3-2 · The Rectangle and Stack, and the Fundamental Geometry of the Fretboard

In 3-1 we looked at the minor pentatonic scale as a sequence of intervals along one string. Now we'll spread it across multiple strings, restricting ourselves to two notes per string.

It turns out that if you take any five-note scale and play it two notes per string, the scale plays out twice over five strings, and then the whole pattern repeats, starting on the same note two octaves higher.

A thought experiment

That geometry is easiest to see on a guitar tuned entirely in fourths: E, A, D, G, C, F. This is not a tuning recommendation. It is just a way of simplifying the picture so the underlying shape becomes visible.

On an all-fourths instrument, moving up one string is exactly the same as moving up five frets, because a fourth is a five-semitone interval. Laying out the scale two notes per string gives this:

Notice that the root on the top string is one fret lower than the root on the bottom string. Five strings of fourths is 25 semitones, which is two octaves plus one fret, so the pattern repeats itself shifted one fret toward the nut. On an all-fourths instrument, you could start anywhere on any string and get the same result.

Two shapes instead of one

That five-string pattern is a useful picture, but it's a lot to hold in mind at once. Rather than memorize the whole thing, break it into two neighboring pieces.

The rectangle sits on two adjacent strings, with the notes on each string separated by three frets. It contains four of the five notes: the root, ♭3, 5, and ♭7. The only note missing is the 4.

The missing 4 is easy to find when you need it. It is two frets above the higher note on the upper string, or two frets below the lower note on the lower string. Don't worry about memorizing that yet.

The stack sits on three adjacent strings, with the notes on each string separated by two frets. It contains all five notes, with the 4 repeated as both the lowest and the highest note in the shape.

In both shapes, pay close attention to where the roots are. Those root locations are the visual anchors you'll use to place these shapes anywhere on the neck.

The warp

Everything above assumes a perfectly uniform tuning, but a real guitar is not quite like that. Most pairs of strings are tuned a perfect fourth apart, but between the G and B strings the interval shrinks to a major third.

This is what I call the warp. The name comes from Jon Finn, an influential guitar teacher at Berklee, who must be a Star Trek fan, because he called it the "warp refraction threshold." Brilliant concept, but the name is a mouthful.

In fretboard diagrams, I represent the warp as a dashed red line between the G and B strings, often with a red arrow showing the direction of the shift.

Memorize this: Whenever a shape crosses the warp, the part of the shape on the higher strings shifts one fret toward the bridge.

A rectangle that sits across the warp stops being a rectangle and becomes a parallelogram. Nothing breaks — the intervals are identical and you are playing the same notes. It just looks different, and the fingering has to adjust.

The stack changes in the same way. On either side of the warp it looks like a box, but it can fall across the warp in two different ways, which gives us two more shapes. Again, nothing breaks. You are still playing the same five notes.

People sometimes ask why we don't simply tune the guitar in all fourths, since that would eliminate all of this. My answer is that the warp changes fingerings in musically interesting ways. Many classic guitar licks depend on it, and so do most chord shapes, especially barre chords. Standard tuning is a compromise, but it's a worthwhile one. Thinking in terms of the warp takes a little practice, and then it becomes second nature.

Exercise 3-3: Pick any starting note on the fretboard and create a minor drone from it, as in 3-1. Find either a rectangle or a stack at that location: if your index finger is on the root, play a rectangle, and if your ring finger is on the root, play a stack (this isn't a rule, just a reliable starting point). Part of the shape may fall off the edge of the fretboard, and that's fine. Once you've found the shape, make short phrases using only the four or five notes inside it. Then keep the same root, shift your hand, and switch from the rectangle to the stack (or vice versa). Don't worry about connecting the two shapes smoothly yet. Focus on how each note sounds against the drone.

Exercise 3-4: Repeat Exercise 3-3 over a major drone, bending the ♭3 each time you play it. In the rectangle, the ♭3 is the highest note. In the stack, it's the lower note on the top string. Bending with the index finger is normally awkward, but because this bend is small, it's very manageable.

Being able to find the ♭3 this quickly is one of the first real payoffs of the rectangle and stack system: every interval is always in the same place relative to the shape, so target notes stop being something you have to search for.