Music teaches us a lot about reality. It shows enough regularity to suggest a simple mathematical model, but also enough irregularity to frustrate our attempts at formalizing it. In this series of essays, I’ll try to describe some of this frustration mixed with fascination. I’m going to talk about the guitar; both because I know more about it and because it’s even more quirky than the piano.

The guitar is a versatile instrument. You can play individual notes of a melody, you can play chords, and you can play the bass line, sometimes all at the same time. All this with just six strings. These six strings are tuned is such a way as to maximize the number of chords that can be played on it. You play chords by making shapes with your left hand (or the right hand if, like Jimi Hendrix, you’re left-handed). It’s a very interesting *optimization problem* that involves equal parts of music theory and human anatomy.

Here are some anatomical constraints: we have five fingers in the string-pressing hand. The thumb is mostly used for grip, although you can sometimes use it to play bass notes on the thickest string by reaching around the neck. That leaves us with four fingers to control six strings. If we want to strum all six strings, we have two options: we can let two or more strings ring free, or use one of the fingers (usually the index finger) to bar multiple strings and use the remaining three fingers to create the chord shape. So the basic chord shape is a three-finger grip. If we can form a chord with three fingers, we can move it up the fretboard using the index finger for barring. As with all musical instruments, the available shapes are limited by anatomy: we can only stretch our fingers so much.

Now for some music theory. The basic chords are triads built from three notes: the root, the third, and the fifth (relative to the root). The intervals between these notes determine the type of the chord. A major triad is build from a major third and a minor third (the sum of these thirds is a perfect fifth–yes, in music 3 + 3 = 5). The C major triad, for instance, consists of three notes: C, E, and G. The distance from C to E is a major third, and the distance from E to G is a minor third. The distance from C to G is a perfect fifth.

Naively, we might think that the guitar should be tuned in thirds, say, the lowest string C, then E, and then G. But what then? What about the three remaining strings? We could repeat C, E, and G, an octave higher. That would be okay if we only wanted to play major triads. But there are also minor triads, with a minor third followed by a major third. C minor triad is C, E♭ (E flat), and G. So maybe we could use that for the tuning? It would allow us to play C minor with no fingers, and C major by pressing two strings with two fingers. Unfortunately, there are many other types of chords that would be very hard to play in that tuning, so this idea is scrapped.

Observe, though, that with six strings, it’s unavoidable that some notes of the triad would have to be doubled (modulo shifting by an octave or two). This introduces more intervals between notes: for instance, the distance from G to the C in the next octave is a fourth. So within a duplicated triad we have the intervals of a major third, minor third, the perfect fifth (their sum), a fourth (from G to C), as well as two sixths (from E to C and from G to E), a few octaves, and so on.

So here’s a new idea: If we tune the strings in fourths, we can easily, without stretching our fingers too much, produce thirds, fourths, and fifths. That’s because we can shorten an interval by pressing the lower sting, or lengthen it by pressing the higher string.

Let’s see how this works. The lowest string on the guitar is E, so that’s where we’ll start. A fourth about it is A, so that’s the next thickest string. Let’s see what intervals we can make using those two.

By pressing the E string at the first fret, we can produce a major third, F to A.

By pressing it at the second fret, we can produce a minor third, F# to A.

By releasing the E string and pressing the A string at the second fret, we can produce a perfect fifth, from E to B.

And, of course, by releasing both strings, we get a perfect fourth, from E to A. That’s a lot of handy intervals within easy reach.

Let’s use this idea to build the simplest guitar chord, E major, which contains E, G#, and B. In principle, the order of these notes and the octaves they are in doesn’t matter, but some combinations sound better than others. We’ll start with the open E string for the root. To start with, let’s assume the tuning in fourths, so the second string is A, the third D, and the fourth G.

We can can press the A string at the second fret to produce the fifth of the triad, B. (We are skipping G# for now, because it’s not easily reachable.)

The next triad note within reach is another E, an octave higher. We can play it by pressing the D string at the second fret.

Now we can finally add the third, G#, by pressing the next string, G, at the first fret.

We now have the root (doubled), the fifth, and the third of the triad.

There are two more strings to go, and we have already used three fingers to press three strings. If we continued tuning strings in fourths, the next string would be C. That’s not part of our triad, and we can’t easily stretch our pinky to reach the next E. So we begrudgingly give up on our rule of fourths, and instead tune the next string a semitone lower than we promised, to a B. B happens to be in our triad, so we’re fine. And a fourth about B is again E, so that works too.

Here are the notes we used in this grip, together with intervals between them.

Notice that the root E is repeated three times, in different octaves. The fifth of the triad, B, appears twice, and the third, G#, only once. As we’ll see later, this arrangement gives us a lot of flexibility when transforming this grip.

The leap of a fifth in the bass, from E to B, is actually very pleasant to the ear — skipping the G# there is advantageous.

Here’s the same grip annotated with root-relative intervals. 1 is the root (E), 3 is the third (G#), and 5 is the fifth (B). It’s very important to remember which is which, in order to understand how to transform this shape to produce other interesting chords.

Not surprisingly, this is called the E shape in the popular CAGED system.

We’ve used only three fingers, which is great, because we’ll be able to use the index finger as a bar to move this triad up the fretboard, if we wish so.

In the process, we have arrived at the standard guitar tuning E, A, D, G, B, E. It is basically in fourth, except for the major third from G to B. This one exception introduces a lot of complexity into chord building on the guitar.

By now, you might have noticed some irregularities in music notation. They have accumulated over the centuries of development. We now use the so called equal temperament system in which the basic interval is a semitone, corresponding to one fret on the guitar. Standard musical intervals can be expressed in semitones, with the additional convenience that they satisfy standard arithmetic. For instance, a minor third is 3 semitones, a major third is 4 semitones, their sum is 7 semitones, corresponding to a perfect fifth. A perfect fourth is 5 semitones, which is an octave (12 semitones) minus the perfect fifth (7 semitones).

We could have motivated our tuning by postulating the distance of two octaves (24 semitones) between the lowest and the highest string. If we divide two octaves between six string (5 intervals), we get 4.8, which is almost the perfect fourth (5 semitones), but not quite. That’s why we introduced the “leap interval” of a major third between the G and the B strings.

Next, I’ll show you how all common chords and the majority of jazz chords can be derived from this single shape by applying various transformations (or, as mathematicians call them, *morphisms*).

## Acknowledgment

I used the excellent free web program chordpic to generate my string diagrams.