Previously, we talked about transforming the basic E chord shape by transposing it up the fretboard. You may be aware that there are other chord shapes, sometimes grouped into the so called CAGED system. I’ll show you how to derive this system “scientifically.” Scientists arrive at new theories by looking at patterns. Sometimes a pattern doesn’t fit exactly, but it makes sense to temporarily ignore the discrepancy, forge ahead with incorrect assumptions, and then introduce subtle corrections to fix them. I know, this is not what they teach you in school, but that’s how it’s done in real life.

Let’s make some simplifying assumptions about guitar tuning. The first is that the top string can be identified with the bottom string: they are both E strings. Strictly speaking, this is not true: they are two octaves apart, and sometimes you can finger them differently, but we are playing scientists who ignore such distinctions. So we’ll consider the bottom E string a duplicate of the top E string. Second assumption is more outrageous: strings are tuned in fourths. Well, this is true in 80% of the cases. The 20% exception is the interval of a major third between the G string and the B string. We’ll just ignore it for the moment. With these assumptions in place, we can think of the strings forming a circle: we glue together the two E strings, and we get a circle of five strings a fourth apart.

In this imaginary world, we can now shift any chord shape sideways, around the circle, without changing its function. Granted, it will shift all the pitches up by a fourth, so shifting the E chord to the right would result in the A chord, another shift would produce the D chord, and so on. Lets try it!

We’ll start with the E chord and shift it to the right.

We get this:

Hurray! Within experimental error, it worked! Granted, this is the A minor chord, not the A major that we were expecting, but still, considering that our assumptions were partially wrong, that’s close enough.

The question is, how can we modify our theory to produce the A major chord? Our problem has its source in the anomaly between the G string and the B string. When we shift a finger between the two, keeping it on the same fret, we are not moving the pitch up by a fourth, we’re moving it by a third. If all shifts were by a fourth, relative intervals wouldn’t change, and we would just transpose a major chord into another major chord.

But not all is lost. It just means that we have to introduce a correction to our theory. In order to preserve relative intervals, when shifting from the G string to the B string, we have to move the finger one fret up the fretboard (or down, in the diagram). It works:

This is indeed the A major chord.

Of course, theory aside, the chord has to also sound right. This one is okay, except that the fifth of the triad is in the bass, which makes it an inverted chord. In guitar notation inversions are often written as slash chords; here it would be A/E, because of the E in the bass. Inverted chords don’t always work in a chord progressions so, in practice, people try to mute the low E and emphasize the root A.

To test our theory further, let’s apply the right shift to A. The fourth above A is D and, indeed we get the D major. Notice the adjustment when moving between the G and the B strings.

Again, the bass part is a little tricky. Here, I just duplicated the high E string grip on the lower E string, which resulted in another inversion D/F#. In practice, people usually mute both the E and the A strings and emphasize the root D.

Another shift to the right and we get the G chord (again, correcting the move from the G string to the B string and mirroring the E string):

This particular variant is often used in folk music. The more popular variant gets rid of the fifth on the B string, since the open B is the third of the triad.

This change leads to the duplication of the third, so people often mute the third in the bass.

Another shift, and we get the C major. Here, the move from the open G string is corrected by pressing the B string at the first fret. All according to our theory.

The third in the bass doesn’t sound good, so it’s usually muted. Another inversion, with the fifth in the bass, sounds better in many contexts, so here’s C/G:

Here’s another variant with, the fifth on the highest E string

We’ll see this variant modified by extension notes (the sevenths and the ninths).

We have just covered all the shapes in the CAGED system (the letters stand for the five major chords). And, indeed, we went full circle, because the next shift produces the F chord (the open G string turns into first fret press on the B string).

This makes perfect sense. If it weren’t for the anomaly, the fifth shift should bring us back to the same shape. But every time a black dot crosses the anomaly, it drops down so, after a full circle, the whole shape drops down one fret. Therefore five shifts to the right equal one shift down. We have just proven a theorem.

Notice that in the first three iterations a triangle shape is formed by three fingers. This shape consists of the fifth, the root, and the third of the triad. This information will come in handy when we discuss chord modifications. To the left of the triangle, we have the root, and to the right, another fifth (except in the D chord, where it’s pushed off the edge). In the G chord, we start seeing part of this triangle peeking on the left (the root and the third), shifted down because of the anomaly. In C/G the triangle is fully reconstituted, albeit one fret down. If you can spot these shapes, you’ll have no problem remembering where to find the third (and, for instance, lower it to make a minor chord) or where to insert the seventh.

This theory can be also visualized by arranging the strings in a radial pattern, the frets forming a spiral. Here’s the diagram for the E chord:

If you rotate the dots counterclockwise, you’ll get the A chord, and so on. The anomaly adjustment happens automatically, because the B string is offset by one step. Also, if no dot moves into the B string, a new dot at the first fret is produced.

Next time, combining the transformations.