains may be hard-wired into interpreting various intervals as happy, sad, angry, and so on.  Above, I assigned each note/interval to a color, and one can see how "close" any two of them are to one another by seeing how similar their colors are.

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:

Harmonic order!

Note that the "color neighbors" here are spread out much more than in the melodic scale I showed above.  There's a method to this seeming madness:  the up/down axis follows the well-known circle of fifths, the upper-left/lower-right axis steps down in major thirds, and the lower-left/upper-right axis steps up in minor thirds.  These steps correspond to the natural harmonies that one would see, for example, by taking a stringed instrument and changing the length of the string by "simple" fractions like 3/2, 4/3, and so on.  Also, most chords (major and minor) can be formed by taking little triangular "clusters" of any three notes in this 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:
  1. Movement away from the center creates tension; movement toward the center gives a sense of resolution.
  2. 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.
  3. 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.
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.
(Note: his phrase "east on the lattice," in my version, is "up.")

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...  :-)


  1. A lot of scholarship has come out of Schoenberg 12 tone works, some of which you probably would find interesting.

    Great post.

    1. Thanks, Laoch. I think I would like Schoenberg, but I know that my ear is nowhere close to being able to appreciate all the subtle mathemagic that I was plotting! :-)

  2. '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.'

    This is beautiful.

    Cygnus, this has got to be one of my favorite of your posts, so far, and that's saying a lot.

    I think my greatest weakness as a storyteller is that I love resolution but find tension very difficult to manufacture with authenticity -- and believe this to be the case often with fiction writers. I find it an enormous challenge to venture into dissonance, what sounds to my particular sensitivities as unnecessary noise. It not only bothers me when I see it done but there is something like physical pain that accompanies the incidence. Musical pain.

    1. Thank you. Everyone has their own natural orbit, and there's nothing wrong with following a path away from the dense asteroid field of popular opinion.

      Gary Garrett pointed to some other site that discussed how the Western musical "ear" has been evolving over the past few centuries in our appreciation for dissonance. Much of the Romantic music of the 1800s, which caused uproar when it was released due to its rule-breaking, can sound blandly "classical" to many ears today.

      It almost sounds cliche to liken that to the "coarsening" of culture that many see happening around us. We've talked before about modern contemporary literature's seeming chase to the bottom, in its desire to heap tragedies and shocking indignities on its characters. Still, we've got to be careful in assigning too much blame to our current age... Randall Munroe has demonstrated that similar gripes seem to appear in every age. :-)

    2. I think about that a lot. Wonder how much of the race to the bottom is contemporary and how much of it is an apparent timeless proclivity -- which I have never and will never understand. But you can't 'force' dignity on people, can you? :)

  3. Mmmh - extremely difficult for me to follow, being not well versed in music - physically, I 'hear' too much, a bit like a bat, meaning that tones can easily hurt me where other people are entranced. (Weird: I can listen to bagpipes without a wince - somebody explained that it has something to do with heartbeat?)
    When I look at your graphic (interesting: I just read an article about people who see numbers in colours - synaesthesia it was called) I can't see the harmony I would have graphically expected - so I am sure I will have to read your post more than once again.

  4. Ah, I was afraid it would be hard to follow. My use of color was pure fluff, by the way... Gary Garrett's original lattice doesn't use it.

    I think it's also the case that most music-theory people don't think in terms of this kind of diagram, either. It's probably more common to refer to lists of "rules" for which chord progressions sound good next to one another, and which don't. The reason I like this diagram so much is that it "builds up" those complicated rules from simple first principles.

    1. When I 'complained' to husband not being able to grasp it (to be honest: I complained about being not willing to invest more time to grasp it) he told me of some texts the (early) Romantic Novalis had written connecting to the Glass Bead Game, but he thinks you will know them. As I follow you only since a few weeks I thought it OK to mention Novalis.

    2. Correction: Novalis wrote about synergy.

    3. Thanks for the pointer to Novalis! I had heard the name before, but knew almost nothing about him. I must remedy that...

      "We dream of a journey through the universe. But is the universe then not in us? We do not know the depths of our spirit. Inward goes the secret path. Eternity with its worlds, the past and the future, is in us or nowhere."

  5. This is wonderful. I love the lattice idea, especially. Do you know any music prof types? I'm sure you could find some who'd enjoy tossing this idea around with you.

    Chromaticism seems like it should be antithetical to Western musical thinking and yet chromatic runs are a trademark of the world's current reigning melodic genius: none other than Sir Paul McCartney. Mozart dug them, too.

    1. If it's good enough for those guys... :-)

      I guess it's not so surprising. I'm no musician, but if you have a piece that contains both major and minor keys, you've already got 10 out of the 12 chromatic semitones to work with. If you're "bluesy" then you might also favor a scale that uses the tritone (bV), which brings it up to 11. Nobody seems to like the poor minor second (bII), though...