Visualizing Harmony: The Lattice Method

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Over the centuries, people have wondered whether there isn't a way to see a chord, without simply looking at the collection of notes that constitute the chord.  The written-out music, of course, shows you the chord.  But for many, this doesn't go far enough to really represent the essential nature of the chord.

Musicians and composers, by dint of sheer practice, get to seeing the sound of the chord by simply looking at the written score.  (“Score” is the technical word for the written music, because it looks as though the page has been scored by, say, a clawed animal.)  You can imagine a musical genius looking hard at the music and hearing it in his or her head at the same time.

In the end, though, there's no getting away from the fact that the only way to see a chord in complete detail is to see the notes.  Stephen Malinowski, an American musician did more than puzzle over this problem.  He got several ideas actually programmed into computer code, and today I want to talk about one of these programs.  (The program, which can be easily obtained by Googling Music Animation Machine and visiting Mr Malinowski’s website, incorporates almost all his various methods of visualizing music; I’m going to talk about just one of the visualization methods built into the program.)

It's called the Lattice Method.  He selects a note, say G, and surrounds it with a hexagon of the most closely related other notes.  Closely-related means other notes that figure with G most often in chords.  You can leave the notes immediately above and immediately below G out, since those rarely find themselves with G in a chord.  Out of the 12 notes on the keyboard, therefore, we select B Flat, B, C, D, E Flat, E.

Let's look at the most common triads that feature G.  These are the usual 3-note chords:

Triads with G as the root (or lowest note)
G major:  G, B, D.  This one can be done with the set of six notes listed above, and G.
G minor:  G, B Flat, D.  This one can be done, too.

Triads with G as the middle note (the “third”)
E Flat major:  E Flat,G, B Flat.  Only uses the six above.
E minor:  E, G, B.  Only uses the six above.

Triads with G as the Fifth (the highest note)
C major:  C, E, G
C minor:  C, E Flat, G.

So, six major and minor triads can be shown with just the closest notes, arranged as a hexagon.  The next step was to make a hexagon around each of the notes in the first hexagon.  Could this be achieved in such a way that the new notes that surround, say, the note D, have the same relationship to D, as that first ring of 6 notes had to G?  The answer is resoundingly yes!   Basically, going to the right, the interval (the distance) between G and D is repeated between D and A, and between A and E, and so on.

Similarly, to the left, the interval from D back to G, G to C, C to F, and so on, are all the same.

The note on the upper right of G is B.  Going along that same direction, we get B, D Sharp/E Flat, then G again.  (The notes repeat, obviously, since there are only 12 of them.)

The note on the upper left of G is E, and going in the same direction, C Sharp / D Flat, B Flat, and back to G.  The completed lattice looks like this:

(I have another image with the numbers here replaced by note-names, but what you will see in the videos will be the numbers.  Note:  Instead of the traditional “Sharp” symbol, the program uses a pound symbol, #, and instead of the traditional “Flat” symbol, the program uses the letter b, which is a barely acceptable substitution.)

Now for the interesting part.

An ordinary major chord, such as G-B-D, appears as an upright triangle.  See at right.  A minor chord, such as G-B Flat-D appears as an inverted triangle!  So if you see a triangle, you know it is a major or minor triad (3-note chord); upright if major, inverted if minor.  This is almost exactly what most of us want to see, when visualizing harmony.

One of my most favorite Bach pieces is a chorale-prelude (or just Choral in German), which is an elaboration of a hymn-tune to be played just on the organ, as a voluntary (that is, a piece between church activity, e.g. while the people wait for the service to begin, or while the minister or priest gets the communion ready, or waits for the collection to be brought up--you know, the usual waiting times).  I put this particular piece through the Music Animation Machine set to display the Lattice, and you can watch it, trying to spot major chords and minor chords.  The chorale is Nun komm’ der Heiden Heiland, or Come thou Saviour of the Gentiles.  (“Come thou long-expected Jesus”, but a different tune than is sung in the US and Britain.)

Next, you can listen for augmented chords, which look like diagonal lines going off to the right and up.

After that, you can listen for diminished chords.  There are two kinds: diminished triads, of just three notes, and diminished sevenths, which have four notes.  Both kinds look like just diagonal lines going off to the upper left.  The triads have three notes, the sevenths have four.

Non-chords have peculiar shapes; a single note looks just like a single circle.  (Of course a single note is represented by multiple circles in the honeycomb pattern, all spaced widely apart.  Even chords, such as a triad, appears as triangles repeated all through the lattice, all colored the same.)

It might make better sense to allow you to explore the Lattice representation by yourself without throwing more detail at you.  What you see, definitely, is what you get.  At the very worst, you can identify the notes you’re hearing.  Remember, G is 1, A is 2, B is 3, (B Flat is 2b,) C is 4, (C Sharp is 4#,) D is 5, E is 6, (E Flat is 6b), F is 7b, F Sharp is 7.  (That’s because G major contains an F Sharp.  Plain old F --or F Natural-- is the stranger, so gets called 7b.)  Here it is:


All these videos were made by me (or at least most of them), but this one is one of my favorites, just because the video came out so well.

Note that when you hear a triad, if the music keeps two of those notes the same but moves the other note one step away, you get a quite different chord!  Bach was a master of this; observe how he does this often.  The bass in this is a “walking bass” of a sort, so that if the upper voices have two notes, the walking bass changes the harmony beautifully.

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